A Measure Of Quantum Correlations That Lies Between Entanglement And Discord

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When a quantum system is divided into two local subsystems, measurements on the two subsystems can exhibit correlations beyond those possible in a classical joint probability distribution; these are partially explained by entanglement, and more generally by a wider class of measures such as the quantum discord. In this work, I introduce a simple thought experiment defining a new measure of quantum correlations, which I call the accord, and write the result as a minimax optimization over unitary matrices. I find the exact result for pure states as a simple function of the Schmidt coefficients and provide a complete proof, and I likewise provide and prove the result for several classes of mixed states, notably including all states of two qubits and the experimentally relevant case of a pure state mixed with colorless noise. I demonstrate that for two qubit states the accord provides a tight lower bound on the discord; for Bell diagonal states it is also an upper bound on entanglement.



The classic example of an entangled quantum state is the singlet state of two spin-1/2 systems,



|Ψ-⟩=12(|↑↓⟩-|↓↑⟩),fragmentsfragmentsfragments|superscriptΨ⟩12fragments(|↑↓⟩|↓↑⟩),\left|\Psi^-\right\rangle=\frac1\sqrt2\big(\left|\uparrow% \downarrow\right\rangle-\left|\downarrow\uparrow\right\rangle\big),| roman_Ψ start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ⟩ = divide start_ARG 1 end_ARG start_ARG square-root start_ARG 2 end_ARG end_ARG ( | ↑ ↓ ⟩ - | ↓ ↑ ⟩ ) , (1) where ↓⟩ket↑ket↓\\downarrow\right\rangle\ ↓ ⟩ is an orthonormal basis for the Hilbert space for each spin. This is a maximally entangled state, meaning that measurements of the two spins, when made along the same spatial axis, will always be perfectly correlated, even if the spins are space-like separated when the measurements occur. The opposite case is a product state, in which the two parts of the system can be described completely independently. Partially entangled states lie between these two extremes, and substantial effort has gone into finding ways of quantifying the precise degree of entanglement and correlation in such states.Plbnio and Virmani (2007); Horodecki et al. (2009); Modi et al. (2012)



One view is that entanglement is a form of nonlocality. If this were true, an entangled state would violate some Bell-type inequalityBell (1964) that is satisfied by any local hidden variable model (LHVM), such as the Clauser-Horne-Shimony-Holt (CHSH) inequalityClauser et al. (1969) for a system of two spin-1/2 subsystems, or similar inequalities involving moreMermin (1990); Ardehali (1992); Son et al. (2006) or higher-dimensionalCollins et al. (2002); Masanes (2003); Son et al. (2006) subspaces. States can be classified by whether or not they violate such an inequality, which all non-product pure states doGisin (1991). The degree of nonlocality can also be quantified, for example by the maximal amount of random noise that can be added to the state such that it still cannot be described by a LHVMKaszlikowski et al. (2000).



Alternatively, the entanglement of a state can be quantified by the number of singlet states, of the form (1), it is equivalent to.Bennett et al. (1996a); Plbnio and Virmani (2007) For example, one can ask how many singlets, m𝑚mitalic_m, can be made from n𝑛nitalic_n copies of the given state in the limit that n𝑛nitalic_n becomes large; the ratio m/n𝑚𝑛m/nitalic_m / italic_n is called the distillable entanglement.Bennett et al. (1996b) Such measures of singlet equivalence are equivalent to the entanglement entropyBennett et al. (1996a); Popescu and Rohrlich (1997) on pure states and satisfy certain axioms;Vedral et al. (1997); Vidal (2000); Donald et al. (2002) these are formally known as entanglement measures. For mixed states there are many inequivalent measures such as entanglement of formationBennett et al. (1996b); Hayden et al. (2001), the aforementioned distillable entanglement, entanglement of purificationTerhal et al. (2002), and logarithmic negativityVidal and Werner (2002) that give different orderings on the set of statesEisert and Plenio (1999); Eltschka et al. (2015).



Quantum correlations can also be understood through their ability to act as a resource for tasks in quantum computation. One prominent example is quantum teleportationBennett et al. (1993), in which an entangled state shared between two subsystems can be used to transfer the state of a particle from one subsystem to the other. The average fidelity for such a transfer is linearly related to the singlet fraction of the shared entangled state, which is its largest overlap with a maximally entangled state with the same subspace dimensionsHorodecki and Horodecki (1996).



Among pure states, entangled states as identified via the entanglement entropy are precisely the same as those that violate Bell-type inequalitiesGisin (1991); Popescu and Rohrlich (1992) and as those that allow teleportation with a greater fidelity than is possible by any classical strategyHorodecki and Horodecki (1996); Banaszek (2000).



For mixed states, this is no longer the case. There are entangled states that admit a LHVM and do not violate the CHSH inequalityWerner (1989) even with a sequence of measurementsPopescu (1995) and similarly there are states that admit a LHVM but can still be used for quantum teleportation with greater fidelity than is possible by any classical strategyPopescu (1994). At the same time, there are computational tasks with quantum advantages over classical algorithms that cannot be explained by entanglementKnill and Laflamme (1998); Lanyon et al. (2008); Bromley et al. (2017), so a different notion of quantumness versus classicality is needed.



The quantum discord, introduced independently by Henderson and VedralHenderson and Vedral (2001) and Ollivier and ZurekOllivier and Zurek (2001), quantifies the notion of nonclassicality in mixed states; given a state shared between two subsystems, the discord computes how much the state of one subsystem is necessarily modified, on average, by a measurement on the other. The discord and its variants, including geometric discordDakić et al. (2010); Luo and Fu (2010), diagonal discordLloyd et al. (2015); Liu et al. (2017), and othersLuo (2008a); Modi et al. (2010, 2012); Spehner (2014); Adesso et al. (2016), are nonzero on most separable statesFerraro et al. (2010). There is strong evidence to suggest that discord is the relevant resource for a variety of quantum computational tasks.Datta et al. (2008); Lanyon et al. (2008); Cavalcanti et al. (2011); Passante et al. (2011); Dakić et al. ; Gu et al. (2012); Adesso et al. (2016); Braun et al. (2018)



In this paper, I present a new measure of quantum correlations, the accord, defined by a simple thought experiment. The rough idea is that entanglement between two subsystems means that there is an inescapable correlation between measurements made on the two; imagining a game in which the holder of one subsystem (Bob) tries to make his measurements as unpredictable as possible to the holder of the other (Alice), the measure is the (rescaled) probability that Alice is able to guess Bob’s measurements correctly, despite Bob’s best efforts to prevent this. This has two primary advantages over existing measures of quantum correlations: first, because it is defined directly in terms of a simple experimental procedure the measure provides clear intuition about the meaning of entanglement and discord; second, as I demonstrate below, it can be computed exactly for wide classes of states.



The organization of the paper is as follows: in section II, I motivate the thought experiment and use it to formally define the accord as a variational optimization over unitary matrices. In section III I evaluate the accord for pure states and prove the result, and in section IV I prove some results for mixed states, including a simple and efficiently computable prescription to compute the measure on all two qubit states. In section V I compare the accord with existing measures from the literature. Finally, in section VI I conclude with a summary and a discussion of the significance of the results.



II The thought experiment



I begin with an example for motivation. Two observers, Alice (A𝐴Aitalic_A) and Bob (B𝐵Bitalic_B), each hold one qubit, realized as a spin-1/2 system, and the two qubits are in some possibly entangled state. Consider in particular the following two pure states:



|Φ+⟩ketsuperscriptΦ\displaystyle\left|\Phi^+\right\rangle| roman_Φ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ⟩ =12(|↑↑⟩+|↓↓⟩),fragmentsfragments12fragments(|↑↑⟩|↓↓⟩),\displaystyle=\frac1\sqrt2\big(\left|\uparrow\uparrow\right\rangle+% \left|\downarrow\downarrow\right\rangle\big),= divide start_ARG 1 end_ARG start_ARG square-root start_ARG 2 end_ARG end_ARG ( | ↑ ↑ ⟩ + | ↓ ↓ ⟩ ) , (2)



|ψsep⟩ketsubscript𝜓sep\displaystyle\left|\psi_\textsep\right\rangle| italic_ψ start_POSTSUBSCRIPT sep end_POSTSUBSCRIPT ⟩ =|↑↑⟩,fragments|↑↑⟩,\displaystyle=\left|\uparrow\uparrow\right\rangle,= | ↑ ↑ ⟩ , (3) where |↑⟩ket↑\left|\uparrow\right\rangle| ↑ ⟩ and |↓⟩ket↓\left|\downarrow\right\rangle| ↓ ⟩ are the eigenstates of the operator Szsubscript𝑆𝑧S_zitalic_S start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT. The first state is maximally entangled and the second is separable, so measurements made by A𝐴Aitalic_A and B𝐵Bitalic_B should be more correlated in the first state; however, if A𝐴Aitalic_A and B𝐵Bitalic_B both naively measure Szsubscript𝑆𝑧S_zitalic_S start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT, their measurements will be perfectly correlated in either case. Likewise, if B𝐵Bitalic_B chooses to measure Sxsubscript𝑆𝑥S_xitalic_S start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT while A𝐴Aitalic_A still chooses to measure Szsubscript𝑆𝑧S_zitalic_S start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT, the measurements in both cases will be completely uncorrelated.



But now suppose that A𝐴Aitalic_A knows the initial state and also knows B𝐵Bitalic_B’s measurement axis. In that case, if B𝐵Bitalic_B chooses to measure Sxsubscript𝑆𝑥S_xitalic_S start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT, when the shared state is |Φ+⟩ketsuperscriptΦ\left|\Phi^+\right\rangle| roman_Φ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ⟩ A𝐴Aitalic_A can choose to also measure Sxsubscript𝑆𝑥S_xitalic_S start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT, in which case their measurements again become perfectly correlated, but when it is |ψsep⟩ketsubscript𝜓sep\left|\psi_\textsep\right\rangle| italic_ψ start_POSTSUBSCRIPT sep end_POSTSUBSCRIPT ⟩, their measurements will be completely uncorrelated no matter what axis A𝐴Aitalic_A chooses for her measurement.



