Jekyll2023-01-17T10:03:24-05:00https://refactorium.com/feed.xmlIntroduction to the structure of free operations in the resource theories of entanglement and magic2023-01-15T00:00:00-05:002023-01-15T00:00:00-05:00https://refactorium.com/blog/2023/01/15/magic-and-entanglement<h2 id="prelude">Prelude</h2>
<p>After three semesters into grad school, I’m realizing that my initial lofty goal of keeping a blog is not going to happen easily. I just prioritize everything else (homework, classes, research, private life) over it. However, I still generate some form of written material - final projects in classes. So, for the next two semesters (of my last three classes), I’m just going to put down here my final projects in the form of blog posts. As final projects are typically done in a rush, I’ll probably clean them up a bit in the process of translating them to markdown from LaTeX.</p>
<p>This particular write-up was for my 2022 Fall semester Quantum Information Theory class taught by Professor Iman Marvian.</p>
<h2 id="abstract">Abstract</h2>
<p>Resource theories formalize the notion of valuable resources. A quantum resource theory identifies “free states” that can be created with “free operations” - these are “easy” relative to assumed capabilities. To create “resource states”, one must either have states that contain “some” of the resource or use resource-generating operations. The theory of entanglement (where the resource is entanglement) and the theory of magic (where the resource is “non-stabilizerness”, or “magic”) share a similar structure through the lens of resource theory. Our main focus is a particular hierarchical structure of free operations, namely, the gap between the operational definition and the axiomatic definition of these classes of operations. For entanglement, this is the gap between the classes of Local Operations and Classical Communication (LOCC) and Separable Operators. In the case of magic, the gap is between the classes of Stabilizer Operators and Completely Stabilizer Preserving operators. These gaps have potential implications for protocols related to entanglement and magic state distillation.</p>
<h2 id="introduction">Introduction</h2>
<p>The majority of physical theories establish laws based on experimental observations of nature and predict behavior for situations that were not yet verified experimentally. Resource theories in contrast have an inherent process-, or engineering focus as they examine what is possible given a demarcation line in operational capabilities. With roots in Carnot’s explorations of how to exploit thermodynamic non-equilibrium as a resource for heat engines, resource theories provide a framework to organize the concepts and questions of availability and quantification of resources, conversion rates, reversibility, and more. At the core of a quantum resource theory, a set of operations is defined to be the <em>free operations</em>, or <em>resource-non-generating</em> (RNG) operations, and a set of states defined to be the <em>free states</em>. The intuition is that RNG operations and free states are the “easy ones”, i.e. assumed to be available to acquire without incurring high costs. This is in contrast to valuable <em>dynamic resources</em> (resource generating operations), i.e. operations generating <em>static resources</em> (resource states) which are “hard” to acquire. <a class="citation" href="#chitambar_quantum_2019">(Chitambar & Gour, 2019)</a>.</p>
<p>While there are numerous quantum resource theories, the two we are interested in in this report are the theories of entanglement and magic. It is well known that quantum entanglement can be used as a <em>resource</em> for communication to extend our capabilities and open up use cases like super-dense coding or quantum teleportation <a class="citation" href="#wilde_quantum_2017">(Wilde, 2017)</a>. These use cases would be impossible with the “free” separable states, and <em>Local Quantum Operations and Classical Communication</em> (LOCC).</p>
<p>Magic is a resource for universal fault-tolerant quantum computation (FTQC). Under the stabilizer formalism <a class="citation" href="#gottesman_stabilizer_1997">(Gottesman, 1997)</a>, large classes of Quantum Error Correction codes can be expressed for both storing and fault tolerantly executing quantum computation in the logical space of these codes. However, stabilizer operations are not universal in the quantum sense, and in-fact are classically efficiently simulatable, as stated by the Gottesman-Knill theorem (missing reference). The term “magic” was introduced in <a class="citation" href="#bravyi_universal_2005">(Bravyi & Kitaev, 2005)</a>, where Bravyi and Kitaev showed that for universal quantum computation, one has to inject the valuable resource of <em>magic states</em> or use non-stabilizer operations.</p>
<p>This report focuses on a particular aspect of resource theories of magic and entanglement, namely, the structure of free operations. Free operations are defined either axiomatically or operationally. The operational definition is usually the more natural, stemming from the assumed limitation on capabilities, and having a clear operational interpretation. Axiomatically defined free operations are more mathematically motivated, they have less clear operational meaning; they are the maximal set of operations that map free states to free states. The question we are interested in is whether the axiomatic class is larger or not than the operational one and if yes, what the implications may be.</p>
<p>In entanglement theory, the class of separable channels (SEP) are the axiomatically defined free operations and the operational ones are the LOCC operations. Chitambar et al. <a class="citation" href="#chitambar_increasing_2012">(Chitambar et al., 2012)</a> found an operationally interpretable measure of entanglement, that shows a 12.5% gap between SEP and LOCC. In the case of magic theory, a recent counterexample from Heimendahl et al. <a class="citation" href="#heimendahl_axiomatic_2022">(Heimendahl et al., 2022)</a> shows that the axiomatically defined class of free operations, Completely Stabilizer Preserving (CSP) is strictly larger than the operationally defined class of stabilizer operations (SO).</p>
<p>The structure of this paper is as follows: in <a href="#fundamental-concepts">Fundamental concepts</a> we will cover the minimum prerequisite of notations and concepts to understand the separation of these two classes in these two theories. In <a href="#the-structure-of-free-operations">The structure of free operations</a> we will then get into the details of the separation of these classes in the two theories and finally, in <a href="#discussion">Discussion</a>, we discuss the potential consequences and draw our conclusions.</p>
<h2 id="fundamental-concepts">Fundamental concepts</h2>
<p>In this section, we review the concepts and mathematical tools in our two theories of interest and establish notation, mostly following Wilde <a class="citation" href="#wilde_quantum_2017">(Wilde, 2017)</a> for the quantum information theoretical and entanglement-related concepts. To introduce the stabilizer formalism, we will follow multiple sources <a class="citation" href="#gottesman_stabilizer_1997">(Gottesman, 1997)</a>, <a class="citation" href="#aaronson_improved_2004">(Aaronson & Gottesman, 2004)</a>, <a class="citation" href="#heimendahl_axiomatic_2022">(Heimendahl et al., 2022)</a>.</p>
<h3 id="state-vector-and-density-matrix-representation-of-quantum-states">State vector and density matrix representation of quantum states</h3>
<p>Quantum mechanics is inherently probabilistic in the sense that measurement outcomes on a system are defined by the amplitude square of a set of complex numbers (<em>probability amplitudes</em>) that describe a state as defined by the Born rule. If we can describe the probability amplitudes of a quantum mechanical system with certainty, we call that state a <em>pure</em> state. However, this is far from realistic, usually, we don’t know perfectly the description of a system. Thus, on top of the inherent “quantum probability distribution”, we might have an uncertainty of these amplitudes themselves, layering a “classical probability distribution” over the quantum probability, which we then call <em>mixed</em> states (or noisy states).</p>
<p>Pure quantum states are represented as normalized vectors (or we can think of them as rays) in a potentially composite Hilbert space, i.e. one that allows for a structure of $\mathcal{H}=\otimes_{i=1}^n \mathcal{H}_i$ with each sub-Hilbert space having dimension $d_i$. We will only discuss cases where $d_i = d_j = d, \forall i, j$, i.e. $\mathcal{H}_i=\mathbb{C}^d$, and where $d$ is prime. When $d=2$, we talk about a Hilbert space of $n$ <em>qubit</em>s. If $d=3$, we call each subsystem a <em>qutrit</em> and $d\gt3$ is the case of <em>qudit</em>s. We denote pure states in the Dirac notation as $\ket{\psi} \in \mathcal{H}$. The members of the computational basis for the $n$ qudit Hilbert space of $d$-dimensional qudits will be labeled by the elements of the vector space $\mathbb{F}_d^n$ over the finite field $\mathbb{F}_d$ (i.e. a $n$ length vector with elements 0 to $d-1$, with addition is modulo $d$). For example $\ket{124} = \ket{1}\otimes \ket{2} \otimes \ket{4} \in (\mathbb{C}^5)^{\otimes 3}$ is a 3-qudit state of dimension 5.</p>
<p>As for mixed states, we don’t know exactly which pure state a system is in, we can represent this lack of knowledge as a probability distribution $P_X$ over possible pure states $\ket{\psi_x}$. This ensemble of states and their probabilities $\{(P_X(x), \ket{\psi_x})\}$ affords a <em>density operator</em> representation $\rho \equiv \sum_x p_X(x) \ket{\psi_x}\bra{\psi_x}$, where $\ket{\psi_x}\bra{\psi_x} \in \mathcal{L}(\mathcal{H})$ is the density operator for the pure state $\ket{\psi_x}$ defined by the outer product of $\ket{\psi_x}$ and $\bra{\psi_x} \equiv \ket{\psi_x}^\dagger$. We denote the set of square linear operators acting on $\mathcal{H}$ with $\mathcal{L}(\mathcal{H})$. A density operator is always positive semi-definite, denoted as $\rho \geq 0$, and has unit trace, $\text{Tr}\{\rho\}=1$. The set of density operators is denoted as $DO(\mathcal{H}) \subset \mathcal{L}(\mathcal{H})$.</p>
<h3 id="describing-change-with-operators-and-quantum-channels">Describing change with operators and quantum channels</h3>
<p>When the change in a system is noiseless, i.e. the system is closed, it is postulated to be a transformation $U \in \mathcal{L}(\mathcal{H})$ on the system that is <em>unitary</em>, i.e. $UU^\dagger=I$. In the vector representation, the effect of $U$ is $\ket{\psi} \rightarrow U\ket{\psi}$, in the density operator formalism, it is $\rho \rightarrow U \rho U^\dagger$. We will make use of the generalizations of the Pauli group, which is generated by the Pauli-X and Pauli-Z operators for a single qubit, with their action on the computational basis:</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>X</mi><mpadded><mi mathvariant="normal">∣</mi><mi>k</mi><mo stretchy="false">⟩</mo></mpadded><mo>=</mo><mpadded><mi mathvariant="normal">∣</mi><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow><mo stretchy="false">⟩</mo></mpadded><mtext> </mtext><mo separator="true">,</mo><mi>Z</mi><mpadded><mi mathvariant="normal">∣</mi><mi>k</mi><mo stretchy="false">⟩</mo></mpadded><mo>=</mo><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mi>k</mi></msup><mpadded><mi mathvariant="normal">∣</mi><mi>k</mi><mo stretchy="false">⟩</mo></mpadded><mo separator="true">,</mo><mi>k</mi><mo>∈</mo><msub><mi mathvariant="double-struck">F</mi><mn>2</mn></msub><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">
X\ket{k}=\ket{k+1} \ \ , Z\ket{k}=(-1)^{k}\ket{k}, k\in \mathbb{F}_2.
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span><span class="mclose">⟩</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="minner"><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">1</span></span><span class="mclose">⟩</span></span><span class="mspace"> </span><span class="mspace"> </span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span><span class="mclose">⟩</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1491em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">−</span><span class="mord">1</span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8991em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span><span class="mclose">⟩</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8389em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathbb">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">.</span></span></span></span></span></p>
<p>The generalized $n$-qudit Pauli-group for $d$ dimension qudits is generated by the generalized $n$-qudit X and Z operators, with action:</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>X</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mpadded><mi mathvariant="normal">∣</mi><mi>k</mi><mo stretchy="false">⟩</mo></mpadded><mo>=</mo><mpadded><mi mathvariant="normal">∣</mi><mrow><mi>k</mi><mo>+</mo><mi>x</mi></mrow><mo stretchy="false">⟩</mo></mpadded><mtext> </mtext><mo separator="true">,</mo><mi>Z</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mpadded><mi mathvariant="normal">∣</mi><mi>k</mi><mo stretchy="false">⟩</mo></mpadded><mo>=</mo><msup><mi>ω</mi><mrow><mi>k</mi><mo>⋅</mo><mi>z</mi></mrow></msup><mpadded><mi mathvariant="normal">∣</mi><mi>k</mi><mo stretchy="false">⟩</mo></mpadded><mo separator="true">,</mo><mi>x</mi><mo separator="true">,</mo><mi>z</mi><mo separator="true">,</mo><mi>k</mi><mo>∈</mo><msubsup><mi mathvariant="double-struck">F</mi><mi>d</mi><mi>n</mi></msubsup><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">
X(x)\ket{k}=\ket{k+x} \ \ , Z(z)\ket{k}=\omega^{k \cdot z}\ket{k}, x,z,k\in \mathbb{F}_d^n.
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span><span class="mclose">⟩</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="minner"><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">x</span></span><span class="mclose">⟩</span></span><span class="mspace"> </span><span class="mspace"> </span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span><span class="mclose">⟩</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.1491em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ω</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8991em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03148em;">k</span><span class="mbin mtight">⋅</span><span class="mord mathnormal mtight" style="margin-right:0.04398em;">z</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span><span class="mclose">⟩</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.04398em;">z</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9614em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathbb">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.7144em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">d</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord">.</span></span></span></span></span></p>
<p>, where $\omega = e^{i2\pi/d}$ is the $d$th root of unity and $k \cdot z = \sum_{i=1}^{n} k_i z_i$ is the inner product between $k$ and $z$.</p>
<p>However, in a realistic scenario, evolution is not perfectly known, and as such, it can be thought of as a probabilistic mixture of certain operators. Formally, we represent the most general type of evolution an experimenter can impart on a system with the <em>quantum channel</em> formalism. A quantum channel is a map $\mathcal{N}_{A \rightarrow B}: \mathcal{L}(\mathcal{H}_A) \rightarrow \mathcal{L}(\mathcal{H}_B)$, that must map density operators to density operators, i.e. we have to make sure that the output operator $\rho_B := \mathcal{N}_{A \rightarrow B}(\rho_A)$ has trace 1 and is positive. This leads to two mandatory properties. The first property a quantum channel has to satisfy is to be trace-preserving (TP), i.e. $\text{Tr}\{\rho_A\}=\text{Tr}\{\rho_B\}$. The second related to positivity is more subtle, namely that $\rho_B \geq 0$ is not enough, because positivity is not closed under the tensor product. Our map has to be <em>completely positive</em> (CP), meaning that for a $\rho_A$ density operator on a system $A$, even if it is extended with an arbitrary reference system $R$ to a density operator $\rho_{AB}$, $(\mathcal{N}_{A \rightarrow B} \otimes id_R) (\rho_{AB}) \geq 0$, i.e. the total system stays positive semi-definite. Both of these properties can be tested via the Choi-Jamiokołwski representation of a channel $\mathcal{E}$:</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>J</mi><mo stretchy="false">(</mo><mi mathvariant="script">E</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><mo stretchy="false">(</mo><mi mathvariant="script">E</mi><mo>⊗</mo><mi>i</mi><msub><mi>d</mi><mi>n</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mpadded><mi mathvariant="normal">∣</mi><msup><mi>ϕ</mi><mo>+</mo></msup><mo stretchy="false">⟩</mo></mpadded><mpadded><mo stretchy="false">⟨</mo><msup><mi>ϕ</mi><mo>+</mo></msup><mi mathvariant="normal">∣</mi></mpadded><mo stretchy="false">)</mo><mo separator="true">,</mo></mrow><annotation encoding="application/x-tex">
J(\mathcal{E}) := (\mathcal{E} \otimes id_n)(\ket{\phi^+}\bra{\phi^+}),
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.09618em;">J</span><span class="mopen">(</span><span class="mord mathcal" style="margin-right:0.08944em;">E</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathcal" style="margin-right:0.08944em;">E</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⊗</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0713em;vertical-align:-0.25em;"></span><span class="mord mathnormal">i</span><span class="mord"><span class="mord mathnormal">d</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mopen">(</span><span class="minner"><span class="mord">∣</span><span class="mord"><span class="mord"><span class="mord mathnormal">ϕ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8213em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">+</span></span></span></span></span></span></span></span></span><span class="mclose">⟩</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen">⟨</span><span class="mord"><span class="mord"><span class="mord mathnormal">ϕ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8213em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">+</span></span></span></span></span></span></span></span></span><span class="mord">∣</span></span><span class="mclose">)</span><span class="mpunct">,</span></span></span></span></span></p>
<p>, where $\ket{\phi^+}:=d^{-n} \sum_{x \in \mathbb{F}_d^n} \ket{xx}$ is the maximally entangled bipartite state on $2n$ $d$-qudits in the computational basis, and $id_n$ is the identity channel acting on the Hilbert space of $n$ $d$-qudits.
