Quantum discord expresses a fundamental non-classicality of correlations more general than quantum entanglement. We combine the no-local-broadcasting theorem, semidefinite-programming characterizations of quantum fidelity and quantum separability, and a recent breakthrough result of Fawzi and Renner about quantum Markov chains to provide a hierarchy of computationally efficient lower bounds to quantum discord. Such a hierarchy converges to the surprisal of measurement recoverability introduced by Seshadreesan and Wilde, and provides a faithful lower bound to quantum discord already at the lowest non-trivial level. Furthermore, the latter constitutes by itself a valid discord-like measure of the quantumness of correlations.
Introduction.-Correlations in quantum mechanics exhibit non-classical features that include non-locality Brunner et al. (2014), steering Wiseman et al. (2007), entanglement Horodecki et al. (2009), and quantum discord Modi et al. (2012). Quantum correlations play a fundamental role in quantum information processing and quantum technologies Nielsen and Chuang (2010), which go from quantum cryptography Gisin et al. (2002) to quantum metrology Giovannetti et al. (2011). While both non-locality and steering are manifestations of entanglement, quantum discord is a more general form of quantumness of correlations that includes entanglement but goes beyond it. In particular, almost all distributed states exhibit discord Ferraro et al. (2010). This fact calls for fully elevating the study of quantum discord to the quantitative level, since just certifying that discord is present may be considered of limited interest. While several approaches to the quantification of discord have been already proposed (see, e.g. Modi et al. (2012); Luo (2008); Modi et al. (2010); Dakić et al. (2010); Luo and Fu (2010); Streltsov et al. (2011); Piani et al. (2011); Girolami and Adesso (2012); Piani and Adesso (2012); Paula et al. (2013); Chang and Luo (2013); Girolami et al. (2014); Piani et al. (2014); Seshadreesan et al. (2014); Seshadreesan and Wilde (2014) and references therein), in this paper we significantly push forward a meaningful, reliable, and practical quantitative approach to the study of quantum discord that is based on fundamental quantum features of quantum correlations, and at the same time is computationally friendly.
Quantum discord was introduced in terms of the minimum amount of correlations, as quantified by mutual information, that is necessarily lost in a local quantum measurement of a bipartite quantum state Ollivier and Zurek (2001); Henderson and Vedral (2001) (see below for exact definitions). It is then clear that it is relatively easy to find upper bounds to quantum discord: the loss of correlations due to any measurement provides some upper bound. Nonetheless, standard quantum discord is not easily computed even in simple cases, and general easily computable lower bounds to it are similarly not known. In this paper we provide a family of lower bounds for the standard quantum discord which can reliably be computed numerically. On the other hand, they have each physical meaning, since they are based on ‘impossibility features’ (i.e., no-go theorems) related to the local manipulation of quantum correlations. Furthermore, such lower bounds satisfy the basic requests that should be imposed on any meaningful measure of quantum correlations Brodutch and Modi (2012); Piani (2012), hence making each quantifier in the hierarchy a valid discord-like quantifier in itself.
One ‘impossibility feature’ associated to quantum discord relates to local broadcasting Piani et al. (2008); Luo and Sun (2010), which can be seen as a generalization of broadcasting Barnum et al. (1996), itself a generalization of cloning Wootters and Zurek (1982): correlations that exhibit quantum discord cannot be freely locally redistributed or shared, and indeed, discord can be exactly interpreted as the asymptotic loss in correlations necessarily associated with such an attempt Streltsov and Zurek (2013); Brandao et al. (2013). A very related ‘impossibility feature’ of discord deals with the ‘local relocation’ of quantum states by classical means, that is, roughly speaking, with the transmission (equivalently, storing) of the quantum information contained in quantum subsystems via classical communication (a classical memory). Indeed, it can be checked through a powerful result by Petz Petz (1986, 1988); Hayden et al. (2004) that the ability to perfectly locally broadcast (equivalently, to perfectly store by classical means) distributed quantum states reduces to the ability to perfectly locally broadcast or classically store correlations, as measured by the quantum mutual information Hayashi (2006); Piani et al. (2008); Luo and Sun (2010), a feat possible-by definition-only in absence of discord. The relation between the above two ‘impossibility features’ is due to the fact that quantum information becomes classical when broadcast to many parties Bae and Acín (2006); Chiribella and DÕAriano (2006); Chiribella (2011); Brandao et al. (2013).