In other words, the state |Φ+⟩ketsuperscriptΦ\left|\Phi^+\right\rangle| roman_Φ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ⟩ can be said to be maximally entangled because no matter what spin component B𝐵Bitalic_B chooses to measure, A𝐴Aitalic_A can always choose one to achieve perfect correlation between their measurements, while the state |ψsep⟩ketsubscript𝜓sep\left|\psi_\textsep\right\rangle| italic_ψ start_POSTSUBSCRIPT sep end_POSTSUBSCRIPT ⟩ is separable because B𝐵Bitalic_B can choose a spin component for which, no matter what component A𝐴Aitalic_A chooses, their measurements will be completely uncorrelated. For a partially entangled state between these two extremes, the degree of entanglement is characterized by how correlated A𝐴Aitalic_A can force their measurements to be by an appropriate choice of measurement axis, even in the worst case of the choice made by B𝐵Bitalic_B.



II.1 Formal statement, version 1



I now formalize the above intuition. The setup is as follows: two observers, Alice (A𝐴Aitalic_A) and Bob (B𝐵Bitalic_B), share many copies of a quantum state, ρ𝜌\rhoitalic_ρ, in the Hilbert space ℋ=ℋA⊗ℋBℋtensor-productsubscriptℋ𝐴subscriptℋ𝐵\mathcalH=\mathcalH_A\otimes\mathcalH_Bcaligraphic_H = caligraphic_H start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⊗ caligraphic_H start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT. The subspaces held by the two observers, ℋAsubscriptℋ𝐴\mathcalH_Acaligraphic_H start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT and ℋBsubscriptℋ𝐵\mathcalH_Bcaligraphic_H start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT, are both d𝑑ditalic_d-dimensional; let d-1⟩Asubscriptket0𝐴⋯subscriptket𝑑1𝐴\0\right\rangle_A,\cdots,\left be an orthonormal basis for ℋAsubscriptℋ𝐴\mathcalH_Acaligraphic_H start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT and subscriptket0𝐵⋯subscriptket𝑑1𝐵\0\right\rangle_B,\cdots,\left 0 ⟩ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT , ⋯ , for ℋBsubscriptℋ𝐵\mathcalH_Bcaligraphic_H start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT. A𝐴Aitalic_A and B𝐵Bitalic_B are each capable of applying any unitary transformation U∈U(d)𝑈𝑈𝑑U\in U(d)italic_U ∈ italic_U ( italic_d ) to their respective subspace, and each has a device for perfect projective measurements of some operator that is diagonal in the specified basis states and nondegenerate, for example n^^𝑛\hatnover^ start_ARG italic_n end_ARG defined by n^|n⟩=n|n⟩^𝑛ket𝑛𝑛ket𝑛\hatn\left|n\right\rangle=n\left|n\right\rangleover^ start_ARG italic_n end_ARG | italic_n ⟩ = italic_n | italic_n ⟩. An application of some U𝑈Uitalic_U before measurement can be thought of as making the measurement in a different basis (eg Sxsubscript𝑆𝑥S_xitalic_S start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT vs Szsubscript𝑆𝑧S_zitalic_S start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT in the example above).



In the example, A𝐴Aitalic_A was able to pick the right measurement basis to guarantee correlations in the maximally entangled state only because she knew both (1) the initial state and (2) B𝐵Bitalic_B’s choice of basis. Likewise, B𝐵Bitalic_B was only able to pick a basis to guarantee a lack of correlation for the separable state because (3) he knew the initial state. For this first formulation of the thought experiment I assume (1)-(3); these assumptions are dangerously strong, but I will show in the second formulation that they are not actually necessary.



I now define the correlation measure by a procedure which for clarity I present in reverse chronological order:



3. For fixed ρ𝜌\rhoitalic_ρ, UAsubscript𝑈𝐴U_Aitalic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT, and UBsubscript𝑈𝐵U_Bitalic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT, A𝐴Aitalic_A measures n^^𝑛\hatnover^ start_ARG italic_n end_ARG after applying UAsubscript𝑈𝐴U_Aitalic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT and B𝐵Bitalic_B measures n^^𝑛\hatnover^ start_ARG italic_n end_ARG after applying UBsubscript𝑈𝐵U_Bitalic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT. The measurement coincidence probability, or MCP, is the probability that the two measurements agree.



2. Prior to step 3, A𝐴Aitalic_A chooses UAsubscript𝑈𝐴U_Aitalic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT to maximize the MCP, given her knowledge of (assumption 1) ρ𝜌\rhoitalic_ρ and (assumption 2) UBsubscript𝑈𝐵U_Bitalic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT.



1. Prior to step 2, B𝐵Bitalic_B chooses UBsubscript𝑈𝐵U_Bitalic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT to minimize A𝐴Aitalic_A’s maximized MCP, given his knowledge (assumption 3) of ρ𝜌\rhoitalic_ρ. He communicates this choice to A𝐴Aitalic_A for use in step 2.



The value of the MCP, given UAsubscript𝑈𝐴U_Aitalic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT, UBsubscript𝑈𝐵U_Bitalic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT, and ρ𝜌\rhoitalic_ρ, is



∑nAP(n^A=nA)×P(n^B=nA|n^A=nA)subscriptsubscript𝑛𝐴𝑃subscript^𝑛𝐴subscript𝑛𝐴𝑃subscript^𝑛𝐵conditionalsubscript𝑛𝐴subscript^𝑛𝐴subscript𝑛𝐴\sum_n_AP(\hatn_A=n_A)\times P(\hatn_B=n_A|\hatn_A=n_A)∑ start_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_P ( over^ start_ARG italic_n end_ARG start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT = italic_n start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ) × italic_P ( over^ start_ARG italic_n end_ARG start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT = italic_n start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT | over^ start_ARG italic_n end_ARG start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT = italic_n start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ) (4) where



P(n^B=nA|n^A=nA)=P(n^A=nA,n^B=nA)/P(n^A=nA)𝑃subscript^𝑛𝐵conditionalsubscript𝑛𝐴subscript^𝑛𝐴subscript𝑛𝐴𝑃formulae-sequencesubscript^𝑛𝐴subscript𝑛𝐴subscript^𝑛𝐵subscript𝑛𝐴𝑃subscript^𝑛𝐴subscript𝑛𝐴P(\hatn_B=n_A|\hatn_A=n_A)=P(\hatn_A=n_A,\hatn_B=n_A)/% P(\hatn_A=n_A)italic_P ( over^ start_ARG italic_n end_ARG start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT = italic_n start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT | over^ start_ARG italic_n end_ARG start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT = italic_n start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ) = italic_P ( over^ start_ARG italic_n end_ARG start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT = italic_n start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT , over^ start_ARG italic_n end_ARG start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT = italic_n start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ) / italic_P ( over^ start_ARG italic_n end_ARG start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT = italic_n start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ) (5) and



P(n^A=nA,n^B=nA)=Tr(|nA,nA⟩⟨nA,nA|(UA⊗UB)ρ(UA⊗UB)†)fragmentsPfragments(subscript^𝑛𝐴subscript𝑛𝐴,subscript^𝑛𝐵subscript𝑛𝐴)Trfragments(|subscript𝑛𝐴,subscript𝑛𝐴⟩fragments⟨subscript𝑛𝐴,subscript𝑛𝐴|fragments(subscript𝑈𝐴tensor-productsubscript𝑈𝐵)ρsuperscriptfragments(subscript𝑈𝐴tensor-productsubscript𝑈𝐵)†)P(\hatn_A=n_A,\hatn_B=n_A)=\textTr\left(\,\left|n_A,n_A% \right\rangle\left\langlen_A,n_A\right|\,(U_A\otimes U_B)\,\rho\,(U_% A\otimes U_B)^\dagger\right)italic_P ( over^ start_ARG italic_n end_ARG start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT = italic_n start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT , over^ start_ARG italic_n end_ARG start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT = italic_n start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ) = Tr ( | italic_n start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT , italic_n start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⟩ ⟨ italic_n start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT , italic_n start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT | ( italic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⊗ italic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ) italic_ρ ( italic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⊗ italic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ) (6) where |nA,nA⟩ketsubscript𝑛𝐴subscript𝑛𝐴\left|n_A,n_A\right\rangle| italic_n start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT , italic_n start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⟩ is shorthand for |nA⟩A⊗|nA⟩Btensor-productsubscriptketsubscript𝑛𝐴𝐴subscriptketsubscript𝑛𝐴𝐵\left|n_A\right\rangle_A\otimes\left|n_A\right\rangle_B| italic_n start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⟩ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⊗ | italic_n start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⟩ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT. Thus the optimized MCP, or OMCP, is



OMCP≡minUB(maxUA(∑n-0d-1⟨n,n|(UA⊗UB)ρ(UA⊗UB)†|n,n⟩)).OMCPsubscriptsubscript𝑈𝐵subscriptsubscript𝑈𝐴superscriptsubscript𝑛0𝑑1quantum-operator-product𝑛𝑛tensor-productsubscript𝑈𝐴subscript𝑈𝐵𝜌superscripttensor-productsubscript𝑈𝐴subscript𝑈𝐵†𝑛𝑛\textOMCP\equiv\min_U_B\!\left(\!\max_U_A\!\left(\sum_n-0^d-1% \left\langlen,n\right|(U_A\otimes U_B)\,\rho\,(U_A\otimes U_B)^% \dagger\left|n,n\right\rangle\right)\!\!\right).OMCP ≡ roman_min start_POSTSUBSCRIPT italic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( roman_max start_POSTSUBSCRIPT italic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( ∑ start_POSTSUBSCRIPT italic_n - 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ⟨ italic_n , italic_n | ( italic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⊗ italic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ) italic_ρ ( italic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⊗ italic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT | italic_n , italic_n ⟩ ) ) . (7)



As I show in section IV.1 below, 1/d≤OMCP≤11𝑑OMCP11/d\leq\textOMCP\leq 11 / italic_d ≤ OMCP ≤ 1, so to compare with other measures it will be useful to also define a rescaled version that runs from 0 to 1 for any d𝑑ditalic_d,



Accord≡dd-1(OMCP-1d).Accord𝑑𝑑1OMCP1𝑑\textAccord\equiv\fracdd-1\left(\textOMCP-\frac1d\right).Accord ≡ divide start_ARG italic_d end_ARG start_ARG italic_d - 1 end_ARG ( OMCP - divide start_ARG 1 end_ARG start_ARG italic_d end_ARG ) . (8) The name is of course a reference both to the similarity to the discord and to the fact that the measure is based on agreement between measurements.



II.2 Formal statement, version 2



The first statement of the thought experiment can be viewed as a game: the first player, A𝐴Aitalic_A, tries to maximize her score by making the the two parties’ measurements agree, while the second player, B𝐵Bitalic_B, tries to minimize A𝐴Aitalic_A’s score by making the measurements uncorrelated. This formulation requires the assumptions (1)-(3) so that both players can make optimal choices of their measurement bases.