Namely, $\mathcal{E}$ is TP if and only if $Tr(J(\mathcal{E})) = 1$, and $\mathcal{E}$ is CP if and only if $J(\mathcal{E}) \geq 0$.</p>
<p>We will use the Kraus representation of a quantum channel $\mathcal{E}$, where, for a set of operators called Kraus operators $\{M_i\}$, such that they are <em>complete</em>, i.e. $\sum_i M_i^\dagger M_i = I$, the channel’s effect is expressed as</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="script">E</mi><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo><mo>=</mo><munder><mo>∑</mo><mi>i</mi></munder><msub><mi>M</mi><mi>i</mi></msub><mi>ρ</mi><msubsup><mi>M</mi><mi>i</mi><mo>†</mo></msubsup><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">
\mathcal{E}(\rho)=\sum_i M_i \rho M_i^\dagger.
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathcal" style="margin-right:0.08944em;">E</span><span class="mopen">(</span><span class="mord mathnormal">ρ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.3277em;vertical-align:-1.2777em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.05em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathnormal">ρ</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.967em;"><span style="top:-2.4231em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.1809em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">†</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2769em;"><span></span></span></span></span></span></span><span class="mord">.</span></span></span></span></span></p>
<p>Measurement of an <em>observable</em> can be described as a quantum channel. An observable $M$ is a Hermitian operator, $M^\dagger=M$, and thus has a spectral decomposition, say $M=\sum_i \lambda_i \ket{\phi_i}\bra{\phi_i}$, where $\lambda_i \in \mathbb{R}$ are the eigenvalues and $\ket{\phi_i}$ are the corresponding eigenstates. When representing a measurement with this observable, the measurement outcomes are the eigenvalues, and after a certain eigenvalue $\lambda_i$ is observed, the system’s state will be projected to the corresponding eigensubspace (of dimension equal to the multiplicity of the eigenvalue), e.g. if $\lambda_i$ has multiplicity 1: $\rho \rightarrow \frac{\ket{\phi_i}\bra{\phi_i}\rho\ket{\phi_i}\bra{\phi_i}}{p_M(i)}$, where $p_M(i)=\text{Tr}\{\ket{\phi_i}\bra{\phi_i}\rho\ket{\phi_i}\bra{\phi_i}\}$ is the probability of observing $\lambda_i$.</p>
<p>Finally, we can always find a <em>Stinespring dilation</em> of a channel $\mathcal{E}_A$ acting on system $A$, which is a unitary $U_{AR}$ on a larger bipartite system $AR$ such that on the subsystem $A$ its effect (i.e. after tracing out system $R$) is exactly the channel $\mathcal{E}_A$:</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi mathvariant="script">E</mi><mi>A</mi></msub><mo stretchy="false">(</mo><msub><mi>ρ</mi><mi>A</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>T</mi><msub><mi>r</mi><mi>B</mi></msub><mspace linebreak="newline"></mspace><mrow><msub><mi>U</mi><mrow><mi>A</mi><mi>R</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>ρ</mi><mi>A</mi></msub><mo>⊗</mo><mpadded><mi mathvariant="normal">∣</mi><mn>0</mn><mo stretchy="false">⟩</mo></mpadded><msub><mpadded><mo stretchy="false">⟨</mo><mn>0</mn><mi mathvariant="normal">∣</mi></mpadded><mi>R</mi></msub><mo stretchy="false">)</mo><msubsup><mi>U</mi><mrow><mi>A</mi><mi>R</mi></mrow><mo>†</mo></msubsup><mspace linebreak="newline"></mspace></mrow></mrow><annotation encoding="application/x-tex">
\mathcal{E}_A(\rho_A) = Tr_B\\{ U_{AR}(\rho_A \otimes \ket{0}\bra{0}_R) U_{AR}^\dagger \\}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathcal" style="margin-right:0.08944em;">E</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0894em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">ρ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">T</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0278em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:1.2667em;vertical-align:-0.2997em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">U</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">A</span><span class="mord mathnormal mtight" style="margin-right:0.00773em;">R</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">ρ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⊗</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="minner"><span class="mord">∣</span><span class="mord"><span class="mord">0</span></span><span class="mclose">⟩</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="minner"><span class="mopen">⟨</span><span class="mord"><span class="mord">0</span></span><span class="mord">∣</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1786em;"><span style="top:-2.4003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.00773em;">R</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2997em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">U</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.967em;"><span style="top:-2.4065em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">A</span><span class="mord mathnormal mtight" style="margin-right:0.00773em;">R</span></span></span></span><span style="top:-3.1809em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mbin mtight">†</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2935em;"><span></span></span></span></span></span></span><span class="mspace newline"></span></span></span></span></span></span></p>
<p>, where $\ket{0}\bra{0}_R$ is an arbitrary pure state picked from the reference subsystem $R$.</p>
<h3 id="entanglement">Entanglement</h3>
<p>A pure quantum state $\ket{\psi}_{AB}$ on a bipartite system $\mathcal{H}_A \otimes \mathcal{H}_B$ is called a <em>tensor product state</em> (TPS) or <em>product state</em> if it can be expressed as a tensor product of two local states: $\ket{\psi} = \ket{\psi}_A \otimes \ket{\psi}_B$. When there are no states local to the subsystems that can completely describe the state of a pure state, it is <em>entangled</em>. A prototypical example is the Bell state, $\frac{\ket{00} + \ket{11}}{\sqrt{2}}$, the qubit case of $\ket{\phi^+}$.</p>
<p>In the case of mixed states, a density operator $\rho$ is called <em>separable</em> if it is a convex combination of pure product states, i.e. $\rho = \sum_x p_X(x) \ket{\psi_x}\bra{\psi_x}_{A} \otimes \ket{\psi_x}\bra{\psi_x}_{B}$. We say that a mixed state is <em>entangled</em> when it is not separable. The set of separable states is convex, i.e. $\forall \rho, \sigma$ separable, for all $\lambda \in [0,1] : \lambda \rho + (1-\lambda) \sigma$ is also separable.</p>
<h3 id="stabilizer-formalism-and-magic">Stabilizer formalism and magic</h3>
<p>The stabilizer formalism is an efficient way to represent certain subspaces and states in a Hilbert space. For the class of states called <em>stabilizer states</em>, which we will define precisely in a moment, simulation and computation with <em>stabilizer operations</em> (SO) become classically simulatable. This result is the Gottesman-Knill theorem <a class="citation" href="#gottesman_stabilizer_1997">(Gottesman, 1997)</a>, and one of its implications is that it identifies a large subset of quantum computing states and operations that are not classical but can be simulated efficiently classically. If one could simulate everything that a quantum computer can do, the field of quantum computing would be in trouble justifying their efforts. Fortunately, this is not the case, as in order to achieve <em>universality</em>, one has to use extra resources: either use operations outside of the stabilizer operations or use a generous amount of special states called <em>magic states</em>. While we will not need magic states in the later exposition,
we remark that Bravyi and Kitaev <a class="citation" href="#bravyi_universal_2005">(Bravyi & Kitaev, 2005)</a> defined magic states as states that are any Clifford transformations of one of these two types of pure magic states:</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mpadded><mi mathvariant="normal">∣</mi><mi>T</mi><mo stretchy="false">⟩</mo></mpadded><mpadded><mo stretchy="false">⟨</mo><mi>T</mi><mi mathvariant="normal">∣</mi></mpadded><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">(</mo><mi>I</mi><mo>+</mo><mfrac><mn>1</mn><msqrt><mn>3</mn></msqrt></mfrac><mo stretchy="false">(</mo><mi>X</mi><mo>+</mo><mi>Z</mi><mo>+</mo><mi>Y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace linebreak="newline"></mspace><mpadded><mi mathvariant="normal">∣</mi><mi>H</mi><mo stretchy="false">⟩</mo></mpadded><mpadded><mo stretchy="false">⟨</mo><mi>H</mi><mi mathvariant="normal">∣</mi></mpadded><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo stretchy="false">(</mo><mi>I</mi><mo>+</mo><mfrac><mn>1</mn><msqrt><mn>2</mn></msqrt></mfrac><mo stretchy="false">(</mo><mi>X</mi><mo>+</mo><mi>Z</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">
\ket{T}\bra{T} = \frac{1}{2}(I + \frac{1}{\sqrt{3}}(X+Z+Y)) \\
\ket{H}\bra{H} = \frac{1}{2}(I + \frac{1}{\sqrt{2}}(X+Z)).
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="minner"><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">T</span></span><span class="mclose">⟩</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen">⟨</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">T</span></span><span class="mord">∣</span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">I</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.2514em;vertical-align:-0.93em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.2028em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9072em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord">3</span></span></span><span style="top:-2.8672em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 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class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.2028em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.9072em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord">2</span></span></span><span style="top:-2.8672em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.1328em;"><span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.93em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mclose">))</span><span class="mord">.</span></span></span></span></span></p>
<p>A state $\ket{\psi} \in \mathcal{H}$ is <em>stabilized</em> by an $n$ qudit Hermitian Pauli operator $P \in \mathcal{P}_n(d)$ (Hermiticity is a requirement for it to be an observable) if the state is an eigenstate of it with +1 eigenvalue, i.e $P\ket{\psi} = \ket{\psi}$. For example $\ket{0}$ is stabilized by $Z$ and $\ket{+}$ is stabilized by $X$.</p>
<p>For two qubits, the situation changes, one might think that for a $\ket{00}$ state, $S_1:=Z_1Z_2$ would be sufficient, however, the +1 eigenspace of $Z_1Z_2$ is two dimensional, both $\ket{00}$ and $\ket{11}$ have +1 eigenvalues. We need then another operator to “filter out” the extra dimension but has $\ket{00}$ as its +1 eigenstate. We find that $S_2:=Z_1I_2$ is a good example. We can also notice that $S_1S_2 = I_1Z_2$, which is also a stabilizer for $\ket{00}$, and naturally $I_1I_2$ is also a stabilizer. The set $\{S_1, S_2, S_1S_2, I\}$ is a <em>group</em>, that is closed under operator multiplication. Also, all of its members commute with each other (i.e. it’s <em>Abelian</em>), thus they can be diagonalized simultaneously. Their shared +1 eigenspace is the <em>stabilized subspace</em>.</p>
<p>In general, we can represent an Abelian group of size $2^k$ with only $k$ independent elements, the <em>generators</em>. An Abelian subgroup of the $n$-qudit Pauli group is called a <em>stabilizer group</em> if it does not contain the element $- \mathbb{I}$ (to ensure that all stabilizers are Hermitian). As each generator “halves” the Hilbert space, we can see that a $k$ dimensional stabilizer group stabilizes an $n-k$ dimensional subspace. When $n=k$, we have a <em>stabilizer state</em>. Single qubit stabilizer states are:</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>Z</mi><mo>:</mo><mpadded><mi mathvariant="normal">∣</mi><mn>0</mn><mo stretchy="false">⟩</mo></mpadded><mtext> </mtext><mo>−</mo><mi>Z</mi><mo>:</mo><mpadded><mi mathvariant="normal">∣</mi><mn>1</mn><mo stretchy="false">⟩</mo></mpadded><mspace linebreak="newline"></mspace><mi>X</mi><mo>:</mo><mpadded><mi mathvariant="normal">∣</mi><mo lspace="0em" rspace="0em">+</mo><mo stretchy="false">⟩</mo></mpadded><mtext> </mtext><mo>−</mo><mi>X</mi><mo>:</mo><mpadded><mi mathvariant="normal">∣</mi><mo lspace="0em" rspace="0em">−</mo><mo stretchy="false">⟩</mo></mpadded><mspace linebreak="newline"></mspace><mi>Y</mi><mo>:</mo><mpadded><mi mathvariant="normal">∣</mi><mi>i</mi><mo stretchy="false">⟩</mo></mpadded><mtext> </mtext><mo>−</mo><mi>Y</mi><mo>:</mo><mpadded><mi mathvariant="normal">∣</mi><mrow><mo>−</mo><mi>i</mi></mrow><mo stretchy="false">⟩</mo></mpadded></mrow><annotation encoding="application/x-tex">
Z: \ket{0} \ -Z: \ket{1} \\
X: \ket{+} \ -X: \ket{-} \\
Y: \ket{i} \ -Y: \ket{-i}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="minner"><span class="mord">∣</span><span class="mord"><span class="mord">0</span></span><span class="mclose">⟩</span></span><span class="mspace"> </span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="minner"><span class="mord">∣</span><span class="mord"><span class="mord">1</span></span><span class="mclose">⟩</span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="minner"><span class="mord">∣</span><span class="mord"><span class="mord">+</span></span><span class="mclose">⟩</span></span><span class="mspace"> </span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="minner"><span class="mord">∣</span><span class="mord"><span class="mord">−</span></span><span class="mclose">⟩</span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="minner"><span class="mord">∣</span><span class="mord"><span class="mord mathnormal">i</span></span><span class="mclose">⟩</span></span><span class="mspace"> </span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="minner"><span class="mord">∣</span><span class="mord"><span class="mord">−</span><span class="mord mathnormal">i</span></span><span class="mclose">⟩</span></span></span></span></span></span></p>
<p>, also $I$ stabilizes all states, and $-I$ stabilizes no states. For two-qubit states, $XX,ZZ$ stabilizes the Bell state, and in general, $\ket{\phi^+}$ is a stabilizer state. The set of $n-qubit$ stabilizer states is denoted as $STAB_n(2)$. Convex combinations of stabilizer states are <em>mixed stabilizer states</em>, and they form <em>stabilizer polytope</em>, denoted $SP_n(2):=convex(STAB_n(2))$. In the single qubit case, this is a simple octahedron on the Bloch sphere as displayed in the figure below.</p>
<center><img src="/assets/images/stab_polytope.png" width="70%" /></center>
<center><small>The stabilizer polytope for single qubit stabilizer states. In higher dimensions, the polytope is more complex.</small></center>
<p>An $n$-qubit Pauli operator can be represented by $2n+1$ bits, with 2 bits for each qubit for the choice of $X=10, Z=01, Y=11, I=00$ and 1 bit for the global phase, the sign $\pm$. Thus a stabilizer state can be described by $n$ $n$-qubit Pauli operators, and as such it requires only $2n^2 + n=O(n^2)$ bits. This polynomial cost is in sharp contrast with the $2^n$ complex numbers required to represent a general state.</p>
<p>Beyond the efficient representation of states, it is possible to efficiently update the state if we limit the set of operations to those that map stabilizer states to stabilizer states. As an $n$-qubit stabilizer state can be represented by $n$ $n$-qubit Pauli operators, we need these operators to map Pauli operators to Pauli operators. The largest set of unitary operators that satisfy this criterion is the Clifford group, $Cl_n(2)$, generated by the $H$, $CNOT$, and the $S=\sqrt{Z}$ gates. Gottesman and Knill showed that a polynomial number of operations is sufficient to simulate Clifford operations or Pauli measurements.</p>
<h2 id="the-structure-of-free-operations">The structure of free operations</h2>
<p>In this section, we will review how the structure of free operations has a gap between axiomatic and operational definitions. In the case of entanglement, after a quick descriptive example for separation from Bennett et al. <a class="citation" href="#bennett_quantum_1999">(Bennett et al., 1999)</a>, we follow Chitambar et al. <a class="citation" href="#chitambar_increasing_2012">(Chitambar et al., 2012)</a> and for magic, the work by Heimendahl et al. <a class="citation" href="#heimendahl_axiomatic_2022">(Heimendahl et al., 2022)</a>.</p>
<h3 id="entanglement-locc-subset-sep">Entanglement: $LOCC \subset SEP$</h3>
<p>As mentioned before, in entanglement theory, the operational and axiomatic classes of RNG operations are the LOCC and the Separable (SEP) channels.</p>
<p><strong>Definition</strong><em>A channel $\mathcal{E}$ on a Hilbert space of $N$ qudits is a <strong>separable channel</strong> if it has a Kraus decomposition with separable Kraus operators, i.e.:
</em></p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="script">E</mi><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo><mo>=</mo><munder><mo>∑</mo><mi>l</mi></munder><msub><mi>A</mi><mi>l</mi></msub><mi>ρ</mi><msubsup><mi>A</mi><mi>l</mi><mo>†</mo></msubsup><mo separator="true">,</mo><msub><mi>A</mi><mi>l</mi></msub><mo>=</mo><msub><mi>M</mi><mrow><mn>1</mn><mo separator="true">,</mo><mi>i</mi></mrow></msub><mo>⊗</mo><msub><mi>M</mi><mrow><mn>2</mn><mo separator="true">,</mo><mi>i</mi></mrow></msub><mo>⊗</mo><mo>…</mo><mo>⊗</mo><msub><mi>M</mi><mrow><mi>N</mi><mo separator="true">,</mo><mi>i</mi></mrow></msub><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">
\mathcal{E}(\rho) = \sum_l A_l \rho A_l^\dagger, A_l = M_{1,i} \otimes M_{2,i} \otimes \ldots \otimes M_{N,i}.