The consideration of the general, non-perfect (for states exhibiting discord) case of the classical transmission/storing of an arbitrary quantum state has recently received renewed attention also thanks to a breakthrough result of Fawzi and Renner Fawzi and Renner (2014) (see also Brandao et al. (2014)) that generalizes the result by Petz. In Seshadreesan and Wilde (2014), Seshadreesan and Wilde explicitly suggested to approach the study of the general quantumness of correlations, and in particular their quantification, in terms of how well distributed quantum states can be locally transmitted or stored by classical means. They introduced a discord-like quantifier, the surprisal of measurement recoverability, which, thanks to the results of Fawzi and Renner (2014), directly translates into a lower bound to the standard quantum discord. Unfortunately, the surprisal of measurement recoverability is in general not easily computable either. In this paper, by considering how well a quantum state can be locally broadcast, we generalize the surprisal of measurement recoverability in such a way to obtain numerically computable (upper and) lower bounds to it, which provably converge to it. Thus, we also obtain computable lower bounds to the standard quantum discord. The hierarchy of lower bounds that we introduce exploits ideas used in the characterization and detection of entanglement via semidefinite programming Doherty et al. (2002, 2004); Doherty (2014). Semidefinite programming optimization techniques Boyd and Vandenberghe (2009) have found many other significant applications in quantum information (see, e.g., Nowakowski and Horodecki (2009); Jain et al. (2011); Kempe et al. (2010); Navascués et al. (2007); Watrous (2009); Johnston and Kribs (2010); Eisert et al. (2007)), and, in recent times, they have been used also in the quantification of steering Skrzypczyk et al. (2014); Piani and Watrous (2014). Here we extend the use of semidefinite programming for the study of quantum correlations to quantum discord.
Preliminaries-We will consider finite-dimensional systems, so that a quantum state corresponds to a d𝑑ditalic_d-dimensional positive semidefinite density matrix ρ𝜌\rhoitalic_ρ which lives in the space L(ℋ)𝐿ℋL(\mathcalH)italic_L ( caligraphic_H ) of linear operators on a Hilbert space ℋ≃ℂdsimilar-to-or-equalsℋsuperscriptℂ𝑑\mathcalH\simeq\mathbbC^dcaligraphic_H ≃ roman_ℂ start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT. The von Neumann entropy associated with ρ𝜌\rhoitalic_ρ is given by S(ρ)=-Tr(ρlogρ)𝑆𝜌Tr𝜌𝜌S(\rho)=-\operatornameTr(\rho\log\rho)italic_S ( italic_ρ ) = - roman_Tr ( italic_ρ roman_log italic_ρ ). We will indicate by Tr\XsubscriptTr\absent𝑋\operatornameTr_\backslash Xroman_Tr start_POSTSUBSCRIPT \ italic_X end_POSTSUBSCRIPT a trace performed over every other system except X𝑋Xitalic_X. In the case we consider a bi- or multi-partite system, with global state ρ𝜌\rhoitalic_ρ, we denote S(X)ρ=S(ρX)𝑆subscript𝑋𝜌𝑆subscript𝜌𝑋S(X)_\rho=S(\rho_X)italic_S ( italic_X ) start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT = italic_S ( italic_ρ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ), where L(ℋX)∋ρX=Tr\X(ρ)contains𝐿subscriptℋ𝑋subscript𝜌𝑋subscriptTr\absent𝑋𝜌L(\mathcalH_X) i\rho_X=\operatornameTr_\backslash X(\rho)italic_L ( caligraphic_H start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ) ∋ italic_ρ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT = roman_Tr start_POSTSUBSCRIPT \ italic_X end_POSTSUBSCRIPT ( italic_ρ ) is the reduced state of system X𝑋Xitalic_X. The fidelity F(σ,ρ)=Trρσρ𝐹𝜎𝜌Tr𝜌𝜎𝜌F(\sigma,\rho)=\operatornameTr\sqrt\sqrt\rho\sigma\sqrt\rhoitalic_F ( italic_σ , italic_ρ ) = roman_Tr square-root start_ARG square-root start_ARG italic_ρ end_ARG italic_σ square-root start_ARG italic_ρ end_ARG end_ARG is a measure of how similar two states ρ𝜌\rhoitalic_ρ and σ𝜎\sigmaitalic_σ are Nielsen and Chuang (2010). It holds 0≤F(σ,ρ)≤10𝐹𝜎𝜌10\leq F(\sigma,\rho)\leq 10 ≤ italic_F ( italic_σ , italic_ρ ) ≤ 1, with F(σ,ρ)=1𝐹𝜎𝜌1F(\sigma,\rho)=1italic_F ( italic_σ , italic_ρ ) = 1 if and only if ρ=σ𝜌𝜎\rho=\sigmaitalic_ρ = italic_σ. We will need the fact that the fidelity can be seen as the solution to the semidefinite programming (SDP) optimization problem Killoran (2012); Watrous (2012)
maximize 12(Tr(X)+Tr(X†))12Tr𝑋Trsuperscript𝑋†\displaystyle\frac12(\operatornameTr(X)+\operatornameTr(X^\dagger))divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( roman_Tr ( italic_X ) + roman_Tr ( italic_X start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ) ) (1a)
subject to (ρXX†σ)≥0.matrix𝜌𝑋superscript𝑋†𝜎0\displaystyle\beginpmatrix\rho&X\\ X^\dagger&\sigma\endpmatrix\geq 0.( start_ARG start_ROW start_CELL italic_ρ end_CELL start_CELL italic_X end_CELL end_ROW start_ROW start_CELL italic_X start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT end_CELL start_CELL italic_σ end_CELL end_ROW end_ARG ) ≥ 0 . (1d)
Another measure of similarity of states is the trace distance Δ(σ,ρ)=12∥σ-ρ∥1Δ𝜎𝜌12subscriptnorm𝜎𝜌1\Delta(\sigma,\rho)=\frac12\|\sigma-\rho\|_1roman_Δ ( italic_σ , italic_ρ ) = divide start_ARG 1 end_ARG start_ARG 2 end_ARG ∥ italic_σ - italic_ρ ∥ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT where ∥ξ∥1=Tr(ξ†ξ)subscriptnorm𝜉1Trsuperscript𝜉†𝜉\|\xi\|_1=\operatornameTr(\sqrt\xi^\dagger\xi)∥ italic_ξ ∥ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = roman_Tr ( square-root start_ARG italic_ξ start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT italic_ξ end_ARG ) is the trace norm Nielsen and Chuang (2010). It holds 0≤Δ(σ,ρ)≤10Δ𝜎𝜌10\leq\Delta(\sigma,\rho)\leq 10 ≤ roman_Δ ( italic_σ , italic_ρ ) ≤ 1, and 1-F(σ,ρ)≤Δ(σ,ρ)≤1-F2(σ,ρ)1𝐹𝜎𝜌Δ𝜎𝜌1superscript𝐹2𝜎𝜌1-F(\sigma,\rho)\leq\Delta(\sigma,\rho)\leq\sqrt1-F^2(\sigma,\rho)1 - italic_F ( italic_σ , italic_ρ ) ≤ roman_Δ ( italic_σ , italic_ρ ) ≤ square-root start_ARG 1 - italic_F start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_σ , italic_ρ ) end_ARG Fuchs and Van De Graaf (1999). Transformations of physical systems are described by completely positive and trace-preserving linear maps, also called channels, from L(ℋin)𝐿subscriptℋinL(\mathcalH_\textrmin)italic_L ( caligraphic_H start_POSTSUBSCRIPT in end_POSTSUBSCRIPT ) to L(ℋout)𝐿subscriptℋoutL(\mathcalH_\textrmout)italic_L ( caligraphic_H start_POSTSUBSCRIPT out end_POSTSUBSCRIPT ), where ℋinsubscriptℋin\mathcalH_\textrmincaligraphic_H start_POSTSUBSCRIPT in end_POSTSUBSCRIPT and ℋoutsubscriptℋout\mathcalH_\textrmoutcaligraphic_H start_POSTSUBSCRIPT out end_POSTSUBSCRIPT are the input and output spaces, respectively Nielsen and Chuang (2010).