The assumptions can be relaxed by viewing the optimization over unitary matrices in equation (7) not as an explicit choice of the optimal change of basis, but rather as post-facto optimization over the observed measurement coincidence probability over a large set of randomly chosen (or otherwise uniformly distributed) unitaries. The correlation measure can thus be defined according to the following procedure:



1. B𝐵Bitalic_B selects some random set of NBsubscript𝑁𝐵N_Bitalic_N start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT unitary transformations.



2. For each UBsubscript𝑈𝐵U_Bitalic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT selected by B𝐵Bitalic_B, A𝐴Aitalic_A selects NAsubscript𝑁𝐴N_Aitalic_N start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT random unitary transformations.



3. For each pair (UB,UA)subscript𝑈𝐵subscript𝑈𝐴(U_B,U_A)( italic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT , italic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ), A𝐴Aitalic_A and B𝐵Bitalic_B apply their respective transformations to many copies of the state ρ𝜌\rhoitalic_ρ and measure n^^𝑛\hatnover^ start_ARG italic_n end_ARG, then record the fraction of the time that the two measurements agree.



4. For each UBsubscript𝑈𝐵U_Bitalic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT, they take the maximum over all the coincidence probabilities from step 3 with that UBsubscript𝑈𝐵U_Bitalic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT.



5. Finally, they take the minimum value from step 4 over all choices of UBsubscript𝑈𝐵U_Bitalic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT.



This procedure evidently leads, in the limit that NAsubscript𝑁𝐴N_Aitalic_N start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT and NBsubscript𝑁𝐵N_Bitalic_N start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT become large, to the exact same final expression given in equation (7), and as promised assumptions (1)-(3) are no longer needed. In principle this formulation allows for a direct experimental probe of entanglement in an unknown state, requiring only the ability to apply random one-subsystem unitaries and to prepare many copies of the desired state, but the number of measurements required is probably too large to be practical compared with a full state tomographyJames et al. (2001).



II.3 Extension to unequal subspace dimensions



The MCP is defined in terms of the probability that the measurements made by A𝐴Aitalic_A and B𝐵Bitalic_B agree, which requires that they be able to make equivalent measurements, ie. that the two subspaces should be isomorphic. It is thus not obvious how to extend the measure to the case of unequal subspace dimensions.



Supposing that the two dimensions are d1>d2subscript𝑑1subscript𝑑2d_1>d_2italic_d start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT >italic_d start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, one option would be to arbitrarily select d2subscript𝑑2d_2italic_d start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT of the d1subscript𝑑1d_1italic_d start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT states as the ones that should match; the result will not depend on which ones are chosen, since whichever party has the subspace of dimension d1subscript𝑑1d_1italic_d start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT can apply a unitary to permute their basis states.



To formalize this, one can use equation (7) for a d12×d12superscriptsubscript𝑑12superscriptsubscript𝑑12d_1^2\times d_1^2italic_d start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT × italic_d start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT density matrix, with d1(d1-d2)subscript𝑑1subscript𝑑1subscript𝑑2d_1(d_1-d_2)italic_d start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( italic_d start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_d start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) rows and columns equal to 0, and with the unitary matrices for whichever party has the smaller subspace restricted to act as the identity on the corresponding d1-d2subscript𝑑1subscript𝑑2d_1-d_2italic_d start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_d start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT dimensions (thus preserving the zero rows and columns in ρ𝜌\rhoitalic_ρ).



In this paper I will not consider this case further.



III Pure states



In the special case that the state ρ𝜌\rhoitalic_ρ is in fact a pure state, ρ=|ψ⟩⟨ψ|𝜌ket𝜓bra𝜓\rho=\left|\psi\right\rangle\left\langle\psi\right|italic_ρ = | italic_ψ ⟩ ⟨ italic_ψ |, equation (7) can be evaluated explicitly, as I now demonstrate.



The first step is to make use of the Schmidt decomposition: given any pure state |ψ⟩ket𝜓\left|\psi\right\rangle| italic_ψ ⟩ in ℋA⊗ℋBtensor-productsubscriptℋ𝐴subscriptℋ𝐵\mathcalH_A\otimes\mathcalH_Bcaligraphic_H start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⊗ caligraphic_H start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT, there exist unitary matrices U~Asubscript~𝑈𝐴\tildeU_Aover~ start_ARG italic_U end_ARG start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT and U~Bsubscript~𝑈𝐵\tildeU_Bover~ start_ARG italic_U end_ARG start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT and nonnegative numbers c0,⋯,cd-1subscript𝑐0⋯subscript𝑐𝑑1\c_0,\cdots,c_d-1\ italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , ⋯ , italic_c start_POSTSUBSCRIPT italic_d - 1 end_POSTSUBSCRIPT satisfying ∑ci=1subscript𝑐𝑖1\sum c_i=1∑ italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT = 1, such that



|ψ⟩=(U~A⊗U~B)∑i=0d-1ci|i⟩A⊗|i⟩B.ket𝜓tensor-productsubscript~𝑈𝐴subscript~𝑈𝐵superscriptsubscript𝑖0𝑑1tensor-productsubscript𝑐𝑖subscriptket𝑖𝐴subscriptket𝑖𝐵\left|\psi\right\rangle=\big(\tildeU_A\otimes\tildeU_B\big)\sum_% i=0^d-1c_i\left|i\right\rangle_A\otimes\left|i\right\rangle_B.| italic_ψ ⟩ = ( over~ start_ARG italic_U end_ARG start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⊗ over~ start_ARG italic_U end_ARG start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ) ∑ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | italic_i ⟩ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⊗ | italic_i ⟩ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT . (9) The coefficients cisubscript𝑐𝑖c_iitalic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT are unique given |ψ⟩ket𝜓\left|\psi\right\rangle| italic_ψ ⟩, although U~Asubscript~𝑈𝐴\tildeU_Aover~ start_ARG italic_U end_ARG start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT and U~Bsubscript~𝑈𝐵\tildeU_Bover~ start_ARG italic_U end_ARG start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT are not.



Using equation (7) with ρ=|ψ⟩⟨ψ|𝜌ket𝜓bra𝜓\rho=\left|\psi\right\rangle\left\langle\psi\right|italic_ρ = | italic_ψ ⟩ ⟨ italic_ψ | with |ψ⟩ket𝜓\left|\psi\right\rangle| italic_ψ ⟩ in this Schmidt-decomposed form, U~Asubscript~𝑈𝐴\tildeU_Aover~ start_ARG italic_U end_ARG start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT and U~Bsubscript~𝑈𝐵\tildeU_Bover~ start_ARG italic_U end_ARG start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT only appear in the combinations UAU~Asubscript𝑈𝐴subscript~𝑈𝐴U_A\tildeU_Aitalic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT over~ start_ARG italic_U end_ARG start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT and UBU~Bsubscript𝑈𝐵subscript~𝑈𝐵U_B\tildeU_Bitalic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT over~ start_ARG italic_U end_ARG start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT; since the OMCP involves optimization over both UAsubscript𝑈𝐴U_Aitalic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT and UBsubscript𝑈𝐵U_Bitalic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT, U~Asubscript~𝑈𝐴\tildeU_Aover~ start_ARG italic_U end_ARG start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT and U~Bsubscript~𝑈𝐵\tildeU_Bover~ start_ARG italic_U end_ARG start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT may each be assumed without loss of generality to be the d×d𝑑𝑑d\times ditalic_d × italic_d identity matrix. In other words, the OMCP depends only on the Schmidt coefficients cisubscript𝑐𝑖\c_i\ italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT and the state |ψ⟩ket𝜓\left|\psi\right\rangle| italic_ψ ⟩ can be assumed without loss of generality to be of the form



|ψ⟩=∑i=0d-1ci|i⟩A|i⟩B.ket𝜓superscriptsubscript𝑖0𝑑1subscript𝑐𝑖subscriptket𝑖𝐴subscriptket𝑖𝐵\left|\psi\right\rangle=\sum_i=0^d-1c_i\left|i\right\rangle_A\left% |i\right\rangle_B.| italic_ψ ⟩ = ∑ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | italic_i ⟩ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT | italic_i ⟩ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT . (10) Equation (7), when evaluated for ρ=|ψ⟩⟨ψ|𝜌ket𝜓bra𝜓\rho=\left|\psi\right\rangle\left\langle\psi\right|italic_ρ = | italic_ψ ⟩ ⟨ italic_ψ | with |ψ⟩ket𝜓\left|\psi\right\rangle| italic_ψ ⟩ from equation (10), gives



OMCP=1d(∑i=0d-1ci)2.OMCP1𝑑superscriptsuperscriptsubscript𝑖0𝑑1subscript𝑐𝑖2\textOMCP=\frac1d\left(\sum_i=0^d-1c_i\right)^2.OMCP = divide start_ARG 1 end_ARG start_ARG italic_d end_ARG ( ∑ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT . (11) In the proceeding sections, I provide an intuitive picture to explain this result, followed by a complete proof.



III.1 Intuitive picture



To build intuition, I begin with the case of d=2𝑑2d=2italic_d = 2. Consider the state



|ψ⟩=c0|00⟩+c1|11⟩;ket𝜓subscript𝑐0ket00subscript𝑐1ket11\left|\psi\right\rangle=c_0\left|00\right\rangle+c_1\left|11\right\rangle;| italic_ψ ⟩ = italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT | 00 ⟩ + italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | 11 ⟩ ; (12) if A𝐴Aitalic_A and B𝐵Bitalic_B each measure immediately without applying a unitary first, their measurements will be in perfect agreement. Thus it is intuitively reasonable that to reduce this coincidence probability, B𝐵Bitalic_B’s goal in the first formulation of the thought experiment, he ought to try to get as far from this basis as possible. Viewing the qubits as spin-1/2 systems with the state originally specified in the Szsubscript𝑆𝑧S_zitalic_S start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT basis, B𝐵Bitalic_B’s optimal measurement axis would be one in the xy𝑥𝑦xyitalic_x italic_y-plane.