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathcal" style="margin-right:0.08944em;">E</span><span class="mopen">(</span><span class="mord mathnormal">ρ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.3521em;vertical-align:-1.3021em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.05em;"><span style="top:-1.8479em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.01968em;">l</span></span></span><span style="top:-3.05em;"><span class="pstrut" 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style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.01968em;">l</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⊗</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">2</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⊗</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⊗</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.9694em;vertical-align:-0.2861em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.10903em;">N</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">i</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mord">.</span></span></span></span></span></p>
<p><em>
The set of separable channels is denoted SEP.
</em></p>
<p>An early example by Bennett et al. <a class="citation" href="#bennett_quantum_1999">(Bennett et al., 1999)</a> shows the separation between these two classes, which shows “non-locality without entanglement”. In the figure below, we can see 9 states in a domino notation. Alice and Bob both have a qutrit and they can be in one of the 9 states. They are each a bipartite, separable state and mutually orthogonal to each other. However, in order to distinguish them, one has to do a non-local measurement. To create the states we have to use Separable channels. Let $\ket{\psi_i}$ denote the 9 states. Then, the measurement that can distinguish these states can be represented by the Kraus representation $S(\rho)=\sum_i S_i \rho S_i^\dagger$, where:</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>S</mi><mi>i</mi></msub><mo>=</mo><msub><mpadded><mi mathvariant="normal">∣</mi><mi>i</mi><mo stretchy="false">⟩</mo></mpadded><mi>A</mi></msub><msub><mpadded><mi mathvariant="normal">∣</mi><mi>i</mi><mo stretchy="false">⟩</mo></mpadded><mi>B</mi></msub><mpadded><mo stretchy="false">⟨</mo><msub><mi>ψ</mi><mi>i</mi></msub><mi mathvariant="normal">∣</mi></mpadded><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">
S_i = \ket{i}_A \ket{i}_B\bra{\psi_i}.
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0576em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0497em;vertical-align:-0.2997em;"></span><span class="minner"><span class="minner"><span class="mord">∣</span><span class="mord"><span class="mord mathnormal">i</span></span><span class="mclose">⟩</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1786em;"><span style="top:-2.4003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">A</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2997em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="minner"><span class="mord">∣</span><span class="mord"><span class="mord mathnormal">i</span></span><span class="mclose">⟩</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1786em;"><span style="top:-2.4003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight" style="margin-right:0.05017em;">B</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2997em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen">⟨</span><span class="mord"><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">ψ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:-0.0359em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span><span class="mord">∣</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">.</span></span></span></span></span></p>
<center><img src="/assets/images/dominos.png" width="100%" /></center>
<center><small>The 9 orthogonal states on the two-qutrit Hilbert space are all separable but only distinguishable by non-local measurements, and indistinguishable locally.</small></center>
<p>An intuition around how local measurements would act here is that they would always “cut” at least one of the dominoes “in half”, which would make those states indistinguishable. It is laboriously derived and carefully argued in <a class="citation" href="#bennett_quantum_1999">(Bennett et al., 1999)</a> that this measurement operator cannot be implemented by LOCC operations (however, it can be prepared locally!), on the other hand, it is clearly a separable channel. Thus, $LOCC \subset SEP$.</p>
<p>The seminal work from Bennett et al. also quantified the gap, but only to the order of $10^{-6}$. A central objective of the approach by <a class="citation" href="#chitambar_increasing_2012">(Chitambar et al., 2012)</a> was to quantify this gap and define a measure of entanglement that can have a significant difference between the two classes. They managed to show a 12.5% gap using a measure that also has an operational interpretation.</p>
<p><strong>Definition</strong> <em>Suppose we have an arbitrary state $\rho \in DO(\mathcal{H})$, and a set of LOCC transformations, each executed with probability $p_i$, converting $\rho$ into $\rho_i$. Then, we call a function $\mu: DO(\mathcal{H}) \rightarrow \mathbb{R}$ an <strong>entanglement monotone</strong>, when it satisfies the following inequality: </em></p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>μ</mi><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo><mo>≤</mo><munder><mo>∑</mo><mi>i</mi></munder><msub><mi>p</mi><mi>i</mi></msub><mi>μ</mi><mo stretchy="false">(</mo><msub><mi>ρ</mi><mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">
\mu(\rho) \leq \sum_i p_i \mu(\rho_i)
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">μ</span><span class="mopen">(</span><span class="mord mathnormal">ρ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.3277em;vertical-align:-1.2777em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.05em;"><span style="top:-1.8723em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.2777em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">p</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathnormal">μ</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">ρ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3117em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></p>
<p>Chitambar et al. defined a monotone that measures the success rate of a particular task: random-EPR pair distillation from W-class states. W-class states are any state that is obtainable from the $\ket{W}=\frac{1}{\sqrt{3}}(\ket{100}+\ket{010}+\ket{001})$ state with LOCC operations and can be parameterized as $\sqrt{x_0}\lvert 000\rangle + \sqrt{x_A}\lvert 100\rangle + \sqrt{x_B}\lvert 010\rangle + \sqrt{x_C}\lvert 001\rangle$. Due to normalization $x_0$ is fully determined by $1=x_0+x_A+x_B+x_C$, thus we will label W-class states as $\vec{x}=\{x_A,x_B,x_C\}$. In random-EPR pair distillation, one uses a protocol to transform an initial W-class state to random singlet states $\ket{\psi}_ij=\frac{1}{\sqrt{2}}(\lvert 01\rangle_{ij} + \lvert 10\rangle_{ij})$ between any two of the 3 parties as shown in the figure below.</p>
<center><img src="/assets/images/epr.png" width="100%" /></center>
<center><small>Random-EPR pair distillation from a W-class state.</small></center>
<p>For $\vec{x}=\{x_A,x_B,x_C\}$, let $\{n_1,n_2,n_3\}$ be a labeling of $\{A,B,C\}$ so that $x_{n_1} \geq x_{n_2} \geq x_{n_3}$. Then, the function</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>κ</mi><mo stretchy="false">(</mo><mover accent="true"><mi>x</mi><mo>⃗</mo></mover><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><mn>2</mn><mo stretchy="false">(</mo><msub><mi>x</mi><msub><mi>n</mi><mn>2</mn></msub></msub><mo>+</mo><msub><mi>x</mi><msub><mi>n</mi><mn>3</mn></msub></msub><mo stretchy="false">)</mo><mo>−</mo><mfrac><mrow><msub><mi>x</mi><msub><mi>n</mi><mn>2</mn></msub></msub><msub><mi>x</mi><msub><mi>n</mi><mn>3</mn></msub></msub></mrow><msub><mi>x</mi><msub><mi>n</mi><mn>1</mn></msub></msub></mfrac></mrow><annotation encoding="application/x-tex">
\kappa(\vec{x}) := 2(x_{n_2} + x_{n_3}) - \frac{x_{n_2}x_{n_3}}{x_{n_1}}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">κ</span><span class="mopen">(</span><span class="mord accent"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.714em;"><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord mathnormal">x</span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="accent-body" style="left:-0.2077em;"><span class="overlay" style="height:0.714em;width:0.471em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.471em" height="0.714em" style="width:0.471em" viewBox="0 0 471 714" preserveAspectRatio="xMinYMin"><path d="M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5 3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11 10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63 -1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1 -7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59 H213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359 c-16-25.333-24-45-24-59z"></path></svg></span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.0001em;vertical-align:-0.2501em;"></span><span class="mord">2</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2501em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.0001em;vertical-align:-0.2501em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2501em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.0437em;vertical-align:-0.9361em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.1076em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2501em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2501em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3173em;"><span style="top:-2.357em;margin-left:0em;margin-right:0.0714em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.143em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2501em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.9361em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>
<p>is an entanglement monotone that strictly decreases on average when a party with maximum component value makes a measurement by Theorem 1 in <a class="citation" href="#chitambar_increasing_2012">(Chitambar et al., 2012)</a>.</p>
<p>The function $\kappa(\vec{x})$ can reach 1 if and only if $x_{n_1}=x_{n_2}$ and $x_0=0$, and thus $p_{AB}+p_{BC}+p_{AC}\leq \kappa(\vec{x})$ for any $\vec{x}$ - thus we have an upper bound on the total probability of the success rate for the random-EPR distillation of $\vec{x}$. Interestingly, by Theorem 2. in (missing reference), for a class of $x_{n_3} \gt x_0=0$ states, the success rate of LOCC operations can get arbitrarily close to this bound, for any $\epsilon\gt0$, $p_{AB}+p_{BC}+p_{AC}\gt\kappa(\vec{x})-\epsilon$, but no LOCC operation can exactly reach $\kappa(\vec{x})$.</p>
<p>The situation is different for SEP channels as there exists a set of separable measurement operators that can randomly distill a three-qubit W-class state with $x_0=0, 1/2 \geq x_A \geq x_B \geq x_C$ with probability one:</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msub><mi>M</mi><mrow><mi>A</mi><mi>B</mi></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><msub><mi>λ</mi><mrow><mi>C</mi><mi>B</mi></mrow></msub><mpadded><mi mathvariant="normal">∣</mi><mn>0</mn><mo stretchy="false">⟩</mo></mpadded><mpadded><mo stretchy="false">⟨</mo><mn>0</mn><mi mathvariant="normal">∣</mi></mpadded><mo>+</mo><mpadded><mi mathvariant="normal">∣</mi><mn>1</mn><mo stretchy="false">⟩</mo></mpadded><mpadded><mo stretchy="false">⟨</mo><mn>1</mn><mi mathvariant="normal">∣</mi></mpadded><mo stretchy="false">)</mo><mo>⊗</mo><mo stretchy="false">(</mo><msub><mi>λ</mi><mrow><mi>C</mi><mi>A</mi></mrow></msub><mpadded><mi mathvariant="normal">∣</mi><mn>0</mn><mo stretchy="false">⟩</mo></mpadded><mpadded><mo stretchy="false">⟨</mo><mn>0</mn><mi mathvariant="normal">∣</mi></mpadded><mo>+</mo><mpadded><mi mathvariant="normal">∣</mi><mn>1</mn><mo stretchy="false">⟩</mo></mpadded><mpadded><mo stretchy="false">⟨</mo><mn>1</mn><mi mathvariant="normal">∣</mi></mpadded><mo stretchy="false">)</mo><mo>⊗</mo><mpadded><mi mathvariant="normal">∣</mi><mn>0</mn><mo stretchy="false">⟩</mo></mpadded><mpadded><mo stretchy="false">⟨</mo><mn>0</mn><mi mathvariant="normal">∣</mi></mpadded><mspace linebreak="newline"></mspace><msub><mi>M</mi><mrow><mi>A</mi><mi>C</mi></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><msub><mi>λ</mi><mrow><mi>B</mi><mi>C</mi></mrow></msub><mpadded><mi mathvariant="normal">∣</mi><mn>0</mn><mo stretchy="false">⟩</mo></mpadded><mpadded><mo stretchy="false">⟨</mo><mn>0</mn><mi mathvariant="normal">∣</mi></mpadded><mo>+</mo><mpadded><mi mathvariant="normal">∣</mi><mn>1</mn><mo stretchy="false">⟩</mo></mpadded><mpadded><mo stretchy="false">⟨</mo><mn>1</mn><mi mathvariant="normal">∣</mi></mpadded><mo stretchy="false">)</mo><mo>⊗</mo><mpadded><mi mathvariant="normal">∣</mi><mn>0</mn><mo stretchy="false">⟩</mo></mpadded><mpadded><mo stretchy="false">⟨</mo><mn>0</mn><mi mathvariant="normal">∣</mi></mpadded><mo>⊗</mo><mo stretchy="false">(</mo><msub><mi>λ</mi><mrow><mi>B</mi><mi>A</mi></mrow></msub><mpadded><mi mathvariant="normal">∣</mi><mn>0</mn><mo stretchy="false">⟩</mo></mpadded><mpadded><mo stretchy="false">⟨</mo><mn>0</mn><mi mathvariant="normal">∣</mi></mpadded><mo>+</mo><mpadded><mi mathvariant="normal">∣</mi><mn>1</mn><mo stretchy="false">⟩</mo></mpadded><mpadded><mo stretchy="false">⟨</mo><mn>1</mn><mi mathvariant="normal">∣</mi></mpadded><mo stretchy="false">)</mo><mspace linebreak="newline"></mspace><msub><mi>M</mi><mrow><mi>B</mi><mi>C</mi></mrow></msub><mo>=</mo><mpadded><mi mathvariant="normal">∣</mi><mn>0</mn><mo stretchy="false">⟩</mo></mpadded><mpadded><mo stretchy="false">⟨</mo><mn>0</mn><mi mathvariant="normal">∣</mi></mpadded><mo>⊗</mo><mo stretchy="false">(</mo><msub><mi>λ</mi><mrow><mi>A</mi><mi>C</mi></mrow></msub><mpadded><mi mathvariant="normal">∣</mi><mn>0</mn><mo stretchy="false">⟩</mo></mpadded><mpadded><mo stretchy="false">⟨</mo><mn>0</mn><mi mathvariant="normal">∣</mi></mpadded><mo>+</mo><mpadded><mi mathvariant="normal">∣</mi><mn>1</mn><mo stretchy="false">⟩</mo></mpadded><mpadded><mo stretchy="false">⟨</mo><mn>1</mn><mi mathvariant="normal">∣</mi></mpadded><mo stretchy="false">)</mo><mo>⊗</mo><mo stretchy="false">(</mo><msub><mi>λ</mi><mrow><mi>A</mi><mi>B</mi></mrow></msub><mpadded><mi mathvariant="normal">∣</mi><mn>0</mn><mo stretchy="false">⟩</mo></mpadded><mpadded><mo stretchy="false">⟨</mo><mn>0</mn><mi mathvariant="normal">∣</mi></mpadded><mo>+</mo><mpadded><mi mathvariant="normal">∣</mi><mn>1</mn><mo stretchy="false">⟩</mo></mpadded><mpadded><mo stretchy="false">⟨</mo><mn>1</mn><mi mathvariant="normal">∣</mi></mpadded><mo stretchy="false">)</mo><mspace linebreak="newline"></mspace><msub><mi>M</mi><mn>000</mn></msub><mo>=</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msubsup><mi>λ</mi><mrow><mi>C</mi><mi>A</mi></mrow><mn>2</mn></msubsup><msubsup><mi>λ</mi><mrow><mi>C</mi><mi>B</mi></mrow><mn>2</mn></msubsup><mo>−</mo><msubsup><mi>λ</mi><mrow><mi>B</mi><mi>A</mi></mrow><mn>2</mn></msubsup><msubsup><mi>λ</mi><mrow><mi>B</mi><mi>C</mi></mrow><mn>2</mn></msubsup><mo>−</mo><msubsup><mi>λ</mi><mrow><mi>A</mi><mi>B</mi></mrow><mn>2</mn></msubsup><msubsup><mi>λ</mi><mrow><mi>A</mi><mi>C</mi></mrow><mn>2</mn></msubsup><msup><mo stretchy="false">)</mo><mrow><mn>1</mn><mi mathvariant="normal">/</mi><mn>2</mn></mrow></msup><mpadded><mi mathvariant="normal">∣</mi><mn>000</mn><mo stretchy="false">⟩</mo></mpadded><mpadded><mo stretchy="false">⟨</mo><mn>000</mn><mi mathvariant="normal">∣</mi></mpadded><mspace linebreak="newline"></mspace><msub><mi>M</mi><mn>111</mn></msub><mo>=</mo><mpadded><mi mathvariant="normal">∣</mi><mn>111</mn><mo stretchy="false">⟩</mo></mpadded><mpadded><mo stretchy="false">⟨</mo><mn>111</mn><mi mathvariant="normal">∣</mi></mpadded></mrow><annotation encoding="application/x-tex">
M_{AB} = (\lambda_{CB} \ket{0}\bra{0} + \ket{1}\bra{1}) \otimes (\lambda_{CA} \ket{0}\bra{0} + \ket{1}\bra{1}) \otimes \ket{0}\bra{0} \\
M_{AC} = (\lambda_{BC} \ket{0}\bra{0} + \ket{1}\bra{1}) \otimes \ket{0}\bra{0} \otimes (\lambda_{BA} \ket{0}\bra{0} + \ket{1}\bra{1}) \\
M_{BC} = \ket{0}\bra{0} \otimes (\lambda_{AC} \ket{0}\bra{0} + \ket{1}\bra{1}) \otimes (\lambda_{AB} \ket{0}\bra{0} + \ket{1}\bra{1}) \\
M_{000} = (1-\lambda_{CA}^2\lambda_{CB}^2-\lambda_{BA}^2\lambda_{BC}^2-\lambda_{AB}^2\lambda_{AC}^2)^{1/2}\ket{000}\bra{000} \\
M_{111} = \ket{111}\bra{111}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">A</span><span class="mord mathnormal mtight" style="margin-right:0.05017em;">B</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05017em;">CB</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mord">∣</span><span class="mord"><span class="mord">0</span></span><span class="mclose">⟩</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen">⟨</span><span class="mord"><span class="mord">0</span></span><span class="mord">∣</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="minner"><span class="mord">∣</span><span class="mord"><span class="mord">1</span></span><span class="mclose">⟩</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen">⟨</span><span class="mord"><span class="mord">1</span></span><span class="mord">∣</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⊗</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">C</span><span class="mord mathnormal mtight">A</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mord">∣</span><span class="mord"><span class="mord">0</span></span><span class="mclose">⟩</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen">⟨</span><span class="mord"><span class="mord">0</span></span><span class="mord">∣</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="minner"><span class="mord">∣</span><span class="mord"><span class="mord">1</span></span><span class="mclose">⟩</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen">⟨</span><span class="mord"><span class="mord">1</span></span><span class="mord">∣</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⊗</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="minner"><span class="mord">∣</span><span class="mord"><span class="mord">0</span></span><span class="mclose">⟩</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen">⟨</span><span class="mord"><span class="mord">0</span></span><span class="mord">∣</span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">A</span><span class="mord mathnormal mtight" style="margin-right:0.07153em;">C</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">BC</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mord">∣</span><span class="mord"><span class="mord">0</span></span><span class="mclose">⟩</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen">⟨</span><span class="mord"><span class="mord">0</span></span><span class="mord">∣</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="minner"><span class="mord">∣</span><span class="mord"><span class="mord">1</span></span><span class="mclose">⟩</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen">⟨</span><span class="mord"><span class="mord">1</span></span><span class="mord">∣</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⊗</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="minner"><span class="mord">∣</span><span class="mord"><span class="mord">0</span></span><span class="mclose">⟩</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen">⟨</span><span class="mord"><span class="mord">0</span></span><span class="mord">∣</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⊗</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05017em;">B</span><span class="mord mathnormal mtight">A</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mord">∣</span><span class="mord"><span class="mord">0</span></span><span class="mclose">⟩</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen">⟨</span><span class="mord"><span class="mord">0</span></span><span class="mord">∣</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="minner"><span class="mord">∣</span><span class="mord"><span class="mord">1</span></span><span class="mclose">⟩</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen">⟨</span><span class="mord"><span class="mord">1</span></span><span class="mord">∣</span></span><span class="mclose">)</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">BC</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="minner"><span class="mord">∣</span><span class="mord"><span class="mord">0</span></span><span class="mclose">⟩</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen">⟨</span><span class="mord"><span class="mord">0</span></span><span class="mord">∣</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⊗</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">A</span><span class="mord mathnormal mtight" style="margin-right:0.07153em;">C</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mord">∣</span><span class="mord"><span class="mord">0</span></span><span class="mclose">⟩</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen">⟨</span><span class="mord"><span class="mord">0</span></span><span class="mord">∣</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="minner"><span class="mord">∣</span><span class="mord"><span class="mord">1</span></span><span class="mclose">⟩</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen">⟨</span><span class="mord"><span class="mord">1</span></span><span class="mord">∣</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">⊗</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">A</span><span class="mord mathnormal mtight" style="margin-right:0.05017em;">B</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mord">∣</span><span class="mord"><span class="mord">0</span></span><span class="mclose">⟩</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen">⟨</span><span class="mord"><span class="mord">0</span></span><span class="mord">∣</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="minner"><span class="mord">∣</span><span class="mord"><span class="mord">1</span></span><span class="mclose">⟩</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen">⟨</span><span class="mord"><span class="mord">1</span></span><span class="mord">∣</span></span><span class="mclose">)</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">000</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1111em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">C</span><span class="mord mathnormal mtight">A</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05017em;">CB</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.1111em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.05017em;">B</span><span class="mord mathnormal mtight">A</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">BC</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1.188em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">A</span><span class="mord mathnormal mtight" style="margin-right:0.05017em;">B</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">λ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8641em;"><span style="top:-2.453em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">A</span><span class="mord mathnormal mtight" style="margin-right:0.07153em;">C</span></span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">1/2</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mord">∣</span><span class="mord"><span class="mord">000</span></span><span class="mclose">⟩</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen">⟨</span><span class="mord"><span class="mord">000</span></span><span class="mord">∣</span></span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.8333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10903em;">M</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:-0.109em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">111</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="minner"><span class="mord">∣</span><span class="mord"><span class="mord">111</span></span><span class="mclose">⟩</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen">⟨</span><span class="mord"><span class="mord">111</span></span><span class="mord">∣</span></span></span></span></span></span></p>
<p>, where $\lambda_{ij}=\sqrt{\frac{1-2x_i}{2x_j}}$.</p>
<p>The example where the 12.5% difference shows up is the following family of states parameterized by $\frac{1}{3} \leq s \leq \frac{1}{2}$:</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><msqrt><mi>s</mi></msqrt><mpadded><mi mathvariant="normal">∣</mi><mn>100</mn><mo stretchy="false">⟩</mo></mpadded><mo>+</mo><msqrt><mfrac><mrow><mn>1</mn><mo>−</mo><mi>s</mi></mrow><mn>2</mn></mfrac></msqrt><mo stretchy="false">(</mo><mpadded><mi mathvariant="normal">∣</mi><mn>010</mn><mo stretchy="false">⟩</mo></mpadded><mo>+</mo><mpadded><mi mathvariant="normal">∣</mi><mn>001</mn><mo stretchy="false">⟩</mo></mpadded><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">
\sqrt{s} \ket{100} + \sqrt{\frac{1-s}{2}}(\ket{010}+\ket{001})
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.0992em;vertical-align:-0.25em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.8492em;"><span class="svg-align" style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span class="mord" style="padding-left:0.833em;"><span class="mord mathnormal">s</span></span></span><span style="top:-2.8092em;"><span class="pstrut" style="height:3em;"></span><span class="hide-tail" style="min-width:0.853em;height:1.08em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="1.08em" viewBox="0 0 400000 1080" preserveAspectRatio="xMinYMin slice"><path d="M95,702 c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14 c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54 c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10 s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429 c69,-144,104.5,-217.7,106.5,-221 l0 -0 c5.3,-9.3,12,-14,20,-14 H400000v40H845.2724 s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7 c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z M834 80h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.1908em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mord">∣</span><span class="mord"><span class="mord">100</span></span><span class="mclose">⟩</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.44em;vertical-align:-0.7884em;"></span><span class="mord sqrt"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6516em;"><span class="svg-align" style="top:-4.4em;"><span class="pstrut" style="height:4.4em;"></span><span class="mord" style="padding-left:1em;"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord mathnormal">s</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span><span style="top:-3.6116em;"><span class="pstrut" style="height:4.4em;"></span><span class="hide-tail" style="min-width:1.02em;height:2.48em;"><svg xmlns="http://www.w3.org/2000/svg" width="400em" height="2.48em" viewBox="0 0 400000 2592" preserveAspectRatio="xMinYMin slice"><path d="M424,2478 c-1.3,-0.7,-38.5,-172,-111.5,-514c-73,-342,-109.8,-513.3,-110.5,-514 c0,-2,-10.7,14.3,-32,49c-4.7,7.3,-9.8,15.7,-15.5,25c-5.7,9.3,-9.8,16,-12.5,20 s-5,7,-5,7c-4,-3.3,-8.3,-7.7,-13,-13s-13,-13,-13,-13s76,-122,76,-122s77,-121,77,-121 s209,968,209,968c0,-2,84.7,-361.7,254,-1079c169.3,-717.3,254.7,-1077.7,256,-1081 l0 -0c4,-6.7,10,-10,18,-10 H400000 v40H1014.6 s-87.3,378.7,-272.6,1166c-185.3,787.3,-279.3,1182.3,-282,1185 c-2,6,-10,9,-24,9 c-8,0,-12,-0.7,-12,-2z M1001 80 h400000v40h-400000z"></path></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.7884em;"><span></span></span></span></span></span><span class="mopen">(</span><span class="minner"><span class="mord">∣</span><span class="mord"><span class="mord">010</span></span><span class="mclose">⟩</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="minner"><span class="mord">∣</span><span class="mord"><span class="mord">001</span></span><span class="mclose">⟩</span></span><span class="mclose">)</span></span></span></span></span></p>
<p>While the SEP probability is 1, the LOCC probability is bound by $\kappa(\vec{x}) = 2(1-s)-\frac{(1-s)^2}{4}$. We plot the two functions in the figure below, where the separation is clearly visible.</p>
<center><img src="/assets/images/kappas.png" width="100%" /></center>
<center><small>Maximum probability of generating EPR pairs from a parameterized W-class state.</small></center>
<h3 id="stabilizer-formalism-so_nd-subset-csp">Stabilizer formalism: $SO_n(d) \subset CSP$</h3>
<p>The operationally defined set of free operations in the stabilizer formalism is the Stabilizer Operators. We will formally define them here. All the above concepts in <a href="#fundamental-concepts">Fundamental concepts</a> including the Pauli group $\mathcal{P}_n(d)$, the stabilizer group $S$, the Clifford group $Cl_n(d)$, and stabilizer states STAB($d,n$) readily generalize from qubits to qudits as described in <a class="citation" href="#heimendahl_axiomatic_2022">(Heimendahl et al., 2022)</a>. Our formal definitions then will be stated using qudits:</p>
<p><strong>Definition</strong>
<em>A quantum channel mapping $n$ $d$-qudits to $m$ $d$-qudits, for prime dimension $d$, is defined as a <strong>Stabilizer operation</strong> when it is a concatenation of operations that fall into one of the following categories:</em></p>
<ul>
<li>preparation of qudits in stabilizer states </li>
<li>application of a Clifford unitary</li>
<li>measurement of a Pauli observable</li>
<li>discarding of qudits</li>
</ul>
<p><em>
Beyond the above, arbitrary random functions of previous measurement results can act as classical logic to decide which quantum operation to do in each step.