Separability and symmetric extensions.-A bipartite state ρABsubscript𝜌𝐴𝐵\rho_ABitalic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT is separable (or unentangled) if it admits the decomposition ρABsep=∑bpb|αb⟩⟨αb|A⊗|βb⟩⟨βb|Bsuperscriptsubscript𝜌𝐴𝐵sepsubscript𝑏tensor-productsubscript𝑝𝑏ketsubscript𝛼𝑏subscriptbrasubscript𝛼𝑏𝐴ketsubscript𝛽𝑏subscriptbrasubscript𝛽𝑏𝐵\rho_AB^\textrmsep=\sum_bp_b|\alpha_b\rangle\langle\alpha_b|_A% \otimes|\beta_b\rangle\langle\beta_b|_Bitalic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT sep end_POSTSUPERSCRIPT = ∑ start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT | italic_α start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ⟩ ⟨ italic_α start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT | start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⊗ | italic_β start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ⟩ ⟨ italic_β start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT | start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT, for pbsubscript𝑝𝑏\p_b\ italic_p start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT a probability distribution, and |αb⟩Asubscriptketsubscript𝛼𝑏𝐴|\alpha_b\rangle_A| italic_α start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ⟩ start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT and |βb⟩Bsubscriptketsubscript𝛽𝑏𝐵|\beta_b\rangle_B| italic_β start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT ⟩ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT (not necessarily orthogonal) vector states for A𝐴Aitalic_A and B𝐵Bitalic_B, respectively. A bipartite state that is not separable is entangled Werner (1989).
Consider systems B1≃B2≃Bsimilar-to-or-equalssubscript𝐵1subscript𝐵2similar-to-or-equals𝐵B_1\simeq B_2\simeq Bitalic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≃ italic_B start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ≃ italic_B. A state ρAB1B2subscript𝜌𝐴subscript𝐵1subscript𝐵2\rho_AB_1B_2italic_ρ start_POSTSUBSCRIPT italic_A italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT such that ρAB1=ρAB2=ρABsubscript𝜌𝐴subscript𝐵1subscript𝜌𝐴subscript𝐵2subscript𝜌𝐴𝐵\rho_AB_1=\rho_AB_2=\rho_ABitalic_ρ start_POSTSUBSCRIPT italic_A italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT = italic_ρ start_POSTSUBSCRIPT italic_A italic_B start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT = italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT, and such that ρAB1B2=VB1B2ρAB1B2VB1B2†subscript𝜌𝐴subscript𝐵1subscript𝐵2subscript𝑉subscript𝐵1subscript𝐵2subscript𝜌𝐴subscript𝐵1subscript𝐵2superscriptsubscript𝑉subscript𝐵1subscript𝐵2†\rho_AB_1B_2=V_B_1B_2\rho_AB_1B_2V_B_1B_2^\daggeritalic_ρ start_POSTSUBSCRIPT italic_A italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT = italic_V start_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_A italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_V start_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT, with VB1B2subscript𝑉subscript𝐵1subscript𝐵2V_B_1B_2italic_V start_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT the swap operator, VB1B2|β⟩B1|β′⟩B2=|β′⟩B1|β⟩B2subscript𝑉subscript𝐵1subscript𝐵2subscriptket𝛽subscript𝐵1subscriptketsuperscript𝛽′subscript𝐵2subscriptketsuperscript𝛽′subscript𝐵1subscriptket𝛽subscript𝐵2V_B_1B_2|\beta\rangle_B_1|\beta^\prime\rangle_B_2=|\beta^% \prime\rangle_B_1|\beta\rangle_B_2italic_V start_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT | italic_β ⟩ start_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT | italic_β start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⟩ start_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT = | italic_β start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ⟩ start_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT | italic_β ⟩ start_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT, is called a (two-)symmetric extension (on B𝐵Bitalic_B) of ρABsubscript𝜌𝐴𝐵\rho_ABitalic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT. If the stronger condition ρAB1B2=ΠB1B2+ρAB1B2ΠB1B2+subscript𝜌𝐴subscript𝐵1subscript𝐵2subscriptsuperscriptΠsubscript𝐵1subscript𝐵2subscript𝜌𝐴subscript𝐵1subscript𝐵2subscriptsuperscriptΠsubscript𝐵1subscript𝐵2\rho_AB_1B_2=\Pi^+_B_1B_2\rho_AB_1B_2\Pi^+_B_1B_2italic_ρ start_POSTSUBSCRIPT italic_A italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT = roman_Π start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_A italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT roman_Π start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT holds, with ΠB1B2+subscriptsuperscriptΠsubscript𝐵1subscript𝐵2\Pi^+_B_1B_2roman_Π start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT the projector onto the symmetric subspace of B1B2subscript𝐵1subscript𝐵2B_1B_2italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, we call ρAB1B2subscript𝜌𝐴subscript𝐵1subscript𝐵2\rho_AB_1B_2italic_ρ start_POSTSUBSCRIPT italic_A italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT a (two-)Bose-symmetric extension (on B𝐵Bitalic_B). The concept can be generalized to k𝑘kitalic_k extensions. Let Bk=B1B2…Bksuperscript𝐵𝑘subscript𝐵1subscript𝐵2…subscript𝐵𝑘B^k=B_1B_2\ldots B_kitalic_B start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT = italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT … italic_B start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT. We say that ρABksubscript𝜌𝐴superscript𝐵𝑘\rho_AB^kitalic_ρ start_POSTSUBSCRIPT italic_A italic_B start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_POSTSUBSCRIPT is a k𝑘kitalic_k-symmetric extension of ρABsubscript𝜌𝐴𝐵\rho_ABitalic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT (on B𝐵Bitalic_B) if: (i) ρABi=Tr\ABi(ρABk)=ρABsubscript𝜌𝐴subscript𝐵𝑖subscriptTr\absent𝐴subscript𝐵𝑖subscript𝜌𝐴superscript𝐵𝑘subscript𝜌𝐴𝐵\rho_AB_i=\operatornameTr_\backslashAB_i(\rho_AB^k)=\rho_ABitalic_ρ start_POSTSUBSCRIPT italic_A italic_B start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT = roman_Tr start_POSTSUBSCRIPT \ italic_A italic_B start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_A italic_B start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) = italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT, for all i=1,…,k𝑖1…𝑘i=1,\dots,kitalic_i = 1 , … , italic_k; (ii) ρABk=VρABkV†subscript𝜌𝐴superscript𝐵𝑘𝑉subscript𝜌𝐴superscript𝐵𝑘superscript𝑉†\rho_AB^k=V\rho_AB^kV^\daggeritalic_ρ start_POSTSUBSCRIPT italic_A italic_B start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = italic_V italic_ρ start_POSTSUBSCRIPT italic_A italic_B start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_V start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT for any unitary V𝑉Vitalic_V that permutes the Bksuperscript𝐵𝑘B^kitalic_B start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT systems. Note that, because of the symmetry (ii), in (i) it is enough to consider the trace over all systems Bisubscript𝐵𝑖B_iitalic_B start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT except an arbitrary one, e.g., B1subscript𝐵1B_1italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT. If the stronger condition (ii’) ρABk=ΠBk+ρABkΠBk+subscript𝜌𝐴superscript𝐵𝑘subscriptsuperscriptΠsuperscript𝐵𝑘subscript𝜌𝐴superscript𝐵𝑘subscriptsuperscriptΠsuperscript𝐵𝑘\rho_AB^k=\Pi^+_B^k\rho_AB^k\Pi^+_B^kitalic_ρ start_POSTSUBSCRIPT italic_A italic_B start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = roman_Π start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_A italic_B start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_POSTSUBSCRIPT roman_Π start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_POSTSUBSCRIPT, with ΠBk+subscriptsuperscriptΠsuperscript𝐵𝑘\Pi^+_B^kroman_Π start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_POSTSUBSCRIPT the projector onto the fully symmetric subspace B+ksubscriptsuperscript𝐵𝑘B^k_+italic_B start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT of Bksuperscript𝐵𝑘B^kitalic_B start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT, holds, we say that ρABksubscript𝜌𝐴superscript𝐵𝑘\rho_AB^kitalic_ρ start_POSTSUBSCRIPT italic_A italic_B start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_POSTSUBSCRIPT is a k𝑘kitalic_k-Bose-symmetric extension of ρABsubscript𝜌𝐴𝐵\rho_ABitalic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT (on B𝐵Bitalic_B). Only separable states admit k𝑘kitalic_k-symmetric extensions for all k𝑘kitalic_k Fannes et al. (1988); Raggio and Werner (1989); Doherty (2014).