I provide two examples: if B𝐵Bitalic_B chooses to measure along x𝑥xitalic_x or along y𝑦yitalic_y, that is equivalent to applying the unitary matrix



UB(x)=12(111-1)orUB(y)=12(11i-i),superscriptsubscript𝑈𝐵𝑥121111orsuperscriptsubscript𝑈𝐵𝑦1211𝑖𝑖U_B^(x)=\frac1\sqrt2\left(\beginarray[]cc1&1\\ 1&-1\endarray\right)\,\,\,\textor\,\,\,\,U_B^(y)=\frac1\sqrt2% \left(\beginarray[]cc1&1\\ i&-i\endarray\right),italic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_x ) end_POSTSUPERSCRIPT = divide start_ARG 1 end_ARG start_ARG square-root start_ARG 2 end_ARG end_ARG ( start_ARRAY start_ROW start_CELL 1 end_CELL start_CELL 1 end_CELL end_ROW start_ROW start_CELL 1 end_CELL start_CELL - 1 end_CELL end_ROW end_ARRAY ) or italic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_y ) end_POSTSUPERSCRIPT = divide start_ARG 1 end_ARG start_ARG square-root start_ARG 2 end_ARG end_ARG ( start_ARRAY start_ROW start_CELL 1 end_CELL start_CELL 1 end_CELL end_ROW start_ROW start_CELL italic_i end_CELL start_CELL - italic_i end_CELL end_ROW end_ARRAY ) , (13) respectively. With these choices, if A𝐴Aitalic_A naively chooses to measure in the Szsubscript𝑆𝑧S_zitalic_S start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT basis, the measurement coincidence probability will be only 50%. However, if A𝐴Aitalic_A instead chooses to use optimal bases, namely



UA(x)=12(111-1),UA(y)=12(11-ii),formulae-sequencesuperscriptsubscript𝑈𝐴𝑥121111superscriptsubscript𝑈𝐴𝑦1211𝑖𝑖U_A^(x)=\frac1\sqrt2\left(\beginarray[]cc1&1\\ 1&-1\endarray\right),\,\,\,\,\,U_A^(y)=\frac1\sqrt2\left(\begin% array[]cc1&1\\ -i&i\endarray\right),italic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_x ) end_POSTSUPERSCRIPT = divide start_ARG 1 end_ARG start_ARG square-root start_ARG 2 end_ARG end_ARG ( start_ARRAY start_ROW start_CELL 1 end_CELL start_CELL 1 end_CELL end_ROW start_ROW start_CELL 1 end_CELL start_CELL - 1 end_CELL end_ROW end_ARRAY ) , italic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_y ) end_POSTSUPERSCRIPT = divide start_ARG 1 end_ARG start_ARG square-root start_ARG 2 end_ARG end_ARG ( start_ARRAY start_ROW start_CELL 1 end_CELL start_CELL 1 end_CELL end_ROW start_ROW start_CELL - italic_i end_CELL start_CELL italic_i end_CELL end_ROW end_ARRAY ) , (14) then the state |ψ⟩ket𝜓\left|\psi\right\rangle| italic_ψ ⟩ becomes in the two cases



|ψ⟩xxsubscriptket𝜓𝑥𝑥\displaystyle\left|\psi\right\rangle_xx| italic_ψ ⟩ start_POSTSUBSCRIPT italic_x italic_x end_POSTSUBSCRIPT =c0+c12(|00⟩+|11⟩)+c0-c12(|01⟩+|10⟩)absentsubscript𝑐0subscript𝑐12ket00ket11subscript𝑐0subscript𝑐12ket01ket10\displaystyle=\fracc_0+c_12\left(\left|00\right\rangle+\left|11% \right\rangle\right)+\fracc_0-c_12\left(\left|01\right\rangle+\left|% 10\right\rangle\right)= divide start_ARG italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG ( | 00 ⟩ + | 11 ⟩ ) + divide start_ARG italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT - italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG ( | 01 ⟩ + | 10 ⟩ ) (15)



|ψ⟩yysubscriptket𝜓𝑦𝑦\displaystyle\left|\psi\right\rangle_yy| italic_ψ ⟩ start_POSTSUBSCRIPT italic_y italic_y end_POSTSUBSCRIPT =c0+c12(|00⟩+|11⟩)+ic0-c12(|01⟩-|10⟩).absentsubscript𝑐0subscript𝑐12ket00ket11𝑖subscript𝑐0subscript𝑐12ket01ket10\displaystyle=\fracc_0+c_12\left(\left|00\right\rangle+\left|11% \right\rangle\right)+i\fracc_0-c_12\left(\left|01\right\rangle-\left% |10\right\rangle\right).= divide start_ARG italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG ( | 00 ⟩ + | 11 ⟩ ) + italic_i divide start_ARG italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT - italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG 2 end_ARG ( | 01 ⟩ - | 10 ⟩ ) . (16) Either way, the probability that A𝐴Aitalic_A and B𝐵Bitalic_B’s measurements will be the same is exactly (c0+c1)2/2superscriptsubscript𝑐0subscript𝑐122(c_0+c_1)^2/2( italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / 2. That the specified UAsubscript𝑈𝐴U_Aitalic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT and UBsubscript𝑈𝐵U_Bitalic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT are optimal is by no means obvious but can be demonstrated by writing fully general unitaries and explicitly performing the optimization.



For d>2𝑑2d>2italic_d >2, some lessons should carry over: (1) B𝐵Bitalic_B’s measurement basis should maximally mix his original basis states, and (2) an optimal choice for A𝐴Aitalic_A is UA=UB∗subscript𝑈𝐴superscriptsubscript𝑈𝐵∗U_A=U_B^\astitalic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT = italic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT. (The second point turns out not to be true for general UBsubscript𝑈𝐵U_Bitalic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT, but it is true when B𝐵Bitalic_B makes an optimal choice.) With this in mind, we consider the state



|ψ⟩=c0|00⟩+⋯cd-1|d-1,d-1⟩.ket𝜓subscript𝑐0ket00⋯subscript𝑐𝑑1ket𝑑1𝑑1\left|\psi\right\rangle=c_0\left|00\right\rangle+\cdots c_d-1\left|d-% 1,d-1\right\rangle.| italic_ψ ⟩ = italic_c start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT | 00 ⟩ + ⋯ italic_c start_POSTSUBSCRIPT italic_d - 1 end_POSTSUBSCRIPT | italic_d - 1 , italic_d - 1 ⟩ . (17) B𝐵Bitalic_B maximally mixes before measuring by applying the change of basis unitary with elements



[UB]jk=ωdjk/dsubscriptdelimited-[]subscript𝑈𝐵𝑗𝑘superscriptsubscript𝜔𝑑𝑗𝑘𝑑\left[U_B\right]_jk=\omega_d^jk/\sqrtd[ italic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ] start_POSTSUBSCRIPT italic_j italic_k end_POSTSUBSCRIPT = italic_ω start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_j italic_k end_POSTSUPERSCRIPT / square-root start_ARG italic_d end_ARG (18) where ωdsubscript𝜔𝑑\omega_ditalic_ω start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT is the d𝑑ditalic_dth root of unity exp(2πi/d)2𝜋𝑖𝑑\exp(2\pi i/d)roman_exp ( 2 italic_π italic_i / italic_d ) and j𝑗jitalic_j and k𝑘kitalic_k run from 0 to d-1𝑑1d-1italic_d - 1, while A𝐴Aitalic_A tries to unmix using UA=UB∗subscript𝑈𝐴superscriptsubscript𝑈𝐵∗U_A=U_B^\astitalic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT = italic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT. The resulting state is



|ψ⟩ket𝜓\displaystyle\left|\psi\right\rangle| italic_ψ ⟩ =1d∑jcj(∑ke-2πjk/d|k⟩)(∑me2πjm/d|m⟩)absent1𝑑subscript𝑗subscript𝑐𝑗subscript𝑘superscript𝑒2𝜋𝑗𝑘𝑑ket𝑘subscript𝑚superscript𝑒2𝜋𝑗𝑚𝑑ket𝑚\displaystyle=\frac1d\sum_jc_j\left(\sum_ke^-2\pi jk/d\left|k% \right\rangle\right)\left(\sum_me^2\pi jm/d\left|m\right\rangle\right)= divide start_ARG 1 end_ARG start_ARG italic_d end_ARG ∑ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_c start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( ∑ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT - 2 italic_π italic_j italic_k / italic_d end_POSTSUPERSCRIPT | italic_k ⟩ ) ( ∑ start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT 2 italic_π italic_j italic_m / italic_d end_POSTSUPERSCRIPT | italic_m ⟩ ) (19)



=1d∑km(∑jcje2πj(m-k)/d)|km⟩.absent1𝑑subscript𝑘𝑚subscript𝑗subscript𝑐𝑗superscript𝑒2𝜋𝑗𝑚𝑘𝑑ket𝑘𝑚\displaystyle=\frac1d\sum_km\left(\sum_jc_je^2\pi j(m-k)/d\right)% \left|km\right\rangle.= divide start_ARG 1 end_ARG start_ARG italic_d end_ARG ∑ start_POSTSUBSCRIPT italic_k italic_m end_POSTSUBSCRIPT ( ∑ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_c start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT 2 italic_π italic_j ( italic_m - italic_k ) / italic_d end_POSTSUPERSCRIPT ) | italic_k italic_m ⟩ . (20) The largest coefficients are those with no destructive interference, m-k=0𝑚𝑘0m-k=0italic_m - italic_k = 0, and these are precisely the ones we wanted to maximize, corresponding to agreement between A𝐴Aitalic_A and B𝐵Bitalic_B’s measurements; those coefficients are all equal, with a value of (∑jcj)/dsubscript𝑗subscript𝑐𝑗𝑑(\sum_jc_j)/d( ∑ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_c start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) / italic_d. The overall probability that the two measurements are equal is the sum of the squares of these coefficients, precisely matching equation (11).



III.2 Proof



I now prove the result. To do so, I rewrite the OMCP for pure states in two equivalent forms:



OMCP =minUB(maxUA(||(UA∘UB)𝐜||2))absentsubscriptsubscript𝑈𝐵subscriptsubscript𝑈𝐴superscriptnormsubscript𝑈𝐴subscript𝑈𝐵𝐜2\displaystyle=\min_U_B\left(\max_U_A\left(|\!|(U_A\circ U_B)% \mathbfc|\!|^2\right)\right)= roman_min start_POSTSUBSCRIPT italic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( roman_max start_POSTSUBSCRIPT italic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( | | ( italic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ∘ italic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ) bold_c | | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ) (21)



=minUB(maxUA(Tr((UAΛUBT)∘(UA∗ΛUB†)))).absentsubscriptsubscript𝑈𝐵subscriptsubscript𝑈𝐴Trsubscript𝑈𝐴Λsuperscriptsubscript𝑈𝐵𝑇superscriptsubscript𝑈𝐴∗Λsuperscriptsubscript𝑈𝐵†\displaystyle=\min_U_B\left(\max_U_A\left(\textTr((U_A\Lambda U_B% ^T)\circ(U_A^\ast\Lambda U_B^\dagger))\right)\right).= roman_min start_POSTSUBSCRIPT italic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( roman_max start_POSTSUBSCRIPT italic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( Tr ( ( italic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT roman_Λ italic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ) ∘ ( italic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT roman_Λ italic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ) ) ) ) . (22) Here ∘\circ∘ is the elementwise, or Hadamard, product, 𝐜𝐜\mathbfcbold_c is a vector whose entries are the Schmidt coefficients cisubscript𝑐𝑖\c_i\ italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , and ΛΛ\Lambdaroman_Λ is a diagonal matrix whose diagonal entries are again the Schmidt coefficients. That these are equivalent to the OMCP is proven in Appendix A. Using these expressions, I prove the result in two steps.