</em></p>
<p>We are taking the stabilizer polytope for $n$-qudits of $d$ dimension, $SP_n(d)$ to be the set of free states. Then, the axiomatically defined RNG operations are the ones that map the stabilizer-polytope to itself. Similar to positivity, in some resource theories, these channels are not closed under tensor product. In order to enforce closeness under tensor product, the set of <em>completely</em>-RNG channels are of interest - in our case the Completely Stabilizer Preserving (CSP) channels, which - for any arbitrary $k$ dimensional extension, maps the $n+k$ dimensional stabilizer polytope to the $m+k$ dimensional stabilizer polytope:</p>
<p><strong>Definition</strong>
<em>A linear superoperator $\mathcal{E}$ mapping $n$ and $m$ qudit spaces of $d$ dimension is <strong>Completely Stabilizer Preserving</strong> if and only if $\mathcal{E} \otimes id_k (SP_{n+k}(d)) \subset SP_{m+k}(d)$ for all $k \in \mathbb{N}$. </em></p>
<p>A CSP operator is also CPTP <a class="citation" href="#heimendahl_axiomatic_2022">(Heimendahl et al., 2022)</a>, thus they are legitimate quantum channels. How can we determine whether a stabilizer-preserving channel $\mathcal{E}$ is also CSP? An interesting result <a class="citation" href="#seddon_quantifying_2019">(Seddon & Campbell, 2019)</a> utilizes the Choi operator to prove that if the Choi operator (which can be looked at as a state) of $\mathcal{E}$ is stabilizer-preserving and trace-preserving, then $\mathcal{E}$ is CSP.</p>
<p>With the definitions out of the way, let’s look at the main result by Heimendahl et al. For the single qudit (qubit) case, $SO_1(d)= CSP_1(d)$. This is intuitively reasonable, as However, starting from two qubits, CSP is strictly larger than SO. Heimendahl et al used the following channel as a counterexample to prove the separation between $SO_2(2)$ and $CSP_2(2)$:</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="normal">Λ</mi><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>ρ</mi><mn>00</mn></msub><mpadded><mi mathvariant="normal">∣</mi><mrow><mo>+</mo><mo>+</mo></mrow><mo stretchy="false">⟩</mo></mpadded><mpadded><mo stretchy="false">⟨</mo><mrow><mo>+</mo><mo>+</mo></mrow><mi mathvariant="normal">∣</mi></mpadded><mo>+</mo><munder><mo>∑</mo><mrow><mi>x</mi><mo>∈</mo><mspace linebreak="newline"></mspace><mrow><mn>01</mn><mo separator="true">,</mo><mn>10</mn><mo separator="true">,</mo><mn>11</mn><mspace linebreak="newline"></mspace></mrow></mrow></munder><msub><mi>ρ</mi><mrow><mi>x</mi><mo separator="true">,</mo><mi>x</mi></mrow></msub><mpadded><mi mathvariant="normal">∣</mi><mi>x</mi><mo stretchy="false">⟩</mo></mpadded><mpadded><mo stretchy="false">⟨</mo><mi>x</mi><mi mathvariant="normal">∣</mi></mpadded><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><munder><mo>∑</mo><mstyle scriptlevel="1"><mtable rowspacing="0.1em" columnalign="center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="1" displaystyle="false"><mrow><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo>∈</mo></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="1" displaystyle="false"><mrow><mn>01</mn><mo separator="true">,</mo><mn>10</mn><mo separator="true">,</mo><mn>11</mn><mspace linebreak="newline"></mspace></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="1" displaystyle="false"><mrow><mi>x</mi><mo mathvariant="normal">≠</mo><mi>y</mi></mrow></mstyle></mtd></mtr></mtable></mstyle></munder><msub><mi>ρ</mi><mrow><mi>x</mi><mo separator="true">,</mo><mi>y</mi></mrow></msub><mpadded><mi mathvariant="normal">∣</mi><mi>x</mi><mo stretchy="false">⟩</mo></mpadded><mpadded><mo stretchy="false">⟨</mo><mi>y</mi><mi mathvariant="normal">∣</mi></mpadded></mrow><annotation encoding="application/x-tex">
\Lambda(\rho) = \rho_{00} \ket{++}\bra{++} + \sum_{x \in \\{01, 10, 11\\}} \rho_{x,x} \ket{x}\bra{x} + \frac{1}{2} \sum_{\substack{x,y \in \\{01, 10, 11\\} \\ x\neq y}} \rho_{x,y}\ket{x}\bra{y}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">Λ</span><span class="mopen">(</span><span class="mord mathnormal">ρ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">ρ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">00</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mord">∣</span><span class="mord"><span class="mord">+</span><span class="mord">+</span></span><span class="mclose">⟩</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen">⟨</span><span class="mord"><span class="mord">+</span><span class="mord">+</span></span><span class="mord">∣</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.4532em;vertical-align:-1.4032em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.05em;"><span style="top:-1.8829em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="mrel mtight">∈</span><span class="mspace mtight newline"></span><span class="mord mtight"><span class="mord mtight">01</span><span class="mpunct mtight">,</span><span class="mord mtight">10</span><span class="mpunct mtight">,</span><span class="mord mtight">11</span><span class="mspace mtight newline"></span></span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.4032em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">ρ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight">x</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mord">∣</span><span class="mord"><span class="mord mathnormal">x</span></span><span class="mclose">⟩</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen">⟨</span><span class="mord"><span class="mord mathnormal">x</span></span><span class="mord">∣</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:3.8604em;vertical-align:-2.5389em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.05em;"><span style="top:-1.2975em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.4807em;"><span style="top:-3.6416em;"><span class="pstrut" style="height:2.7em;"></span><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span><span class="mrel mtight">∈</span></span></span><span style="top:-2.8027em;"><span class="pstrut" style="height:2.7em;"></span><span class="mord mtight"><span class="mord mtight"><span class="mord mtight">01</span><span class="mpunct mtight">,</span><span class="mord mtight">10</span><span class="mpunct mtight">,</span><span class="mord mtight">11</span><span class="mspace mtight newline"></span></span></span></span><span style="top:-1.9138em;"><span class="pstrut" style="height:2.7em;"></span><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="mrel mtight"><span class="mrel mtight"><span class="mord vbox mtight"><span class="thinbox mtight"><span class="rlap mtight"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="inner"><span class="mord mtight"><span class="mrel mtight"></span></span></span><span class="fix"></span></span></span></span></span><span class="mrel mtight">=</span></span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.9806em;"><span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.05em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.5389em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord"><span class="mord mathnormal">ρ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1514em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span><span class="mpunct mtight">,</span><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mord">∣</span><span class="mord"><span class="mord mathnormal">x</span></span><span class="mclose">⟩</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen">⟨</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span><span class="mord">∣</span></span></span></span></span></span></p>
<p>, where $\rho_{x,y}=\bra{x} \rho \ket{y}$.</p>
<p>This channel can be written as the following three operations:</p>
<ol>
<li> measure a projective measurement that distinguishes between $\ket{00}$ and the subspace spanned by $\ket{01},\ket{10},\ket{11}$, i.e. $\ket{00}^\perp$. This operation cannot be implemented using stabilizer operations. This makes the density matrix block-diagonal: </li>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo fence="true">(</mo><mtable rowspacing="0.16em" columnalign="center center center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>ρ</mi><mn>00</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>ρ</mi><mrow><mn>01</mn><mo separator="true">,</mo><mn>01</mn></mrow></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>ρ</mi><mrow><mn>01</mn><mo separator="true">,</mo><mn>10</mn></mrow></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>ρ</mi><mrow><mn>01</mn><mo separator="true">,</mo><mn>11</mn></mrow></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>ρ</mi><mrow><mn>10</mn><mo separator="true">,</mo><mn>01</mn></mrow></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>ρ</mi><mrow><mn>10</mn><mo separator="true">,</mo><mn>10</mn></mrow></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>ρ</mi><mrow><mn>10</mn><mo separator="true">,</mo><mn>11</mn></mrow></msub></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>ρ</mi><mrow><mn>11</mn><mo separator="true">,</mo><mn>01</mn></mrow></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>ρ</mi><mrow><mn>11</mn><mo separator="true">,</mo><mn>10</mn></mrow></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>ρ</mi><mrow><mn>11</mn><mo separator="true">,</mo><mn>11</mn></mrow></msub></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow><annotation encoding="application/x-tex">
\begin{pmatrix}
\rho_{00} & 0 & 0 & 0 \\
0 & \rho_{01,01} & \rho_{01,10} & \rho_{01,11} \\
0 & \rho_{10,01} & \rho_{10,10} & \rho_{10,11} \\
0 & \rho_{11,01} & \rho_{11,10} & \rho_{11,11}
\end{pmatrix}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.8em;vertical-align:-2.15em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.65em;"><span style="top:-4.65em;"><span class="pstrut" style="height:6.8em;"></span><span style="width:0.875em;height:4.800em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.875em" height="4.800em" viewBox="0 0 875 4800"><path d="M863,9c0,-2,-2,-5,-6,-9c0,0,-17,0,-17,0c-12.7,0,-19.3,0.3,-20,1
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style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-1.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.15em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.65em;"><span style="top:-4.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">ρ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">01</span><span class="mpunct mtight">,</span><span class="mord mtight">01</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">ρ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span><span class="mpunct mtight">,</span><span class="mord mtight">01</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span><span style="top:-1.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">ρ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">11</span><span class="mpunct mtight">,</span><span class="mord mtight">01</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.15em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.65em;"><span style="top:-4.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">ρ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">01</span><span class="mpunct mtight">,</span><span class="mord mtight">10</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">ρ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span><span class="mpunct mtight">,</span><span class="mord mtight">10</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span><span style="top:-1.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">ρ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">11</span><span class="mpunct mtight">,</span><span class="mord mtight">10</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.15em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.65em;"><span style="top:-4.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">ρ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">01</span><span class="mpunct mtight">,</span><span class="mord mtight">11</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">ρ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span><span class="mpunct mtight">,</span><span class="mord mtight">11</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span><span style="top:-1.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">ρ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">11</span><span class="mpunct mtight">,</span><span class="mord mtight">11</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.15em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.65em;"><span style="top:-4.65em;"><span class="pstrut" style="height:6.8em;"></span><span style="width:0.875em;height:4.800em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.875em" height="4.800em" viewBox="0 0 875 4800"><path d="M76,0c-16.7,0,-25,3,-25,9c0,2,2,6.3,6,13c21.3,28.7,42.3,60.3,
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<li>Apply partial dephasing with probability 1/2. Off-diagonal terms become half their size: </li>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo fence="true">(</mo><mtable rowspacing="0.16em" columnalign="center center center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>ρ</mi><mn>00</mn></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>ρ</mi><mrow><mn>01</mn><mo separator="true">,</mo><mn>01</mn></mrow></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>ρ</mi><mrow><mn>01</mn><mo separator="true">,</mo><mn>10</mn></mrow></msub><mi mathvariant="normal">/</mi><mn>2</mn></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>ρ</mi><mrow><mn>01</mn><mo separator="true">,</mo><mn>11</mn></mrow></msub><mi mathvariant="normal">/</mi><mn>2</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>ρ</mi><mrow><mn>10</mn><mo separator="true">,</mo><mn>01</mn></mrow></msub><mi mathvariant="normal">/</mi><mn>2</mn></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>ρ</mi><mrow><mn>10</mn><mo separator="true">,</mo><mn>10</mn></mrow></msub></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>ρ</mi><mrow><mn>10</mn><mo separator="true">,</mo><mn>11</mn></mrow></msub><mi mathvariant="normal">/</mi><mn>2</mn></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>ρ</mi><mrow><mn>11</mn><mo separator="true">,</mo><mn>01</mn></mrow></msub><mi mathvariant="normal">/</mi><mn>2</mn></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><msub><mi>ρ</mi><mrow><mn>11</mn><mo separator="true">,</mo><mn>10</mn></mrow></msub><mi mathvariant="normal">/</mi><mn>2</mn></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><msub><mi>ρ</mi><mrow><mn>11</mn><mo separator="true">,</mo><mn>11</mn></mrow></msub></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow><annotation encoding="application/x-tex">
\begin{pmatrix}
\rho_{00} & 0 & 0 & 0 \\
0 & \rho_{01,01} & \rho_{01,10}/2 & \rho_{01,11}/2 \\
0 & \rho_{10,01}/2 & \rho_{10,10} & \rho_{10,11}/2 \\
0 & \rho_{11,01}/2 & \rho_{11,10}/2 & \rho_{11,11}
\end{pmatrix}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.8em;vertical-align:-2.15em;"></span><span class="minner"><span class="mopen"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.65em;"><span style="top:-4.65em;"><span class="pstrut" style="height:6.8em;"></span><span style="width:0.875em;height:4.800em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.875em" height="4.800em" viewBox="0 0 875 4800"><path d="M863,9c0,-2,-2,-5,-6,-9c0,0,-17,0,-17,0c-12.7,0,-19.3,0.3,-20,1
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style="height:0.15em;"><span></span></span></span></span></span></span></span></span><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-1.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.15em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.65em;"><span style="top:-4.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">ρ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">01</span><span class="mpunct mtight">,</span><span class="mord mtight">01</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">ρ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span><span class="mpunct mtight">,</span><span class="mord mtight">01</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mord">/2</span></span></span><span style="top:-1.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">ρ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">11</span><span class="mpunct mtight">,</span><span class="mord mtight">01</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mord">/2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.15em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.65em;"><span style="top:-4.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">ρ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">01</span><span class="mpunct mtight">,</span><span class="mord mtight">10</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mord">/2</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">ρ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span><span class="mpunct mtight">,</span><span class="mord mtight">10</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span><span style="top:-1.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">ρ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">11</span><span class="mpunct mtight">,</span><span class="mord mtight">10</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mord">/2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.15em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.65em;"><span style="top:-4.81em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">ρ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">01</span><span class="mpunct mtight">,</span><span class="mord mtight">11</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mord">/2</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">ρ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">10</span><span class="mpunct mtight">,</span><span class="mord mtight">11</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span><span class="mord">/2</span></span></span><span style="top:-1.21em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"><span class="mord mathnormal">ρ</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3011em;"><span style="top:-2.55em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">11</span><span class="mpunct mtight">,</span><span class="mord mtight">11</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2861em;"><span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.15em;"><span></span></span></span></span></span></span></span><span class="mclose"><span class="delimsizing mult"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.65em;"><span style="top:-4.65em;"><span class="pstrut" style="height:6.8em;"></span><span style="width:0.875em;height:4.800em;"><svg xmlns="http://www.w3.org/2000/svg" width="0.875em" height="4.800em" viewBox="0 0 875 4800"><path d="M76,0c-16.7,0,-25,3,-25,9c0,2,2,6.3,6,13c21.3,28.7,42.3,60.3,
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c242.7,-294.7,395.3,-681.7,458,-1161c21.3,-164.7,33.3,-350.7,36,-558
l0,-1344c-2,-159.3,-10,-310.7,-24,-454c-53.3,-528,-210,-949.7,
-470,-1265c-4.7,-6,-9.7,-11.7,-15,-17c-0.7,-0.7,-6.7,-1,-18,-1z" /></svg></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.15em;"><span></span></span></span></span></span></span></span></span></span></span></span>
<li>Depending on the outcome of the measurement, apply Hadamard on both qubits if $\ket{00}$ was observed. At this point, we get the expression for $\Lambda(\rho)$. </li>
</ol>
<p>It is possible to show that all three steps are necessary for the channel to be CSP but not SO. In a bit more intuitive explanation, in the geometry of the space of channels, after the first step, we are outside of the CSP class (which is also a polytope). The dephasing step removes sufficient “magic” from the channel to make it CSP and the Hadamard at the end sets the direction the CSP polytope is being approached from to make the channel a vertex.</p>
<p>We will introduce the notion of a Clifford dilation for the sketch of the proof.