No local broadcasting.-The no-local-broadcasting theorem Piani et al. (2008); Luo and Sun (2010) states that there exists a broadcasting channel ΛB→B1B2subscriptΛ→𝐵subscript𝐵1subscript𝐵2\Lambda_B\rightarrow B_1B_2roman_Λ start_POSTSUBSCRIPT italic_B → italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT such that TrB1(ΛB→B1B2[ρAB])=TrB2(ΛB→B1B2[ρAB])=ρABsubscriptTrsubscript𝐵1subscriptΛ→𝐵subscript𝐵1subscript𝐵2delimited-[]superscript𝜌𝐴𝐵subscriptTrsubscript𝐵2subscriptΛ→𝐵subscript𝐵1subscript𝐵2delimited-[]superscript𝜌𝐴𝐵superscript𝜌𝐴𝐵\operatornameTr_B_1(\Lambda_B\rightarrow B_1B_2[\rho^AB])=% \operatornameTr_B_2(\Lambda_B\rightarrow B_1B_2[\rho^AB])=\rho^ABroman_Tr start_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( roman_Λ start_POSTSUBSCRIPT italic_B → italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT [ italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ] ) = roman_Tr start_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( roman_Λ start_POSTSUBSCRIPT italic_B → italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT [ italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ] ) = italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT if and only if ρABsuperscript𝜌𝐴𝐵\rho^ABitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT is quantum-classical, i.e., of the form
ρAB=∑bpbρbA⊗|b⟩⟨b|B,subscript𝜌𝐴𝐵subscript𝑏tensor-productsubscript𝑝𝑏superscriptsubscript𝜌𝑏𝐴ket𝑏subscriptbra𝑏𝐵\rho_AB=\sum_bp_b\rho_b^A\otimes|b\rangle\langle b|_B,italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ⊗ | italic_b ⟩ ⟨ italic_b | start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT , (2)
with orthogonal |b⟩ket𝑏|b\rangle| italic_b ⟩s, and pbsubscript𝑝𝑏\p_b\ italic_p start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT a probability distribution. Notice that here we focus on one-sided local broadcasting Luo and Sun (2010), rather than two-sided local broadcasting Piani et al. (2008). If local broadcasting is possible, then: (i) it can be realized with a symmetric broadcasting channel, whose output is symmetric among the outputs; (ii) an arbitrary number k𝑘kitalic_k of extensions can be obtained, simply by |b⟩↦|b⟩⊗kmaps-toket𝑏superscriptket𝑏tensor-productabsent𝑘|b\rangle\mapsto|b\rangle^\otimes k| italic_b ⟩ ↦ | italic_b ⟩ start_POSTSUPERSCRIPT ⊗ italic_k end_POSTSUPERSCRIPT, for |b⟩ket𝑏|b\rangle| italic_b ⟩ as in (2), i.e., with output into the fully symmetric subspace B+ksubscriptsuperscript𝐵𝑘B^k_+italic_B start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT, so that the broadcasting channel has actually Bose-symmetric output (see Fig. 1).