Step 1: OMCP≥(∑ci)2/dOMCPsuperscriptsubscript𝑐𝑖2𝑑\textOMCP\geq(\sum c_i)^2/dOMCP ≥ ( ∑ italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / italic_d



For any fixed UBsubscript𝑈𝐵U_Bitalic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT,



maxUA(||(UA∘UB)𝐜||2)≥||(UB∗∘UB)𝐜||2subscriptsubscript𝑈𝐴superscriptnormsubscript𝑈𝐴subscript𝑈𝐵𝐜2superscriptnormsuperscriptsubscript𝑈𝐵∗subscript𝑈𝐵𝐜2\max_U_A\left(|\!|(U_A\circ U_B)\mathbfc|\!|^2\right)\geq|\!|(U_B% ^\ast\circ U_B)\mathbfc|\!|^2roman_max start_POSTSUBSCRIPT italic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( | | ( italic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ∘ italic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ) bold_c | | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ≥ | | ( italic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∘ italic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ) bold_c | | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT (23) since UB∗superscriptsubscript𝑈𝐵∗U_B^\astitalic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT is included as a possible UAsubscript𝑈𝐴U_Aitalic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT on the left-hand side, and thus



minUB(maxUA(||(UA∘UB)𝐜||2))≥minUB(||(UB∗∘UB)𝐜||2).subscriptsubscript𝑈𝐵subscriptsubscript𝑈𝐴superscriptnormsubscript𝑈𝐴subscript𝑈𝐵𝐜2subscriptsubscript𝑈𝐵superscriptnormsuperscriptsubscript𝑈𝐵∗subscript𝑈𝐵𝐜2\min_U_B\left(\max_U_A\left(|\!|(U_A\circ U_B)\mathbfc|\!|^2% \right)\right)\geq\min_U_B\left(|\!|(U_B^\ast\circ U_B)\mathbfc|\! discord server % |^2\right).roman_min start_POSTSUBSCRIPT italic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( roman_max start_POSTSUBSCRIPT italic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( | | ( italic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ∘ italic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ) bold_c | | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ) ≥ roman_min start_POSTSUBSCRIPT italic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( | | ( italic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∘ italic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ) bold_c | | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) . (24) It therefore suffices to show that



||(UB∗∘UB)𝐜||2≥1d(∑i=0d-1ci)2superscriptnormsuperscriptsubscript𝑈𝐵∗subscript𝑈𝐵𝐜21𝑑superscriptsuperscriptsubscript𝑖0𝑑1subscript𝑐𝑖2|\!|(U_B^\ast\circ U_B)\mathbfc|\!|^2\geq\frac1d\left(\sum_i=0% ^d-1c_i\right)^2| | ( italic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∘ italic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ) bold_c | | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ divide start_ARG 1 end_ARG start_ARG italic_d end_ARG ( ∑ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT (25) for all UBsubscript𝑈𝐵U_Bitalic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT.



To do so, I use the lemmaTaskara and Gumus (2013)



Tr(A)Tr(B)=dTr(A∘B)-∑i=1d-1∑j=i+1d(aii-ajj)(bii-bjj)Tr𝐴Tr(B)𝑑Tr𝐴𝐵superscriptsubscript𝑖1𝑑1superscriptsubscript𝑗𝑖1𝑑subscript𝑎𝑖𝑖subscript𝑎𝑗𝑗subscript𝑏𝑖𝑖subscript𝑏𝑗𝑗\textTr(A)\textTr(B)=d\,\textTr(A\circ B)-\sum_i=1^d-1\sum_j=i+1^% d(a_ii-a_jj)(b_ii-b_jj)Tr ( italic_A ) Tr(B) = italic_d Tr ( italic_A ∘ italic_B ) - ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_j = italic_i + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i italic_i end_POSTSUBSCRIPT - italic_a start_POSTSUBSCRIPT italic_j italic_j end_POSTSUBSCRIPT ) ( italic_b start_POSTSUBSCRIPT italic_i italic_i end_POSTSUBSCRIPT - italic_b start_POSTSUBSCRIPT italic_j italic_j end_POSTSUBSCRIPT ) (26) (in Appendix B I present an alternate proof to the one in reference Taskara and Gumus, 2013) with equation (22), finding that for any unitary U𝑈Uitalic_U



||(U∗∘U)𝐜||2=Tr((U∗ΛUT)∘(UΛU†))superscriptnormsuperscript𝑈∗𝑈𝐜2Trsuperscript𝑈∗Λsuperscript𝑈𝑇𝑈Λsuperscript𝑈†\displaystyle|\!|(U^\ast\!\circ U)\mathbfc|\!|^2=\textTr\!\left((U^% \ast\Lambda U^T)\circ(U\Lambda U^\dagger)\right)| | ( italic_U start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ∘ italic_U ) bold_c | | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = Tr ( ( italic_U start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT roman_Λ italic_U start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ) ∘ ( italic_U roman_Λ italic_U start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ) ) =1d[Tr(U∗ΛUT)Tr(UΛU†)+∑i=1d-1∑j=i+1d|(UΛU†)ii-(UΛU†)jj|2]≥Tr(Λ)2d.absent1𝑑delimited-[]Trsuperscript𝑈∗Λsuperscript𝑈𝑇Tr𝑈Λsuperscript𝑈†superscriptsubscript𝑖1𝑑1superscriptsubscript𝑗𝑖1𝑑superscriptsubscript𝑈Λsuperscript𝑈†𝑖𝑖subscript𝑈Λsuperscript𝑈†𝑗𝑗2TrsuperscriptΛ2𝑑\displaystyle=\frac1d\!\left[\textTr\!\left(U^\ast\Lambda U^T\right)% \textTr\!\left(U\Lambda U^\dagger\right)+\sum_i=1^d-1\sum_j=i+1^d% \!\big(U\Lambda U^\dagger)_ii-(U\Lambda U^\dagger)_jj\big^2% \right]\!\geq\!\frac\textTr\!\left(\Lambda\right)^2d.= divide start_ARG 1 end_ARG start_ARG italic_d end_ARG [ Tr ( italic_U start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT roman_Λ italic_U start_POSTSUPERSCRIPT italic_T end_POSTSUPERSCRIPT ) Tr ( italic_U roman_Λ italic_U start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ) + ∑ start_POSTSUBSCRIPT italic_i = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_j = italic_i + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT | ( italic_U roman_Λ italic_U start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ) start_POSTSUBSCRIPT italic_i italic_i end_POSTSUBSCRIPT - ( italic_U roman_Λ italic_U start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ) start_POSTSUBSCRIPT italic_j italic_j end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ] ≥ divide start_ARG Tr ( roman_Λ ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_d end_ARG . (27) The trace of ΛΛ\Lambdaroman_Λ is just the sum of the Schmidt coefficients, thus proving equation (25); note that as demonstrated in the previous section the bound in equation (25) is achieved by the matrix UBsubscript𝑈𝐵U_Bitalic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT given in equation (18) above.



■■\blacksquare■ (Step 1)



Step 2: OMCP≤(∑ci)2/dOMCPsuperscriptsubscript𝑐𝑖2𝑑\textOMCP\leq(\sum c_i)^2/dOMCP ≤ ( ∑ italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / italic_d



For any fixed unitary matrix UB0superscriptsubscript𝑈𝐵0U_B^0italic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT,



minUB(maxUA(||(UA∘UB)𝐜||2))≤maxUA(||(UA∘UB0)𝐜||2),subscriptsubscript𝑈𝐵subscriptsubscript𝑈𝐴superscriptnormsubscript𝑈𝐴subscript𝑈𝐵𝐜2subscriptsubscript𝑈𝐴superscriptnormsubscript𝑈𝐴superscriptsubscript𝑈𝐵0𝐜2\min_U_B\left(\max_U_A\left(|\!|(U_A\circ U_B)\mathbfc|\!|^2% \right)\right)\leq\max_U_A\left(|\!|(U_A\circ U_B^0)\mathbfc|\!|^% 2\right),roman_min start_POSTSUBSCRIPT italic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( roman_max start_POSTSUBSCRIPT italic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( | | ( italic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ∘ italic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ) bold_c | | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ) ≤ roman_max start_POSTSUBSCRIPT italic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( | | ( italic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ∘ italic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT ) bold_c | | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) , (28) so it suffices to find some UB0superscriptsubscript𝑈𝐵0U_B^0italic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT such that



||(UA∘UB0)𝐜||2≤1d(∑i=0d-1ci)2superscriptnormsubscript𝑈𝐴superscriptsubscript𝑈𝐵0𝐜21𝑑superscriptsuperscriptsubscript𝑖0𝑑1subscript𝑐𝑖2|\!|(U_A\circ U_B^0)\mathbfc|\!|^2\leq\frac1d\left(\sum_i=0^d% -1c_i\right)^2| | ( italic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ∘ italic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT ) bold_c | | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≤ divide start_ARG 1 end_ARG start_ARG italic_d end_ARG ( ∑ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT (29) for all UAsubscript𝑈𝐴U_Aitalic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT. Unsurprisingly, this is again achieved by the UBsubscript𝑈𝐵U_Bitalic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT given in equation (18), as I show now. With that choice of UB0superscriptsubscript𝑈𝐵0U_B^0italic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT, we get



||(U∘UB0)𝐜||2=1d∑jkcjck[∑lei(2π/d)l(k-j)Ulj∗Ulk].superscriptnorm𝑈superscriptsubscript𝑈𝐵0𝐜21𝑑subscript𝑗𝑘subscript𝑐𝑗subscript𝑐𝑘delimited-[]subscript𝑙superscript𝑒𝑖2𝜋𝑑𝑙𝑘𝑗superscriptsubscript𝑈𝑙𝑗∗subscript𝑈𝑙𝑘|\!|(U\circ U_B^0)\mathbfc|\!|^2=\frac1d\sum_jkc_jc_k\left[% \sum_le^i(2\pi/d)l(k-j)U_lj^\astU_lk\right].| | ( italic_U ∘ italic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT ) bold_c | | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = divide start_ARG 1 end_ARG start_ARG italic_d end_ARG ∑ start_POSTSUBSCRIPT italic_j italic_k end_POSTSUBSCRIPT italic_c start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_c start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT [ ∑ start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT italic_i ( 2 italic_π / italic_d ) italic_l ( italic_k - italic_j ) end_POSTSUPERSCRIPT italic_U start_POSTSUBSCRIPT italic_l italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_U start_POSTSUBSCRIPT italic_l italic_k end_POSTSUBSCRIPT ] . (30) The expression in square brackets can be written as an inner product between two vectors:



∑lei(2π/d)l(k-j)Ulj∗Ulksubscript𝑙superscript𝑒𝑖2𝜋𝑑𝑙𝑘𝑗superscriptsubscript𝑈𝑙𝑗∗subscript𝑈𝑙𝑘\displaystyle\sum_le^i(2\pi/d)l(k-j)U_lj^\astU_lk∑ start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT italic_e start_POSTSUPERSCRIPT italic_i ( 2 italic_π / italic_d ) italic_l ( italic_k - italic_j ) end_POSTSUPERSCRIPT italic_U start_POSTSUBSCRIPT italic_l italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT italic_U start_POSTSUBSCRIPT italic_l italic_k end_POSTSUBSCRIPT =⟨v|w⟩,absentinner-product𝑣𝑤\displaystyle=\langle v|w\rangle,= ⟨ italic_v | italic_w ⟩ , (31)



vlsubscript𝑣𝑙\displaystyle v_litalic_v start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT =ei(2π/d)ljUlj,absentsuperscript𝑒𝑖2𝜋𝑑𝑙𝑗subscript𝑈𝑙𝑗\displaystyle=e^i(2\pi/d)ljU_lj,= italic_e start_POSTSUPERSCRIPT italic_i ( 2 italic_π / italic_d ) italic_l italic_j end_POSTSUPERSCRIPT italic_U start_POSTSUBSCRIPT italic_l italic_j end_POSTSUBSCRIPT , (32)



wlsubscript𝑤𝑙\displaystyle w_litalic_w start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT =ei(2π/d)lkUlk.absentsuperscript𝑒𝑖2𝜋𝑑𝑙𝑘subscript𝑈𝑙𝑘\displaystyle=e^i(2\pi/d)lkU_lk.= italic_e start_POSTSUPERSCRIPT italic_i ( 2 italic_π / italic_d ) italic_l italic_k end_POSTSUPERSCRIPT italic_U start_POSTSUBSCRIPT italic_l italic_k end_POSTSUBSCRIPT . (33) Using the Cauchy-Schwarz inequality, this inner product satisfies |⟨v|w⟩|≤||v||||w||inner-product𝑣𝑤norm𝑣norm𝑤|\langle v|w\rangle|\leq|\!|v|\!|\,|\!|w|\!|| ⟨ italic_v | italic_w ⟩ | ≤ | | italic_v | | | | italic_w | |, and since U𝑈Uitalic_U is unitary, each row and column of U𝑈Uitalic_U is a normalized vector so that ||v||=||w||=1norm𝑣norm𝑤1|\!|v|\!|=|\!|w|\!|=1| | italic_v | | = | | italic_w | | = 1. Thus



||(U∘UB)𝐜||2≤1d∑jkcjck=1d(∑i=0d-1ci)2,superscriptnorm𝑈subscript𝑈𝐵𝐜21𝑑subscript𝑗𝑘subscript𝑐𝑗subscript𝑐𝑘1𝑑superscriptsuperscriptsubscript𝑖0𝑑1subscript𝑐𝑖2|\!|(U\circ U_B)\mathbfc|\!|^2\leq\frac1d\sum_jkc_jc_k=\frac1% d\left(\sum_i=0^d-1c_i\right)^2,| | ( italic_U ∘ italic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ) bold_c | | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≤ divide start_ARG 1 end_ARG start_ARG italic_d end_ARG ∑ start_POSTSUBSCRIPT italic_j italic_k end_POSTSUBSCRIPT italic_c start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_c start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG italic_d end_ARG ( ∑ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , (34) completing the proof of equation (29). The bound in that equation is achieved by UA=UB0∗subscript𝑈𝐴superscriptsuperscriptsubscript𝑈𝐵0∗U_A=U_B^0^\astitalic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT = italic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT. ■■\blacksquare■ (Step 2)



In combination, the two inequalities OMCP≥(∑ci)2/dOMCPsuperscriptsubscript𝑐𝑖2𝑑\textOMCP\geq(\sum c_i)^2/dOMCP ≥ ( ∑ italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / italic_d and OMCP≤(∑ci)2/dOMCPsuperscriptsubscript𝑐𝑖2𝑑\textOMCP\leq(\sum c_i)^2/dOMCP ≤ ( ∑ italic_c start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT / italic_d prove equation (11).



IV Mixed states



I now turn to the more general case of mixed states. The problem of calculating entanglement measures on mixed states is notoriously difficult, with only a few, such as negativity, being computationally tractable in general.Huang (2014) The OMCP/accord, too, is quite difficult to evaluate for general mixed states, and I do not have a general solution analogous to equation (11) for pure states. However, substantial analytical progress is still possible. In particular, I prove universal upper and lower bounds on the OMCP, I prove that it is convex, and I present exact results for several particularly important classes of mixed states, most notably all states of two qubits and all mixtures of a pure state with colorless noise; in doing so I show how the accord compares with well-known measures such as entanglement of formation, discord, and the singlet fraction.



IV.1 Upper and lower bounds



The upper bound of the OMCP is exactly 1, since it is defined as an optimized probability. This is achieved by maximally entangled pure states as demonstrated in the previous section.



The lower bound is 1/d1𝑑1/d1 / italic_d. This is the probability that A𝐴Aitalic_A and B𝐵Bitalic_B’s measurements agree when their shared state ρ𝜌\rhoitalic_ρ is completely uncorrelated. The presence of correlations should not decrease this probability if A𝐴Aitalic_A makes a good choice of basis, so it is natural to conjecture that for any ρ𝜌\rhoitalic_ρ, OMCP≥1/dOMCP1𝑑\textOMCP\geq 1/dOMCP ≥ 1 / italic_d. Furthermore, this probability of measurement coincidence should be achievable by a strategy that does not depend at all on any correlations between the two subsystems that might exist. This is indeed the case.



Proof: Let UBsubscript𝑈𝐵U_Bitalic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT be fixed. Then the probability that B𝐵Bitalic_B measures outcome i𝑖iitalic_i is P(n^B=i)=⟨i|UBρBUB†|i⟩𝑃subscript^𝑛𝐵𝑖quantum-operator-product𝑖subscript𝑈𝐵subscript𝜌𝐵superscriptsubscript𝑈𝐵†𝑖P(\hatn_B=i)=\left\langlei\right|U_B\rho_BU_B^\dagger\left|i\right\rangleitalic_P ( over^ start_ARG italic_n end_ARG start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT = italic_i ) = ⟨ italic_i | italic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT | italic_i ⟩ where ρBsubscript𝜌𝐵\rho_Bitalic_ρ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT is the reduced density matrix on B𝐵Bitalic_B given by the partial trace of ρ𝜌\rhoitalic_ρ over subsystem A𝐴Aitalic_A. Assume without loss of generality that P(n^B=0)≥P(1)≥⋯≥P(d-1)𝑃subscript^𝑛𝐵0𝑃1⋯𝑃𝑑1P(\hatn_B=0)\geq P(1)\geq\cdots\geq P(d-1)italic_P ( over^ start_ARG italic_n end_ARG start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT = 0 ) ≥ italic_P ( 1 ) ≥ ⋯ ≥ italic_P ( italic_d - 1 ).



Likewise, P(n^A=i)=⟨i|UAρAUA†|i⟩𝑃subscript^𝑛𝐴𝑖quantum-operator-product𝑖subscript𝑈𝐴subscript𝜌𝐴superscriptsubscript𝑈𝐴†𝑖P(\hatn_A=i)=\left\langlei\right|U_A\rho_AU_A^\dagger\left|i\right\rangleitalic_P ( over^ start_ARG italic_n end_ARG start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT = italic_i ) = ⟨ italic_i | italic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT italic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT | italic_i ⟩. When UAsubscript𝑈𝐴U_Aitalic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT is the identity matrix, there is some ordering of the probabilities, P(n^A=σ0)≥P(σ1)≥⋯≥P(σd-1)𝑃subscript^𝑛𝐴subscript𝜎0𝑃subscript𝜎1⋯𝑃subscript𝜎𝑑1P(\hatn_A=\sigma_0)\geq P(\sigma_1)\geq\cdots\geq P(\sigma_d-1)italic_P ( over^ start_ARG italic_n end_ARG start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT = italic_σ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) ≥ italic_P ( italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ≥ ⋯ ≥ italic_P ( italic_σ start_POSTSUBSCRIPT italic_d - 1 end_POSTSUBSCRIPT ), where σ𝜎\sigmaitalic_σ is some permutation. A𝐴Aitalic_A can then choose UAsubscript𝑈𝐴U_Aitalic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT to be the permutation matrix for σ-1superscript𝜎1\sigma^-1italic_σ start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT, in which case P(n^A=0)≥P(1)≥⋯≥P(d-1)𝑃subscript^𝑛𝐴0𝑃1⋯𝑃𝑑1P(\hatn_A=0)\geq P(1)\geq\cdots\geq P(d-1)italic_P ( over^ start_ARG italic_n end_ARG start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT = 0 ) ≥ italic_P ( 1 ) ≥ ⋯ ≥ italic_P ( italic_d - 1 ).



For that choice of UAsubscript𝑈𝐴U_Aitalic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT, the probability that A𝐴Aitalic_A and B𝐵Bitalic_B’s measurements agree is



∑i=0d-1P(n^A=i)×P(n^B=i)≥∑i=0d-1P(n^A=i)×1dsuperscriptsubscript𝑖0𝑑1𝑃subscript^𝑛𝐴𝑖𝑃subscript^𝑛𝐵𝑖superscriptsubscript𝑖0𝑑1𝑃subscript^𝑛𝐴𝑖1𝑑\sum_i=0^d-1P(\hatn_A=i)\times P(\hatn_B=i)\geq\sum_i=0^d-1P(% \hatn_A=i)\times\frac1d∑ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT italic_P ( over^ start_ARG italic_n end_ARG start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT = italic_i ) × italic_P ( over^ start_ARG italic_n end_ARG start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT = italic_i ) ≥ ∑ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT italic_P ( over^ start_ARG italic_n end_ARG start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT = italic_i ) × divide start_ARG 1 end_ARG start_ARG italic_d end_ARG (35) where the inequality follows by viewing each side as a weighted sum of the probabilities on subsystem A𝐴Aitalic_A; going from the left-hand side to the right-hand side increases the weights given to the smaller probabilities and decreases the weights given to the greater ones. The factor of 1/d1𝑑1/d1 / italic_d can then be pulled out of the sum, and the sum on probabilities for subsystem A𝐴Aitalic_A of course gives 1.