A superoperator $\mathcal{E}$ mapping $n$ and $m$ qudit spaces of $d$ dimension has a Clifford dilation if there exists a number $k$, a stabilizer state $\ket{s} \in SP_k(d)$ and a Clifford unitary $U$ on $n+k$ qudits such that:
<span class="kdmath">$\mathcal{E}(\rho) = Tr_{m+1,...n+k}\\{U(\rho \otimes \ket{s}\bra{s})U^\dagger\\}$</span></p>
<p>The channels that have a Clifford dilation are the ones that have no measurements or classical randomness.</p>
<p>The sketch of the proof is the following:</p>
<ol>
<li>it is shown that $\Lambda$ is an <em>almost-diagonal channel</em>, meaning that it belongs to $AD_2$ that is defined by its effect on the computational basis states: </li>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="script">E</mi><mo stretchy="false">(</mo><mpadded><mi mathvariant="normal">∣</mi><mn>00</mn><mo stretchy="false">⟩</mo></mpadded><mpadded><mo stretchy="false">⟨</mo><mn>00</mn><mi mathvariant="normal">∣</mi></mpadded><mo stretchy="false">)</mo><mo>=</mo><mpadded><mi mathvariant="normal">∣</mi><mrow><mo>+</mo><mo>+</mo></mrow><mo stretchy="false">⟩</mo></mpadded><mpadded><mo stretchy="false">⟨</mo><mrow><mo>+</mo><mo>+</mo></mrow><mi mathvariant="normal">∣</mi></mpadded><mo separator="true">,</mo><mi mathvariant="script">E</mi><mo stretchy="false">(</mo><mpadded><mi mathvariant="normal">∣</mi><mi>x</mi><mo stretchy="false">⟩</mo></mpadded><mpadded><mo stretchy="false">⟨</mo><mi>x</mi><mi mathvariant="normal">∣</mi></mpadded><mo stretchy="false">)</mo><mo>=</mo><mpadded><mi mathvariant="normal">∣</mi><mi>x</mi><mo stretchy="false">⟩</mo></mpadded><mpadded><mo stretchy="false">⟨</mo><mi>x</mi><mi mathvariant="normal">∣</mi></mpadded><mo separator="true">,</mo><mi>x</mi><mo>∈</mo><msup><mpadded><mi mathvariant="normal">∣</mi><mn>00</mn><mo stretchy="false">⟩</mo></mpadded><mo>⊥</mo></msup></mrow><annotation encoding="application/x-tex">
\mathcal{E}(\ket{00}\bra{00})=\ket{++}\bra{++}, \mathcal{E}(\ket{x}\bra{x})=\ket{x}\bra{x}, x \in \ket{00}^\perp
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathcal" style="margin-right:0.08944em;">E</span><span class="mopen">(</span><span class="minner"><span class="mord">∣</span><span class="mord"><span class="mord">00</span></span><span class="mclose">⟩</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen">⟨</span><span class="mord"><span class="mord">00</span></span><span class="mord">∣</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="minner"><span class="mord">∣</span><span class="mord"><span class="mord">+</span><span class="mord">+</span></span><span class="mclose">⟩</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen">⟨</span><span class="mord"><span class="mord">+</span><span class="mord">+</span></span><span class="mord">∣</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathcal" style="margin-right:0.08944em;">E</span><span class="mopen">(</span><span class="minner"><span class="mord">∣</span><span class="mord"><span class="mord mathnormal">x</span></span><span class="mclose">⟩</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen">⟨</span><span class="mord"><span class="mord mathnormal">x</span></span><span class="mord">∣</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="minner"><span class="mord">∣</span><span class="mord"><span class="mord mathnormal">x</span></span><span class="mclose">⟩</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="minner"><span class="mopen">⟨</span><span class="mord"><span class="mord mathnormal">x</span></span><span class="mord">∣</span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1.239em;vertical-align:-0.25em;"></span><span class="minner"><span class="minner"><span class="mord">∣</span><span class="mord"><span class="mord">00</span></span><span class="mclose">⟩</span></span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.989em;"><span style="top:-3.2029em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mrel mtight">⊥</span></span></span></span></span></span></span></span></span></span></span></span>
<li>This property is then used to show that $\Lambda$ is extremal in $CS_2$. From here, if $\Lambda$ is a stabilizer operation, then it would follow that it is an extremal point in $SO_2$ as well. This is due to convex subsets inheriting the extremality of points from their supersets. </li>
<li>It is shown that extremal points in $SO_2$ do not have a Clifford dilation.</li>
<li>However, it can be proved that $\Lambda$ cannot have a Clifford dilation - thus it must be that $\Lambda \in CSP_2$ but $\Lambda \notin SO_2$</li>
</ol>
<p>For larger qubit and qudit dimensions we refer the reader to <a class="citation" href="#heimendahl_axiomatic_2022">(Heimendahl et al., 2022)</a>. For the full picture though, it is worth mentioning that for larger dimensional qudit states even $CSP_n(d)$ might not be maximal in terms of free operations. For $d$ odd dimensions, resource theories of magic use the Wigner function representation of a function. This is a quasi-probability distribution in phase-space and its total negativity is a magic monotone called <em>mana</em>. Thus, the set of free states can be enlarged to include those with positive Wigner functions, and channels that preserve the positivity of the Wigner function and do so closed under the tensor product are called the <em>completely</em>-Wigner-positivity-preserving ($CWPP_n,m(d)$) channels. Thus in case of odd $d\gt2$ dimensions, the full hierarchy is $SO_{n,m}(d) \subset CSP_{n,m} \subseteq CWPP_{n,m}(d)$, where the last containment is conjectured to be proper by Heimendahl et al.</p>
<h2 id="discussion">Discussion</h2>
<p>Starting from the basics, we reviewed how the structure of the free operations in the theories of entanglement and magic are structured. To explore entanglement, we looked at an early example by Bennett et al. that demonstrated measurement that is separable but not LOCC. Following Chitambar et al., we explored an entanglement monotone specifically designed to measure the maximum success of random-EPR distillation from W-class states, and how it can show a 12.5% gap between LOCC and SEP.</p>
<p>For the theory of magic, Heimendahl et al. did not work with magic monotones, instead, for 2-qubit states, they showed through a specific counter-example that is a completely stabilizer preserving operation but not a stabilizer operation. One could say that their result is similar in spirit to the Bennett et al. example, where careful arguments around a specific set of counterexamples demonstrate the separation of the two classes. Future work might explore actually quantifying the gap in this theory as well for magic distillation tasks. Further implications of the CSP / SO separation could be that the classically simulatable class of channels will become larger, beyond the Gottesman-Knill theorem.</p>
<h2 id="comments">Comments</h2>
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</script>PreludeVisualizing commutative structure in groups2021-10-20T00:00:00-04:002021-10-20T00:00:00-04:00https://refactorium.com/blog/2021/10/20/visualizing-commutatitivity<h2 id="introduction">Introduction</h2>
<p>I built a simple visualization tool, the commutativity plot, that I used to better understand commutativity in finite groups as I was learning about group theory this semester. By the end of this post, I aim to summarize a couple of concepts leading up to two theorems illustrated with this visualization. Namely, we’ll touch on group action, orbits and stabilizers of an element, the permutation representation of groups, conjugation, conjugacy classes, centralizers, and we’ll derive the formula for the number of conjugacy classes. We will also look at why the probability that two randomly picked elements commute in a finite group is a maximum of 5/8 in non-abelian groups, which I find fascinating.</p>
<p>Disclaimer: the function implementation and Mathematica syntax used is very naive, and blows up with larger groups. I am open to PRs, contributions, suggestions for improvements!</p>
<h2 id="groups-acting-on-sets">Groups acting on sets</h2>
<p>One of the most important things you can do with a group is to have it act on a set. The group action has to be valid operations on the set. For example, if the set is the group itself, the action can be left/right multiplication or conjugation (defined later) as well. When we’re talking about an action of an element $g$ of the group $G$ on a set element $a$ in abstract, i.e. without specifying exactly what we mean, we denote it by $g \centerdot a$.</p>
<p>If we apply a group action defined by all elements of $G$ on an element $a$, we get a set: $ O_a = \{ g \centerdot a \lvert g \in G \} $, the <a href="https://proofwiki.org/wiki/Definition:Orbit_(Group_Theory)">orbit</a> of $a$. The elements in $G$ that keep $a$ the same are called the <em>stabilizer</em> of $a$ in $G$, denoted $G_S(a)=\{g \in G \lvert g \centerdot a = a\}$. The stabilizer of any element is a subgroup of $G$. In fact the orbit of $a$ is an equivalence class for $a$, and the group action divides $G$ into disjoint equivalence classes. It is also provable that the order of (the number of elements in) the orbit equals to the <a href="https://en.wikipedia.org/wiki/Index_of_a_subgroup">index</a> of the stabilizer $\lvert G : G_a\lvert$.</p>
<h2 id="permutation-representation">Permutation representation</h2>
<p>By <a href="https://en.wikipedia.org/wiki/Cayley%27s_theorem">Cayley’s theorem</a>, every group of order $n$ is isomorphic to some subgroup of $S_n$, the symmetric group of order $n$. We are going to assume that for the group in question you can generate the permutation representation, specifically the <a href="https://mathworld.wolfram.com/PermutationCycle.html">permutation cycle notation</a> of the elements. For example, for the quaternion group $Q_8$, we can list the elements of the group itself using the <a href="https://reference.wolfram.com/language/Quaternions/tutorial/Quaternions.html"><code class="language-plaintext highlighter-rouge">Quaternions</code> package</a>:</p>
<div class="language-mathematica highlighter-rouge"><div class="highlight"><pre class="highlight"><code><span class="o"><<</span><span class="nv">Quaternions`</span><span class="w">
</span><span class="c">(* Code to generate a group based on its generators and composition </span><span class="err">
</span><span class="c">rule, this is brute force, can easily blow up if your generators are </span><span class="err">
</span><span class="c">not closed under multiplication, or are too big *)</span><span class="w">
</span><span class="nv">GenerateGroupElements</span><span class="p">[</span><span class="nv">generators</span><span class="o">_,</span><span class="w"> </span><span class="nv">compose</span><span class="o">_</span><span class="p">]</span><span class="w"> </span><span class="o">:=</span><span class="w">
</span><span class="nb">FixedPoint</span><span class="p">[</span><span class="nb">Union</span><span class="p">[</span><span class="nf">#</span><span class="o">,</span><span class="w"> </span><span class="nv">compose</span><span class="w"> </span><span class="o">@@@</span><span class="w"> </span><span class="nb">Tuples</span><span class="p">[</span><span class="nf">#</span><span class="o">,</span><span class="w"> </span><span class="m">2</span><span class="p">]]</span><span class="w"> </span><span class="o">&,</span><span class="w"> </span><span class="nv">generators</span><span class="p">]</span><span class="o">;</span><span class="w">
</span><span class="c">(* Q8 group = <i,j,k>, composition: quaternion product *)</span><span class="w">
</span><span class="nv">quaternionGroupElements</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nv">GenerateGroupElements</span><span class="p">[</span><span class="w">
</span><span class="p">{</span><span class="nv">Quaternion</span><span class="p">[</span><span class="m">0</span><span class="o">,</span><span class="w"> </span><span class="m">1</span><span class="o">,</span><span class="w"> </span><span class="m">0</span><span class="o">,</span><span class="w"> </span><span class="m">0</span><span class="p">]</span><span class="o">,</span><span class="w"> </span><span class="c">(* i *)</span><span class="w">
</span><span class="nv">Quaternion</span><span class="p">[</span><span class="m">0</span><span class="o">,</span><span class="w"> </span><span class="m">0</span><span class="o">,</span><span class="w"> </span><span class="m">1</span><span class="o">,</span><span class="w"> </span><span class="m">0</span><span class="p">]</span><span class="o">,</span><span class="w"> </span><span class="c">(* j *)</span><span class="w">
</span><span class="nv">Quaternion</span><span class="p">[</span><span class="m">0</span><span class="o">,</span><span class="w"> </span><span class="m">0</span><span class="o">,</span><span class="w"> </span><span class="m">0</span><span class="o">,</span><span class="w"> </span><span class="m">1</span><span class="p">]}</span><span class="o">,</span><span class="w"> </span><span class="c">(* k *)</span><span class="w">
</span><span class="p">(</span><span class="nf">#1</span><span class="w"> </span><span class="o">**</span><span class="w"> </span><span class="nf">#2</span><span class="p">)</span><span class="w"> </span><span class="o">&</span><span class="w"> </span><span class="c">(* quaternion product *)</span><span class="w">
</span><span class="p">]</span><span class="o">;</span><span class="w">
</span><span class="c">(* A nice example of Cayley's theorem, where we are acting on G by itself with </span><span class="err">
</span><span class="c">left multiplication *)</span><span class="w">
</span><span class="nv">quaternionGroupPermutations</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="p">(</span><span class="nv">el</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nf">#</span><span class="o">;</span><span class="w">
</span><span class="nb">FindPermutation</span><span class="p">[</span><span class="w">
</span><span class="nv">quaternionGroupElements</span><span class="o">,</span><span class="w"> </span><span class="p">(</span><span class="nv">el</span><span class="w"> </span><span class="o">**</span><span class="w"> </span><span class="nf">#</span><span class="p">)</span><span class="w"> </span><span class="o">&</span><span class="w"> </span><span class="o">/@</span><span class="w">
</span><span class="nv">quaternionGroupElements</span><span class="p">]</span><span class="w">
</span><span class="p">)</span><span class="w"> </span><span class="o">&</span><span class="w"> </span><span class="o">/@</span><span class="w"> </span><span class="nv">quaternionGroupElements</span><span class="o">;</span><span class="w">
</span><span class="c">(* storing the result in a PermutationGroup object *)</span><span class="w">
</span><span class="nv">QuaternionGroup</span><span class="w"> </span><span class="o">=</span><span class="w"> </span><span class="nb">PermutationGroup</span><span class="p">[</span><span class="nv">quaternionGroupPermutations</span><span class="w"> </span><span class="p">]</span><span class="w">
</span></code></pre></div></div>
<p>The above results in:</p>
<div class="language-mathematica highlighter-rouge"><div class="highlight"><pre class="highlight"><code><span class="nb">PermutationGroup</span><span class="p">[{</span><span class="nb">Cycles</span><span class="p">[{}]</span><span class="o">,</span><span class="w">
</span><span class="nb">Cycles</span><span class="p">[{{</span><span class="m">1</span><span class="o">,</span><span class="w"> </span><span class="m">2</span><span class="p">}</span><span class="o">,</span><span class="w"> </span><span class="p">{</span><span class="m">3</span><span class="o">,</span><span class="w"> </span><span class="m">4</span><span class="p">}</span><span class="o">,</span><span class="w"> </span><span class="p">{</span><span class="m">5</span><span class="o">,</span><span class="w"> </span><span class="m">6</span><span class="p">}</span><span class="o">,</span><span class="w"> </span><span class="p">{</span><span class="m">7</span><span class="o">,</span><span class="w"> </span><span class="m">8</span><span class="p">}}]</span><span class="o">,</span><span class="w">
</span><span class="nb">Cycles</span><span class="p">[{{</span><span class="m">1</span><span class="o">,</span><span class="w"> </span><span class="m">4</span><span class="o">,</span><span class="w"> </span><span class="m">2</span><span class="o">,</span><span class="w"> </span><span class="m">3</span><span class="p">}</span><span class="o">,</span><span class="w"> </span><span class="p">{</span><span class="m">5</span><span class="o">,</span><span class="w"> </span><span class="m">8</span><span class="o">,</span><span class="w"> </span><span class="m">6</span><span class="o">,</span><span class="w"> </span><span class="m">7</span><span class="p">}}]</span><span class="o">,</span><span class="w">
</span><span class="nb">Cycles</span><span class="p">[{{</span><span class="m">1</span><span class="o">,</span><span class="w"> </span><span class="m">3</span><span class="o">,</span><span class="w"> </span><span class="m">2</span><span class="o">,</span><span class="w"> </span><span class="m">4</span><span class="p">}</span><span class="o">,</span><span class="w"> </span><span class="p">{</span><span class="m">5</span><span class="o">,</span><span class="w"> </span><span class="m">7</span><span class="o">,</span><span class="w"> </span><span class="m">6</span><span class="o">,</span><span class="w"> </span><span class="m">8</span><span class="p">}}]</span><span class="o">,</span><span class="w">
</span><span class="nb">Cycles</span><span class="p">[{{</span><span class="m">1</span><span class="o">,</span><span class="w"> </span><span class="m">6</span><span class="o">,</span><span class="w"> </span><span class="m">2</span><span class="o">,</span><span class="w"> </span><span class="m">5</span><span class="p">}</span><span class="o">,</span><span class="w"> </span><span class="p">{</span><span class="m">3</span><span class="o">,</span><span class="w"> </span><span class="m">7</span><span class="o">,</span><span class="w"> </span><span class="m">4</span><span class="o">,</span><span class="w"> </span><span class="m">8</span><span class="p">}}]</span><span class="o">,</span><span class="w">
</span><span class="nb">Cycles</span><span class="p">[{{</span><span class="m">1</span><span class="o">,</span><span class="w"> </span><span class="m">5</span><span class="o">,</span><span class="w"> </span><span class="m">2</span><span class="o">,</span><span class="w"> </span><span class="m">6</span><span class="p">}</span><span class="o">,</span><span class="w"> </span><span class="p">{</span><span class="m">3</span><span class="o">,</span><span class="w"> </span><span class="m">8</span><span class="o">,</span><span class="w"> </span><span class="m">4</span><span class="o">,</span><span class="w"> </span><span class="m">7</span><span class="p">}}]</span><span class="o">,</span><span class="w">
</span><span class="nb">Cycles</span><span class="p">[{{</span><span class="m">1</span><span class="o">,</span><span class="w"> </span><span class="m">8</span><span class="o">,</span><span class="w"> </span><span class="m">2</span><span class="o">,</span><span class="w"> </span><span class="m">7</span><span class="p">}</span><span class="o">,</span><span class="w"> </span><span class="p">{</span><span class="m">3</span><span class="o">,</span><span class="w"> </span><span class="m">6</span><span class="o">,</span><span class="w"> </span><span class="m">4</span><span class="o">,</span><span class="w"> </span><span class="m">5</span><span class="p">}}]</span><span class="o">,</span><span class="w">
</span><span class="nb">Cycles</span><span class="p">[{{</span><span class="m">1</span><span class="o">,</span><span class="w"> </span><span class="m">7</span><span class="o">,</span><span class="w"> </span><span class="m">2</span><span class="o">,</span><span class="w"> </span><span class="m">8</span><span class="p">}</span><span class="o">,</span><span class="w"> </span><span class="p">{</span><span class="m">3</span><span class="o">,</span><span class="w"> </span><span class="m">5</span><span class="o">,</span><span class="w"> </span><span class="m">4</span><span class="o">,</span><span class="w"> </span><span class="m">6</span><span class="p">}}]}]</span><span class="w">
</span></code></pre></div></div>
<p>The permutation representation for groups is akin to the matrix representation of operators on a vector space in a given basis. We can see in this chosen labeling that $1$ keeps the elements in place, $-1$ is a quadruple of transpositions (swaps), and $i$ permutes $12345678$ into $43128756$.</p>
<p>One thing we get with this representation is stable, deterministic ordering between permutations, which will allow for seeing subgroup structure clearly in larger groups.</p>
<h2 id="conjugacy-classes">Conjugacy classes</h2>
<p>Fun things happen when a group $G$ acts on itself!
Two elements, $g, h \in G$ commute when $hg = gh$. This is the same as saying $g=h^{-1}gh$, i.e. that $g$ is fixed by conjugation by $h$.</p>
<p>If we have $G$ act on itself by conjugation, the orbit of an element $g$, a.k.a the disjoint equivalence class, is called the <a href="https://en.wikipedia.org/wiki/Conjugacy_class">conjugacy class</a> of $g$. The stabilizer subgroup under conjugation for $g$ is the subgroup that fixes $g$ by conjugation. What does that mean? It means that it’s all the elements $g$ commutes with. There is a special name for that: the <a href="https://mathworld.wolfram.com/Centralizer.html">centralizer</a> of $g$ in $G$, denoted $C_G(g)$. The elements that commute with everything are called the <a href="https://en.wikipedia.org/wiki/Center_(group_theory)">center</a> of $G$, $Z(G)$ - it is easy to see that the center is the intersection of all the centralizers and hence it is also a subgroup.</p>
<p>It is easy to prove (p 125 in <a class="citation" href="#dummit2003abstract">(Dummit & Foote, 2003)</a>) that conjugation preserves the length of the cycles of a permutation, i.e. all the conjugates within a conjugacy class have the same cycle lengths. In Mathematica, you could write a function called <code class="language-plaintext highlighter-rouge">CycleLengths</code> that does this calculation.</p>
<div class="language-mathematica highlighter-rouge"><div class="highlight"><pre class="highlight"><code><span class="w">
</span><span class="gp">In[1]:=</span><span class="w"> </span><span class="nv">CycleLengths</span><span class="p">[</span><span class="nv">c</span><span class="o">_</span><span class="p">]</span><span class="o">:=</span><span class="nb">Sort</span><span class="p">[</span><span class="nb">Length</span><span class="o">/@</span><span class="nv">c</span><span class="p">[[</span><span class="m">1</span><span class="p">]]]</span><span class="o">;</span><span class="w">
</span><span class="nv">CycleLengths</span><span class="p">[</span><span class="nb">Cycles</span><span class="p">[{{</span><span class="m">1</span><span class="o">,</span><span class="m">2</span><span class="p">}</span><span class="o">,</span><span class="p">{</span><span class="m">3</span><span class="o">,</span><span class="m">4</span><span class="p">}</span><span class="o">,</span><span class="p">{</span><span class="m">5</span><span class="o">,</span><span class="m">6</span><span class="p">}</span><span class="o">,</span><span class="p">{</span><span class="m">7</span><span class="o">,</span><span class="m">8</span><span class="p">}}]]</span><span class="w">
</span><span class="nv">CycleLengths</span><span class="p">[</span><span class="nb">Cycles</span><span class="p">[{{</span><span class="m">1</span><span class="o">,</span><span class="m">2</span><span class="p">}</span><span class="o">,</span><span class="p">{</span><span class="m">3</span><span class="o">,</span><span class="m">4</span><span class="o">,</span><span class="m">5</span><span class="p">}}]]</span><span class="w">
</span><span class="gp">Out[2]=</span><span class="w"> </span><span class="p">{</span><span class="m">2</span><span class="o">,</span><span class="m">2</span><span class="o">,</span><span class="m">2</span><span class="o">,</span><span class="m">2</span><span class="p">}</span><span class="w">
</span><span class="gp">Out[3]=</span><span class="w"> </span><span class="p">{</span><span class="m">2</span><span class="o">,</span><span class="m">3</span><span class="p">}</span><span class="w">
</span></code></pre></div></div>
<p>However, even though elements in the same conjugacy class have the same cycle lengths, it doesn’t necessarily work the other direction. To see this there are two examples in mind: abelian groups and the $Q_8$ quaternion group.</p>
<p>As everything commutes with everything in abelian groups, the conjugacy class for each element is going to contain only that element. But in a cyclic group of prime order for example all non-trivial elements are of order $p$, their permutation representation looks like $(1234…p)$. How is that possible? Well, there are no other elements to conjugate them into each other…or to put it another way all elements fix each other by conjugation.</p>
<p>In the quaternion group $Q_8$, $i, j, k$ have the same cycle lengths but you can only get as far as an element and its inverse in the conjugacy classes. There are no elements in the group that can do the conjugation.</p>
<p>To summarize - conjugacy classes split up $G$ into disjoint sets, and while cycle lengths are the same within a conjugacy class, the property of being in a conjugacy class is inherently connected to the group you are in.</p>
<h2 id="the-commutativity-plot-commuting-visibly">The commutativity plot: commuting visibly</h2>
<p>To visualize all the ideas above, I coded up a function called <code class="language-plaintext highlighter-rouge">CommutativityPlot</code> in <a href="https://gist.github.com/balopat/3e363e7ca4492fd77703ff80a14830bf">this gist</a>. If we order the elements of the group by their conjugacy class and then their natural sorting order (defined by Mathematica), and we plot whether they commute or not, we’ll get a graph plot like this for our quaternion group, $Q_8$:</p>
<center>
<img src="/assets/images/q8-commute.png" width="60%" />
</center>
<p>Let’s dissect this!</p>
<p>Each row and column contains the elements that are either commute (blue) or don’t commute (red) with the given element corresponding to that row. The blue squares in a row (or column) are the centralizer $C_G(g)$ for each $g$. If the full row (column) is blue, that means that the $g$ element is in the center, $Z(G)$.</p>
<p>We know that every element commutes with:</p>
<ul>
<li>itself $\rightarrow$ the diagonal is always going to be blue</li>
<li>the identity $\rightarrow$ the first row and column will always be blue</li>
</ul>
<p>In the case of $Q_8$, some of the elements are not self-inverse, e.g. $i^{-1} = -i$, and we know that the elements commute with their inverse. In the current labeling, the inverses are next to each other, that’s why we see the 2x2 blue squares on the diagonal.</p>
<p>Also, notice the yellow grid. As we organized the elements by conjugacy class, a conjugacy class will be a contiguous interval of rows and columns. The intersection of these regions is where the inter-class commutation relations are visible. We can see intra-class commutativity relations outside of the block diagonal squares.</p>
<p>As I mentioned in the previous part, cyclic groups are abelian, every element commutes with each other, thus in the commutativity plot we will have</p>
<ul>
<li>a fully blue plot</li>
<li>conjugacy classes of size 1.</li>
</ul>
<p>As an example see below the (pretty boring) commutativity plot for the $C_4$ group.</p>
<center>
<img src="/assets/images/c4-commute.png" width="60%" />
</center>
<h2 id="number-of-conjugacy-classes">Number of conjugacy classes</h2>
<p>Within a conjugacy class (i.e. between two yellow lines), the number of blue squares is going to be the number of elements in the group! Why?</p>
<p>To see this, let’s line up our concepts next to each other in the different “languages” we talk about them:</p>
<table>
<thead>
<tr>
<th>Conjugation</th>
<th>Group acting on <em>itself</em> by conjugation</th>
<th>commutativity plot</th>
</tr>
</thead>
<tbody>
<tr>
<td>element $a \in G$</td>
<td>the target $a \in G$ of the group action</td>
<td>a row / a column</td>
</tr>
<tr>
<td>centralizer of $a$, $C_G(a)$, things that commute with $a$</td>
<td>stabilizer of $a$, $S_G(a)$</td>
<td>the elements corresponding to the blue squares in a row / column</td>
</tr>
<tr>
<td>conjugacy class of $a$</td>
<td>orbit of $a$, $O_a$</td>
<td>the rows/cols within the same yellow grid interval as $a$’s row/col</td>
</tr>
<tr>
<td>number of conjugates / size of conjugacy class</td>
<td>size of orbit</td>
<td>the number of rows/cols within the same yellow grid interval as $a$’s row/col</td>
</tr>
<tr>
<td>number of conjugacy classes</td>
<td>number of orbits</td>
<td>number of yellow grid intervals</td>
</tr>
</tbody>
</table>
<p>Now, the centralizer for each element is a subgroup, in fact, the stabilizer for the element when we consider the group acting on itself by conjugation. As we noted, the size of the orbit is exactly the index of the stabilizer, which means exactly that the blue squares will add up to $\lvert G \lvert$ within the yellow lines. In the case of $Q_8$ above, if we add up all the blue squares and divide it by $G$, then we get the number of conjugacy classes, which is 5!</p>
<p>Hopefully, now it is more clear why the equation holds for the number of conjugacy classes:</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>k</mi><mo>=</mo><mfrac><mrow><munder><mo>∑</mo><mrow><mi>g</mi><mo>∈</mo><mi>G</mi></mrow></munder><mi mathvariant="normal">∣</mi><msub><mi>C</mi><mi>G</mi></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mi mathvariant="normal">∣</mi></mrow><mrow><mi mathvariant="normal">∣</mi><mi>G</mi><mi mathvariant="normal">∣</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">
k = \frac{\sum_{g \in G} |C_G(g)|}{|G|}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.5118em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5758em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣</span><span class="mord mathnormal">G</span><span class="mord">∣</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.8258em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1786em;"><span style="top:-2.4003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">g</span><span class="mrel mtight">∈</span><span class="mord mathnormal mtight">G</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.4358em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">G</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mclose">)</span><span class="mord">∣</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>
<p>Another example is the Pauli group (or Heisenberg-Weyl group), which is of fundamental importance in quantum mechanics and quantum computing. It is generated by the Pauli matrices $X, Y, Z$:</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>X</mi><mo>=</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow><mi>Y</mi><mo>=</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mi>i</mi></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mi>i</mi></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow><mi>Z</mi><mo>=</mo><mrow><mo fence="true">(</mo><mtable rowspacing="0.16em" columnalign="center center" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>1</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="false"><mn>0</mn></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="false"><mrow><mo>−</mo><mn>1</mn></mrow></mstyle></mtd></mtr></mtable><mo fence="true">)</mo></mrow></mrow><annotation encoding="application/x-tex">
X = \begin{pmatrix} 0 & 1 \\
1 & 0
\end{pmatrix}
Y = \begin{pmatrix} 0 & -i \\
i & 0
\end{pmatrix}
Z = \begin{pmatrix} 1 & 0 \\
0 & -1
\end{pmatrix}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord mathnormal">i</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.4em;vertical-align:-0.95em;"></span><span class="minner"><span class="mopen delimcenter" style="top:0em;"><span class="delimsizing size3">(</span></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span><span class="arraycolsep" style="width:0.5em;"></span><span class="arraycolsep" style="width:0.5em;"></span><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.45em;"><span style="top:-3.61em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0</span></span></span><span style="top:-2.41em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.95em;"><span></span></span></span></span></span></span></span><span class="mclose delimcenter" style="top:0em;"><span class="delimsizing size3">)</span></span></span></span></span></span></span></p>
<p>These 3 matrices generate 16 elements, so our permutation representation will be the result of permuting 16 symbols, they get pretty lengthy.</p>
<center>
<img src="/assets/images/pauli-commute.png" width="90%" />
</center>
<p>We can see that the center of the Pauli group is $\{\pm I, \pm i I\}$, each element there has its own conjugacy class, and then, interestingly each element has only one conjugate, which is -1 times the element itself.</p>
<h2 id="the-probability-that-two-elements-commute">The probability that two elements commute</h2>
<p>The commutativity plot almost intuitively leads to this question: how dense can the blue squares be? In probabilistic terms, the proportion of the blue vs the full area is equivalent to the probability that two random elements commute from $G$. Of course, we are interested in non-abelian groups, as abelian groups have a boring full-blue plot. In the case of $S_6$, the plot becomes very large (6! x 6!), but we can see that it is very sparse (also has some cool structure in there):</p>
<center>
<a href="/assets/images/s6-commute.png" target="blank"><img src="/assets/images/s6-commute.png" width="60%" /></a>
</center>
<p>The Pauli group above and the quaternion group have much more blue in them. Let’s quantify this probability exactly.</p>
<div class="language-mathematica highlighter-rouge"><div class="highlight"><pre class="highlight"><code><span class="nv">CommutationProbability</span><span class="p">[</span><span class="nv">group</span><span class="o">_</span><span class="p">]</span><span class="w"> </span><span class="o">:=</span><span class="w">
</span><span class="nb">Count</span><span class="p">[</span><span class="nb">Tuples</span><span class="p">[</span><span class="nb">GroupElements</span><span class="p">[</span><span class="nv">group</span><span class="p">]</span><span class="o">,</span><span class="w"> </span><span class="m">2</span><span class="p">]</span><span class="o">,</span><span class="w">
</span><span class="nv">x</span><span class="o">_</span><span class="w"> </span><span class="o">/;</span><span class="w">
</span><span class="nb">PermutationProduct</span><span class="p">[</span><span class="nv">x</span><span class="p">[[</span><span class="m">1</span><span class="p">]]</span><span class="o">,</span><span class="w"> </span><span class="nv">x</span><span class="p">[[</span><span class="m">2</span><span class="p">]]]</span><span class="w"> </span><span class="o">==</span><span class="w">
</span><span class="nb">PermutationProduct</span><span class="p">[</span><span class="nv">x</span><span class="p">[[</span><span class="m">2</span><span class="p">]]</span><span class="o">,</span><span class="w"> </span><span class="nv">x</span><span class="p">[[</span><span class="m">1</span><span class="p">]]]]</span><span class="o">/</span><span class="nb">GroupOrder</span><span class="p">[</span><span class="nv">group</span><span class="p">]</span><span class="o">^</span><span class="m">2</span><span class="w">
</span></code></pre></div></div>
<p>With the function above (again, very slow, brute force implementation, careful), we can see that our groups have the following probabilities of two of their elements to commute:</p>
<ul>
<li>abelian groups of course 100%</li>
<li>Symmetric group of order 6: 11/720 - pretty low!</li>
<li>Quaternion group, Dihedral group of order 4, and the Pauli group: 5/8</li>
</ul>
<p>Can we go higher? It turns out that two random elements in a non-abelian group can’t have more than a 5/8 chance to commute!</p>
<p>The proof is relatively simple: given that the centralizer $C_G(g)$ for any element $g \in G$ in a non-abelian group is a proper subgroup, it can only be of order $\lvert G \lvert/2$ at most (due to <a href="https://en.wikipedia.org/wiki/Lagrange%27s_theorem_(group_theory)">Lagrange</a> the subgroup’s order has to divide the group’s order). But, the same is true for the center of the group itself, $Z(G)$ is a subgroup of all the centralizers, and as such, it must be up to half the size of the centralizers, and as such, it must be that $\lvert Z(G) \lvert \leq \lvert G \lvert /4$. We can see in the Pauli group that the center has 4 elements, out of the 16, we are hitting this limit.</p>
<p>Now, let’s use these bounds. We’ll simply add all the blue squares - that’s going to be all the orders of the centralizers of all the elements, and divide by all the squares, which is simply $\lvert G \lvert^2$.</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>g</mi><mo separator="true">,</mo><mi>h</mi><mo>∈</mo><mi>G</mi><mtext> commute</mtext><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><munder><mo>∑</mo><mrow><mi>g</mi><mo>∈</mo><mi>G</mi></mrow></munder><mi mathvariant="normal">∣</mi><msub><mi>C</mi><mi>G</mi></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mi mathvariant="normal">∣</mi></mrow><mrow><mi mathvariant="normal">∣</mi><mi>G</mi><msup><mi mathvariant="normal">∣</mi><mn>2</mn></msup></mrow></mfrac></mrow><annotation encoding="application/x-tex">
p(g,h \in G \text{ commute}) = \frac{\sum_{g \in G} |C_G(g)|}{|G|^2}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">h</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">G</span><span class="mord text"><span class="mord"> commute</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.5118em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.5758em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣</span><span class="mord mathnormal">G</span><span class="mord"><span class="mord">∣</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.8258em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.1786em;"><span style="top:-2.4003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">g</span><span class="mrel mtight">∈</span><span class="mord mathnormal mtight">G</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.4358em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">G</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mclose">)</span><span class="mord">∣</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>