This shows that for any ρ𝜌\rhoitalic_ρ and any UBsubscript𝑈𝐵U_Bitalic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT, there exists some UAsubscript𝑈𝐴U_Aitalic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT (in fact, some permutation matrix) such that MCP≥1/dMCP1𝑑\textMCP\geq 1/dMCP ≥ 1 / italic_d. Thus the maximum over all UAsubscript𝑈𝐴U_Aitalic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT is also at least this large, and since this is true for all UBsubscript𝑈𝐵U_Bitalic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT the minimum over UBsubscript𝑈𝐵U_Bitalic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT is as well. In other words, for any state ρ𝜌\rhoitalic_ρ, OMCP≥1/dOMCP1𝑑\textOMCP\geq 1/dOMCP ≥ 1 / italic_d.



Thus 1/d≤OMCP≤11𝑑OMCP11/d\leq\textOMCP\leq 11 / italic_d ≤ OMCP ≤ 1, and hence the accord of equation (8) runs from 0 to 1 as intended.



These upper and lower bounds apply to all states; for any given state, it is possible to find tighter bounds. For example, an upper bound may be found by diagonalizing the density matrix and using convexity (proven in the next section) along with the result for pure states.



IV.2 Convexity



The OMCP is convex; in other words, for any set of normalized density matrices ρisubscript𝜌𝑖\\rho_i\ italic_ρ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT and corresponding weights pisubscript𝑝𝑖\p_i\ italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , OMCP(∑piρi)≤∑piOMCP(ρi)OMCPsubscript𝑝𝑖subscript𝜌𝑖subscript𝑝𝑖OMCPsubscript𝜌𝑖\textOMCP(\sum p_i\rho_i)\leq\sum p_i\textOMCP(\rho_i)OMCP ( ∑ italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ≤ ∑ italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT OMCP ( italic_ρ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ).



To see this, consider some fixed UBsubscript𝑈𝐵U_Bitalic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT. Then in the inner optimization in equation (7), there is some optimal UA(ρi)superscriptsubscript𝑈𝐴subscript𝜌𝑖U_A^(\rho_i)italic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT for each ρisubscript𝜌𝑖\rho_iitalic_ρ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT. Unless it is possible for all of these UAsubscript𝑈𝐴U_Aitalic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT to coincide, the optimal choice of UAsubscript𝑈𝐴U_Aitalic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT for the total density matrix will be suboptimal for one or more ρisubscript𝜌𝑖\rho_iitalic_ρ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT, so that



maxUA(MCP(∑piρi))≤∑pi×maxUA(MCP(ρi)).subscriptsubscript𝑈𝐴MCPsubscript𝑝𝑖subscript𝜌𝑖subscript𝑝𝑖subscriptsubscript𝑈𝐴MCPsubscript𝜌𝑖\max_U_A\left(\textMCP\left(\sum p_i\rho_i\right)\right)\leq\sum p_% i\times\max_U_A\big(\textMCP(\rho_i)\big).roman_max start_POSTSUBSCRIPT italic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( MCP ( ∑ italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ) ≤ ∑ italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT × roman_max start_POSTSUBSCRIPT italic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( MCP ( italic_ρ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) ) . (36) Since this is true for any fixed UBsubscript𝑈𝐵U_Bitalic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT, it is also true when the result is minimized over UBsubscript𝑈𝐵U_Bitalic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT.



IV.3 Classical states



Classical states are those for which there exists a complete set of projective measurements that leave the state invariant; these are precisely the states of the formLuo (2008a); Modi et al. (2010); Spehner (2014); Adesso et al. (2016)



ρ=∑i,j=0d-1aij|ψiA⟩⟨ψiA|⊗|ψjB⟩⟨ψjB|𝜌superscriptsubscript𝑖𝑗0𝑑1tensor-productsubscript𝑎𝑖𝑗ketsuperscriptsubscript𝜓𝑖𝐴brasuperscriptsubscript𝜓𝑖𝐴ketsuperscriptsubscript𝜓𝑗𝐵brasuperscriptsubscript𝜓𝑗𝐵\rho=\sum_i,j=0^d-1a_ij\left|\psi_i^A\right\rangle\left\langle% \psi_i^A\right|\otimes\left|\psi_j^B\right\rangle\left\langle\psi_% j^B\right|italic_ρ = ∑ start_POSTSUBSCRIPT italic_i , italic_j = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT italic_a start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT | italic_ψ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ⟩ ⟨ italic_ψ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT | ⊗ | italic_ψ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ⟩ ⟨ italic_ψ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT | (37) where ψiA⟩ketsuperscriptsubscript𝜓𝑖𝐴\left\\psi_i^A\right\rangle\right\ italic_ψ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ⟩ is some orthonormal basis of ℋAsubscriptℋ𝐴\mathcalH_Acaligraphic_H start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT and likewise for B𝐵Bitalic_B. That is, ρ𝜌\rhoitalic_ρ is diagonal in a basis of orthogonal separable states. In this case, the OMCP is exactly 1/d1𝑑1/d1 / italic_d, corresponding to random chance and a total lack of correlation.



Proof: Substituting ρ𝜌\rhoitalic_ρ into the MCP from equation (7), the MCP factorizes for each term in the sum on i𝑖iitalic_i and j𝑗jitalic_j:



MCP=∑i,j=0d-1aij∑n|⟨n|UA|ψiA⟩|2×|⟨n|UB|ψjB⟩|2.MCPsuperscriptsubscript𝑖𝑗0𝑑1subscript𝑎𝑖𝑗subscript𝑛superscriptquantum-operator-product𝑛subscript𝑈𝐴superscriptsubscript𝜓𝑖𝐴2superscriptquantum-operator-product𝑛subscript𝑈𝐵superscriptsubscript𝜓𝑗𝐵2\textMCP=\sum_i,j=0^d-1a_ij\sum_n\big\left\langlen\right|U_A% \left|\psi_i^A\right\rangle\big^2\times\big\left\langlen\right% |U_B\left|\psi_j^B\right\rangle\big^2.MCP = ∑ start_POSTSUBSCRIPT italic_i , italic_j = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT italic_a start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | ⟨ italic_n | italic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT | italic_ψ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ⟩ | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT × | ⟨ italic_n | italic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT | italic_ψ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ⟩ | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT . (38)



Because the states |ψiA⟩ketsuperscriptsubscript𝜓𝑖𝐴\left|\psi_i^A\right\rangle| italic_ψ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ⟩ form an orthonormal basis for ℋAsubscriptℋ𝐴\mathcalH_Acaligraphic_H start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT, there exists a change of basis matrix U~Asubscript~𝑈𝐴\tildeU_Aover~ start_ARG italic_U end_ARG start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT such that |ψiA⟩=U~A|i⟩Aketsuperscriptsubscript𝜓𝑖𝐴subscript~𝑈𝐴subscriptket𝑖𝐴\left|\psi_i^A\right\rangle=\tildeU_A\left|i\right\rangle_A| italic_ψ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ⟩ = over~ start_ARG italic_U end_ARG start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT | italic_i ⟩ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT for all i𝑖iitalic_i, where |i⟩Asubscriptket𝑖𝐴\left|i\right\rangle_A| italic_i ⟩ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT is an element of the same standard basis as |n⟩Asubscriptket𝑛𝐴\left|n\right\rangle_A| italic_n ⟩ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT. Since UAsubscript𝑈𝐴U_Aitalic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT is optimized over, UAU~Asubscript𝑈𝐴subscript~𝑈𝐴U_A\tildeU_Aitalic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT over~ start_ARG italic_U end_ARG start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT can be renamed to UAsubscript𝑈𝐴U_Aitalic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT; in other words, we can assume without loss of generality that |ψiA⟩=|i⟩Aketsuperscriptsubscript𝜓𝑖𝐴subscriptket𝑖𝐴\left|\psi_i^A\right\rangle=\left|i\right\rangle_A| italic_ψ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ⟩ = | italic_i ⟩ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT, in which case the expression ⟨n|UA|ψiA⟩quantum-operator-product𝑛subscript𝑈𝐴superscriptsubscript𝜓𝑖𝐴\left\langlen\right|U_A\left|\psi_i^A\right\rangle⟨ italic_n | italic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT | italic_ψ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ⟩ is nothing but the matrix element UniAsubscriptsuperscript𝑈𝐴𝑛𝑖U^A_niitalic_U start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n italic_i end_POSTSUBSCRIPT. Making this substitution and following the same steps for B𝐵Bitalic_B, the OMCP becomes



minUB(maxUA(∑i,jaij∑n|UniA|2×|UnjB|2)).subscriptsubscript𝑈𝐵subscriptsubscript𝑈𝐴subscript𝑖𝑗subscript𝑎𝑖𝑗subscript𝑛superscriptsubscriptsuperscript𝑈𝐴𝑛𝑖2superscriptsubscriptsuperscript𝑈𝐵𝑛𝑗2\min_U_B\left(\max_U_A\left(\sum_i,ja_ij\sum_n\bigU^A_ni% \big^2\times\bigU^B_nj\big^2\right)\right).roman_min start_POSTSUBSCRIPT italic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( roman_max start_POSTSUBSCRIPT italic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( ∑ start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | italic_U start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n italic_i end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT × | italic_U start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n italic_j end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ) . (39) Now suppose that UBsubscript𝑈𝐵U_Bitalic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT is any unitary for which all elements are equal in magnitude, |UnjB|2=1/dsuperscriptsubscriptsuperscript𝑈𝐵𝑛𝑗21𝑑\bigU^B_nj\big^2=1/d| italic_U start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n italic_j end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = 1 / italic_d. Then the expression to optimize is just



1d∑i,jaij∑n|UniA|2,1𝑑subscript𝑖𝑗subscript𝑎𝑖𝑗subscript𝑛superscriptsubscriptsuperscript𝑈𝐴𝑛𝑖2\frac1d\sum_i,ja_ij\sum_n\bigU^A_ni\big^2,divide start_ARG 1 end_ARG start_ARG italic_d end_ARG ∑ start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | italic_U start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n italic_i end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , (40) and the inner sum is exactly the norm of the n𝑛nitalic_nth row of UAsubscript𝑈𝐴U_Aitalic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT, which is 1. This leaves (∑aij)/dsubscript𝑎𝑖𝑗𝑑(\sum a_ij)/d( ∑ italic_a start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ) / italic_d, and because ρ𝜌\rhoitalic_ρ is a normalized density matrix the sum on a𝑎aitalic_a is 1 as well. In other words, there exists a UBsubscript𝑈𝐵U_Bitalic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT for which



maxUA(∑i,jaij∑n|UniA|2×|UniB|2)=1dsubscriptsubscript𝑈𝐴subscript𝑖𝑗subscript𝑎𝑖𝑗subscript𝑛superscriptsubscriptsuperscript𝑈𝐴𝑛𝑖2superscriptsubscriptsuperscript𝑈𝐵𝑛𝑖21𝑑\max_U_A\left(\sum_i,ja_ij\sum_n\bigU^A_ni\big^2\times% \bigU^B_ni\big^2\right)=\frac1droman_max start_POSTSUBSCRIPT italic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( ∑ start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT italic_a start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | italic_U start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n italic_i end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT × | italic_U start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n italic_i end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) = divide start_ARG 1 end_ARG start_ARG italic_d end_ARG (41) and thus the minimum over UBsubscript𝑈𝐵U_Bitalic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT is no larger than this. So OMCP≤1/dOMCP1𝑑\textOMCP\leq 1/dOMCP ≤ 1 / italic_d.



The opposite inequality, that OMCP≥1/dOMCP1𝑑\textOMCP\geq 1/dOMCP ≥ 1 / italic_d, has already been shown above to hold for any state ρ𝜌\rhoitalic_ρ. The two inequalities, taken together, prove that OMCP=1/dOMCP1𝑑\textOMCP=1/dOMCP = 1 / italic_d as claimed.



■■\blacksquare■



To summarize, the OMCP achieves its minimum possible value, 1/d1𝑑1/d1 / italic_d, when ρ𝜌\rhoitalic_ρ is diagonal in a basis of orthogonal separable states. One might think that this implies that the OMCP achieves its minimum value on all separable states, but that is not the case, as I show below.



IV.4 Pure states with colorless noise



One particularly experimentally relevant class of mixed states consists of pure states mixed with colorless noise; the latter is represented by the maximally mixed state, 𝟙/d21superscript𝑑2\mathbbm1/d^2blackboard_1 / italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT for 𝟙1\mathbbm1blackboard_1 the d2×d2superscript𝑑2superscript𝑑2d^2\times d^2italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT × italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT identity matrix. Such a state is written as



ρ=x|ψ⟩⟨ψ|+(1-x)𝟙d2𝜌𝑥ket𝜓bra𝜓1𝑥1superscript𝑑2\rho=x\left|\psi\right\rangle\left\langle\psi\right|+(1-x)\frac\mathbbm1% d^2italic_ρ = italic_x | italic_ψ ⟩ ⟨ italic_ψ | + ( 1 - italic_x ) divide start_ARG blackboard_1 end_ARG start_ARG italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG (42) where I assume x∈[0,1]𝑥01x\in[0,1]italic_x ∈ [ 0 , 1 ]. The evaluation of the OMCP is actually quite easy: since the second term is invariant under conjugation by any unitary transformation, the OMCP becomes



x×minUB(maxUA(∑n⟨n,n|(UA⊗UB)|ψ⟩⟨ψ|(UA⊗UB)†|n,n⟩))+(1-x)d=x×OMCP(|ψ⟩)+(1-x)d.𝑥subscriptsubscript𝑈𝐵subscriptsubscript𝑈𝐴subscript𝑛quantum-operator-product𝑛𝑛tensor-productsubscript𝑈𝐴subscript𝑈𝐵𝜓quantum-operator-product𝜓superscripttensor-productsubscript𝑈𝐴subscript𝑈𝐵†𝑛𝑛1𝑥𝑑𝑥OMCPket𝜓1𝑥𝑑x\times\min_U_B\!\left(\!\max_U_A\!\left(\!\sum_n\left\langlen,n% \right|(U_A\otimes U_B)\,\left|\psi\right\rangle\left\langle\psi\right% |\,(U_A\otimes U_B)^\dagger\left|n,n\right\rangle\right)\!\!\right)+% \frac(1-x)d=x\times\textOMCP\left(\left|\psi\right\rangle\right)+\frac% (1-x)d.italic_x × roman_min start_POSTSUBSCRIPT italic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( roman_max start_POSTSUBSCRIPT italic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( ∑ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ⟨ italic_n , italic_n | ( italic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⊗ italic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ) | italic_ψ ⟩ ⟨ italic_ψ | ( italic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⊗ italic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT | italic_n , italic_n ⟩ ) ) + divide start_ARG ( 1 - italic_x ) end_ARG start_ARG italic_d end_ARG = italic_x × OMCP ( | italic_ψ ⟩ ) + divide start_ARG ( 1 - italic_x ) end_ARG start_ARG italic_d end_ARG . (43) The OMCP for |ψ⟩ket𝜓\left|\psi\right\rangle| italic_ψ ⟩ is just the pure state result as computed in section III.



IV.5 Isotropic states



Isotropic states are those that are invariant under conjugation by any transformation of the form U⊗U∗tensor-product𝑈superscript𝑈∗U\otimes U^\astitalic_U ⊗ italic_U start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT.Horodecki and Horodecki (1999); Terhal and Horodecki (2000) These are states of the form



ρ=p|Φ+⟩⟨Φ+|+(1-p)𝟙-|Φ+⟩⟨Φ+|d2-1𝜌𝑝ketsuperscriptΦbrasuperscriptΦ1𝑝1ketsuperscriptΦbrasuperscriptΦsuperscript𝑑21\rho=p\left|\Phi^+\right\rangle\left\langle\Phi^+\right|+(1-p)\frac% \mathbbm1-\leftd^2-1italic_ρ = italic_p | roman_Φ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ⟩ ⟨ roman_Φ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT | + ( 1 - italic_p ) divide start_ARG blackboard_1 - | roman_Φ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ⟩ ⟨ roman_Φ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT | end_ARG start_ARG italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 1 end_ARG (44) where p∈[0,1]𝑝01p\in[0,1]italic_p ∈ [ 0 , 1 ] and |Φ+⟩ketsuperscriptΦ\left|\Phi^+\right\rangle| roman_Φ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ⟩ is the maximally entangled state



|Φ+⟩=1d∑n=0d-1|nn⟩.ketsuperscriptΦ1𝑑superscriptsubscript𝑛0𝑑1ket𝑛𝑛\left|\Phi^+\right\rangle=\frac1\sqrtd\sum_n=0^d-1\left|nn% \right\rangle.| roman_Φ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ⟩ = divide start_ARG 1 end_ARG start_ARG square-root start_ARG italic_d end_ARG end_ARG ∑ start_POSTSUBSCRIPT italic_n = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT | italic_n italic_n ⟩ . (45) The isotropic states are notable because they allow for substantial analytical progress in calculating the entanglement of formation for any d𝑑ditalic_d.Terhal and Vollbrecht (2000)



For p≥1/d2𝑝1superscript𝑑2p\geq 1/d^2italic_p ≥ 1 / italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, this is in the form of a pure state plus colorless noise, as discussed in the previous section, with x=(p-1/d2)/(1-1/d2)𝑥𝑝1superscript𝑑211superscript𝑑2x=(p-1/d^2)/(1-1/d^2)italic_x = ( italic_p - 1 / italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) / ( 1 - 1 / italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ). The OMCP for a maximally entangled state is 1, so the OMCP on the isotropic state with p≥1/d2𝑝1superscript𝑑2p\geq 1/d^2italic_p ≥ 1 / italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT is 1/d+(p-1/d2)(1-1/d)/(1-1/d2)1𝑑𝑝1superscript𝑑211𝑑11superscript𝑑21/d+(p-1/d^2)(1-1/d)/(1-1/d^2)1 / italic_d + ( italic_p - 1 / italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ( 1 - 1 / italic_d ) / ( 1 - 1 / italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ).



The case of p<1/d2𝑝1superscript𝑑2p<1/d^2italic_p <1 / italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT must be treated separately. Consider the case of x<0𝑥0x<0italic_x <0 in equation (42). When calculating the OMCP, the result is the same as in equation (43) except that because x𝑥xitalic_x is negative, when it is pulled out of the optimizations the minimization over UBsubscript𝑈𝐵U_Bitalic_U start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT becomes a maximization and the maximization over UAsubscript𝑈𝐴U_Aitalic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT becomes as minimization. The result, in the case that |ψ⟩=|Φ+⟩ket𝜓ketsuperscriptΦ\left|\psi\right\rangle=\left|\Phi^+\right\rangle| italic_ψ ⟩ = | roman_Φ start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ⟩, is that the first term is exactly 0.



To see this, first note that an OMCP of 1 means that no matter what basis B𝐵Bitalic_B selects for his measurement, A𝐴Aitalic_A can always select a basis to guarantee that their measurements agree. Instead of selecting this UAsubscript𝑈𝐴U_Aitalic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT, A𝐴Aitalic_A first applies this UAsubscript𝑈𝐴U_Aitalic_U start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT and then the permutation matrix sending |0⟩A↦|1⟩A↦⋯↦|d-1⟩A↦|0⟩Amaps-tosubscriptket0𝐴subscriptket1𝐴maps-to⋯maps-tosubscriptket𝑑1𝐴maps-tosubscriptket0𝐴\left|0\right\rangle_A\mapsto\left|1\right\rangle_A\mapsto\cdots% \mapsto\left|d-1\right\rangle_A\mapsto\left|0\right\rangle_A| 0 ⟩ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ↦ | 1 ⟩ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ↦ ⋯ ↦ | italic_d - 1 ⟩ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ↦ | 0 ⟩ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT, thus guaranteeing that her measurement will never agree with B𝐵Bitalic_B’s.



Thus in the case of p<1/d2𝑝1superscript𝑑2p<1/d^2italic_p <1 / italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, the result is just (1-x)/d1𝑥𝑑(1-x)/d( 1 - italic_x ) / italic_d or 1/d+(1/d2-p)/(d-1/d)1𝑑1superscript𝑑2𝑝𝑑1𝑑1/d+(1/d^2-p)/(d-1/d)1 / italic_d + ( 1 / italic_d start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_p ) / ( italic_d - 1 / italic_d ). Putting both cases together, the result is



OMCP=1d+|p-1d2|×p-\frac1d^2\right discord server

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