<p>For elements in the center, the centralizer is the group itself (they commute with every element), so we can separate those out from the sum:</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>g</mi><mo separator="true">,</mo><mi>h</mi><mo>∈</mo><mi>G</mi><mtext> commute</mtext><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><munder><mo>∑</mo><mrow><mi>g</mi><mo>∈</mo><mi>Z</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow></munder><mi mathvariant="normal">∣</mi><mi>G</mi><mi mathvariant="normal">∣</mi></mrow><mrow><mi mathvariant="normal">∣</mi><mi>G</mi><msup><mi mathvariant="normal">∣</mi><mn>2</mn></msup></mrow></mfrac><mo>+</mo><mfrac><mrow><munder><mo>∑</mo><mrow><mi>g</mi><mo mathvariant="normal">∉</mo><mi>Z</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow></munder><mi mathvariant="normal">∣</mi><msub><mi>C</mi><mi>G</mi></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mi mathvariant="normal">∣</mi></mrow><mrow><mi mathvariant="normal">∣</mi><mi>G</mi><msup><mi mathvariant="normal">∣</mi><mn>2</mn></msup></mrow></mfrac></mrow><annotation encoding="application/x-tex">
p(g,h \in G \text{ commute}) = \frac{\sum_{g \in Z(G)} |G|}{|G|^2} + \frac{\sum_{g \notin Z(G)} |C_G(g)|}{|G|^2}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">h</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">G</span><span class="mord text"><span class="mord"> commute</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.5507em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6147em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣</span><span class="mord mathnormal">G</span><span class="mord"><span class="mord">∣</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.8647em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2253em;"><span style="top:-2.4003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">g</span><span class="mrel mtight">∈</span><span class="mord mathnormal mtight" style="margin-right:0.07153em;">Z</span><span class="mopen mtight">(</span><span class="mord mathnormal mtight">G</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.4747em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">∣</span><span class="mord mathnormal">G</span><span class="mord">∣</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.5507em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6147em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣</span><span class="mord mathnormal">G</span><span class="mord"><span class="mord">∣</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7401em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.8647em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2253em;"><span style="top:-2.4003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">g</span><span class="mrel mtight"><span class="mord mtight"><span class="mrel mtight">∈</span></span><span class="mord vbox mtight"><span class="thinbox mtight"><span class="llap mtight"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="inner"><span class="mord mtight"><span class="mord mtight">/</span><span class="mspace mtight" style="margin-right:0.0651em;"></span></span></span><span class="fix"></span></span></span></span></span><span class="mord mathnormal mtight" style="margin-right:0.07153em;">Z</span><span class="mopen mtight">(</span><span class="mord mathnormal mtight">G</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.4747em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">G</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mclose">)</span><span class="mord">∣</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>
<p>Let’s divide in with the order of $G$:</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>g</mi><mo separator="true">,</mo><mi>h</mi><mo>∈</mo><mi>G</mi><mtext> commute</mtext><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><munder><mo>∑</mo><mrow><mi>g</mi><mo>∈</mo><mi>Z</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow></munder><mi mathvariant="normal">∣</mi><mi>G</mi><mi mathvariant="normal">∣</mi><mi mathvariant="normal">/</mi><mi mathvariant="normal">∣</mi><mi>G</mi><mi mathvariant="normal">∣</mi></mrow><mrow><mi mathvariant="normal">∣</mi><mi>G</mi><mi mathvariant="normal">∣</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><munder><mo>∑</mo><mrow><mi>g</mi><mo mathvariant="normal">∉</mo><mi>Z</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow></munder><mi mathvariant="normal">∣</mi><msub><mi>C</mi><mi>G</mi></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mi mathvariant="normal">∣</mi><mi mathvariant="normal">/</mi><mi mathvariant="normal">∣</mi><mi>G</mi><mi mathvariant="normal">∣</mi></mrow><mrow><mi mathvariant="normal">∣</mi><mi>G</mi><mi mathvariant="normal">∣</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">
p(g,h \in G \text{ commute}) = \frac{\sum_{g \in Z(G)} |G|/|G|}{|G|} + \frac{\sum_{g \notin Z(G)} |C_G(g)|/|G|}{|G|}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">h</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">G</span><span class="mord text"><span class="mord"> commute</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.5507em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6147em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣</span><span class="mord mathnormal">G</span><span class="mord">∣</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.8647em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2253em;"><span style="top:-2.4003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">g</span><span class="mrel mtight">∈</span><span class="mord mathnormal mtight" style="margin-right:0.07153em;">Z</span><span class="mopen mtight">(</span><span class="mord mathnormal mtight">G</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.4747em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">∣</span><span class="mord mathnormal">G</span><span class="mord">∣/∣</span><span class="mord mathnormal">G</span><span class="mord">∣</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.5507em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6147em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣</span><span class="mord mathnormal">G</span><span class="mord">∣</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.8647em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2253em;"><span style="top:-2.4003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">g</span><span class="mrel mtight"><span class="mord mtight"><span class="mrel mtight">∈</span></span><span class="mord vbox mtight"><span class="thinbox mtight"><span class="llap mtight"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="inner"><span class="mord mtight"><span class="mord mtight">/</span><span class="mspace mtight" style="margin-right:0.0651em;"></span></span></span><span class="fix"></span></span></span></span></span><span class="mord mathnormal mtight" style="margin-right:0.07153em;">Z</span><span class="mopen mtight">(</span><span class="mord mathnormal mtight">G</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.4747em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">∣</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3283em;"><span style="top:-2.55em;margin-left:-0.0715em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">G</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mclose">)</span><span class="mord">∣/∣</span><span class="mord mathnormal">G</span><span class="mord">∣</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>
<p>And let’s use our bounds: $\lvert Z(G) \lvert / \lvert G \lvert \leq 1/4$ and $\lvert C_G(g) \lvert / \lvert G \lvert \leq 1/2$ for all $g \in G$:</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>g</mi><mo separator="true">,</mo><mi>h</mi><mo>∈</mo><mi>G</mi><mtext> commute</mtext><mo stretchy="false">)</mo><mo>≤</mo><mfrac><mrow><munder><mo>∑</mo><mrow><mi>g</mi><mo>∈</mo><mi>Z</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow></munder><mn>1</mn></mrow><mrow><mi mathvariant="normal">∣</mi><mi>G</mi><mi mathvariant="normal">∣</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><munder><mo>∑</mo><mrow><mi>g</mi><mo mathvariant="normal">∉</mo><mi>Z</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow></munder><mn>1</mn><mi mathvariant="normal">/</mi><mn>2</mn></mrow><mrow><mi mathvariant="normal">∣</mi><mi>G</mi><mi mathvariant="normal">∣</mi></mrow></mfrac><mo>=</mo><mspace linebreak="newline"></mspace><mo>=</mo><mfrac><mrow><mi mathvariant="normal">∣</mi><mi>Z</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mi mathvariant="normal">∣</mi></mrow><mrow><mi mathvariant="normal">∣</mi><mi>G</mi><mi mathvariant="normal">∣</mi></mrow></mfrac><mo>+</mo><mfrac><mrow><mi mathvariant="normal">∣</mi><mi>G</mi><mi mathvariant="normal">∣</mi><mo>−</mo><mi mathvariant="normal">∣</mi><mi>Z</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mi mathvariant="normal">∣</mi></mrow><mrow><mi mathvariant="normal">∣</mi><mi>G</mi><mi mathvariant="normal">∣</mi></mrow></mfrac><mn>1</mn><mi mathvariant="normal">/</mi><mn>2</mn><mo>≤</mo><mn>1</mn><mi mathvariant="normal">/</mi><mn>4</mn><mo>+</mo><mfrac><mrow><mi mathvariant="normal">∣</mi><mi>G</mi><mi mathvariant="normal">∣</mi><mo>−</mo><mi mathvariant="normal">∣</mi><mi>Z</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mi mathvariant="normal">∣</mi></mrow><mrow><mi mathvariant="normal">∣</mi><mi>G</mi><mi mathvariant="normal">∣</mi></mrow></mfrac><mn>1</mn><mi mathvariant="normal">/</mi><mn>2</mn></mrow><annotation encoding="application/x-tex">
p(g,h \in G \text{ commute}) \leq \frac{\sum_{g \in Z(G)} 1 }{|G|} + \frac{\sum_{g \notin Z(G)} 1/2}{|G|} = \\
=\frac{|Z(G)|}{|G|} + \frac{|G|-|Z(G)|}{|G|}1/2 \leq 1/4 + \frac{|G|-|Z(G)|}{|G|}1/2
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">h</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">G</span><span class="mord text"><span class="mord"> commute</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.5507em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6147em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣</span><span class="mord mathnormal">G</span><span class="mord">∣</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.8647em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2253em;"><span style="top:-2.4003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">g</span><span class="mrel mtight">∈</span><span class="mord mathnormal mtight" style="margin-right:0.07153em;">Z</span><span class="mopen mtight">(</span><span class="mord mathnormal mtight">G</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.4747em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.5507em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6147em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣</span><span class="mord mathnormal">G</span><span class="mord">∣</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.8647em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mop"><span class="mop op-symbol small-op" style="position:relative;top:0em;">∑</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.2253em;"><span style="top:-2.4003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">g</span><span class="mrel mtight"><span class="mord mtight"><span class="mrel mtight">∈</span></span><span class="mord vbox mtight"><span class="thinbox mtight"><span class="llap mtight"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="inner"><span class="mord mtight"><span class="mord mtight">/</span><span class="mspace mtight" style="margin-right:0.0651em;"></span></span></span><span class="fix"></span></span></span></span></span><span class="mord mathnormal mtight" style="margin-right:0.07153em;">Z</span><span class="mopen mtight">(</span><span class="mord mathnormal mtight">G</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.4747em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">1/2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.3669em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.363em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣</span><span class="mord mathnormal">G</span><span class="mord">∣</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mopen">(</span><span class="mord mathnormal">G</span><span class="mclose">)</span><span class="mord">∣</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.363em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣</span><span class="mord mathnormal">G</span><span class="mord">∣</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣</span><span class="mord mathnormal">G</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mopen">(</span><span class="mord mathnormal">G</span><span class="mclose">)</span><span class="mord">∣</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord">1/2</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1/4</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.363em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣</span><span class="mord mathnormal">G</span><span class="mord">∣</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣</span><span class="mord mathnormal">G</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mopen">(</span><span class="mord mathnormal">G</span><span class="mclose">)</span><span class="mord">∣</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord">1/2</span></span></span></span></span></p>
<p>Here we can use the bounds again!</p>
<p><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>g</mi><mo separator="true">,</mo><mi>h</mi><mo>∈</mo><mi>G</mi><mtext> commute</mtext><mo stretchy="false">)</mo><mo>≤</mo><mn>1</mn><mi mathvariant="normal">/</mi><mn>4</mn><mo>+</mo><mfrac><mrow><mi mathvariant="normal">∣</mi><mi>G</mi><mi mathvariant="normal">∣</mi><mo>−</mo><mi mathvariant="normal">∣</mi><mi>Z</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mi mathvariant="normal">∣</mi></mrow><mrow><mi mathvariant="normal">∣</mi><mi>G</mi><mi mathvariant="normal">∣</mi></mrow></mfrac><mn>1</mn><mi mathvariant="normal">/</mi><mn>2</mn><mo>≤</mo><mn>1</mn><mi mathvariant="normal">/</mi><mn>4</mn><mo>+</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mn>1</mn><mi mathvariant="normal">/</mi><mn>4</mn><mo stretchy="false">)</mo><mo>∗</mo><mn>1</mn><mi mathvariant="normal">/</mi><mn>2</mn><mo>=</mo><mn>5</mn><mi mathvariant="normal">/</mi><mn>8</mn></mrow><annotation encoding="application/x-tex">
p(g,h \in G \text{ commute}) \leq 1/4 + \frac{|G|-|Z(G)|}{|G|}1/2 \leq 1/4 + (1-1/4) * 1/2 = 5/8
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">p</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">h</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">∈</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">G</span><span class="mord text"><span class="mord"> commute</span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1/4</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.363em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣</span><span class="mord mathnormal">G</span><span class="mord">∣</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣</span><span class="mord mathnormal">G</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mord">∣</span><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="mopen">(</span><span class="mord mathnormal">G</span><span class="mclose">)</span><span class="mord">∣</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord">1/2</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1/4</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1/4</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">∗</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1/2</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">5/8</span></span></span></span></span></p>
<p>When I first saw this, my mind was blown - group theory seems like this area of infinite possibilities, but it seems like it contains a lot more structure than we’d think at first.</p>
<h2 id="conclusion">Conclusion</h2>
<p>We demonstrated a simple tool to visualize commutation relationships within finite groups. It leverages the permutation representation of groups, which allows for a natural ordering that simplifies grouping conjugation classes together. We demonstrated proofs aided by this visual language for two theorems. I hope you enjoyed it, learned something from it!</p>
<h3 id="further-things-to-explore-and-improve">Further things to explore and improve</h3>
<p>This is probably just the tip of the iceberg. If I had an infinite amount of time I would explore a couple of ideas:</p>
<ul>
<li>Are these “game-of-life” type structures/patterns that show up in the plot of any interest? Can we derive anything from patterns formed on the plot from this particular ordering? Can other orderings result in different insights?</li>
<li>What else can we visualize in this plot:
<ul>
<li>Can we represent normalizers?</li>
<li>Can we represent subgroup structure in the commutation plot?</li>
</ul>
</li>
<li>Improve the tool:
<ul>
<li>Make the tool interactive</li>
<li>create a Python/Javascript version of it, so it doesn’t depend on non-opensource software</li>
</ul>
</li>
</ul>
<p>If you find issues with the post, please open an issue or PR to fix it!</p>
<h2 id="comments">Comments</h2>
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