Consider then Bose-symmetric broadcast maps ΛB→B+ksubscriptΛ→𝐵subscriptsuperscript𝐵𝑘\Lambda_B\rightarrow B^k_+roman_Λ start_POSTSUBSCRIPT italic_B → italic_B start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT end_POSTSUBSCRIPT with output in the fully symmetric subspace B+ksubscriptsuperscript𝐵𝑘B^k_+italic_B start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT, and the induced maps ΛBSym+(k)=Tr\B1∘ΛB→B+ksubscriptsuperscriptΛsubscriptSym𝑘𝐵subscriptTr\absentsubscript𝐵1subscriptΛ→𝐵subscriptsuperscript𝐵𝑘\Lambda^\textrmSym_+(k)_B=\operatornameTr_\backslash B_1\circ% \Lambda_B\rightarrow B^k_+roman_Λ start_POSTSUPERSCRIPT Sym start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ( italic_k ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT = roman_Tr start_POSTSUBSCRIPT \ italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∘ roman_Λ start_POSTSUBSCRIPT italic_B → italic_B start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT end_POSTSUBSCRIPT, where ∘\circ∘ denotes composition. We say that any map ΛBSym+(k)subscriptsuperscriptΛsubscriptSym𝑘𝐵\Lambda^\textrmSym_+(k)_Broman_Λ start_POSTSUPERSCRIPT Sym start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ( italic_k ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT that admits such a representation is k𝑘kitalic_k-Bose-symmetric extendible. The no-local-broadcasting theorem can then be recast as the fact that, for any ρABsubscript𝜌𝐴𝐵\rho_ABitalic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT that is not quantum-classical, F(ρAB,ΛBSym+(k)[ρAB])<1𝐹subscript𝜌𝐴𝐵subscriptsuperscriptΛsubscriptSym𝑘𝐵delimited-[]subscript𝜌𝐴𝐵1F(\rho_AB,\Lambda^\textrmSym_+(k)_B[\rho_AB])<1italic_F ( italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT , roman_Λ start_POSTSUPERSCRIPT Sym start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ( italic_k ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT [ italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ] ) <1 for any k≥2𝑘2k\geq 2italic_k ≥ 2 and any k𝑘kitalic_k-Bose-symmetric extendible ΛBSym+(k)subscriptsuperscriptΛsubscriptSym𝑘𝐵\Lambda^\textrmSym_+(k)_Broman_Λ start_POSTSUPERSCRIPT Sym start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ( italic_k ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT.
We now recall that every k𝑘kitalic_k-Bose-symmetric extendible channel is close to an entanglement-breaking (EB)-also called measure-and-prepare-map Horodecki et al. (2003)
ΛBEB[⋅]=∑yTr(MyB⋅)|βy⟩⟨βy|B,fragmentssubscriptsuperscriptΛEB𝐵fragments[⋅]subscript𝑦Trfragments(subscriptsuperscript𝑀𝐵𝑦⋅)|subscript𝛽𝑦⟩⟨subscript𝛽𝑦subscript|𝐵,\Lambda^\textEB_B[\cdot]=\sum_y\operatornameTr(M^B_y\cdot)|\beta% _y\rangle\langle\beta_y|_B,roman_Λ start_POSTSUPERSCRIPT EB end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT [ ⋅ ] = ∑ start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT roman_Tr ( italic_M start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ⋅ ) | italic_β start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ⟩ ⟨ italic_β start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT | start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT , (3)
where MyBsubscriptsuperscript𝑀𝐵𝑦\M^B_y\ italic_M start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT is a positive-operator-valued measure (POVM) and |βy⟩Bsubscriptketsubscript𝛽𝑦𝐵|\beta_y\rangle_B| italic_β start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ⟩ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPTs are normalized vector states, not necessarily orthogonal. Entanglement-breaking maps have the defining property that, for any given ρABsubscript𝜌𝐴𝐵\rho_ABitalic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT, (idA⊗ΛBEB)[ρAB]tensor-productsubscriptid𝐴subscriptsuperscriptΛEB𝐵delimited-[]subscript𝜌𝐴𝐵(\rm id_A\otimes\Lambda^\textEB_B)[\rho_AB]( roman_id start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⊗ roman_Λ start_POSTSUPERSCRIPT EB end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ) [ italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ] is separable. More precisely, one can prove that for any k𝑘kitalic_k-Bose-symmetric extendible ΛBSym+(k)subscriptsuperscriptΛsubscriptSym𝑘𝐵\Lambda^\textrmSym_+(k)_Broman_Λ start_POSTSUPERSCRIPT Sym start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ( italic_k ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT there is an entanglement breaking map ΛBEBsubscriptsuperscriptΛEB𝐵\Lambda^\textEB_Broman_Λ start_POSTSUPERSCRIPT EB end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT close to it in the so-called diamond-norm distance; more precisely Chiribella (2011):
supρABΔ(ΛBSym+(k)[ρAB],ΛBEB[ρAB])≤|B|k,subscriptsupremumsubscript𝜌𝐴𝐵ΔsubscriptsuperscriptΛsubscriptSym𝑘𝐵delimited-[]subscript𝜌𝐴𝐵subscriptsuperscriptΛEB𝐵delimited-[]subscript𝜌𝐴𝐵𝐵𝑘\sup_\rho_AB\Delta\left(\Lambda^\textrmSym_+(k)_B[\rho_AB],% \Lambda^\textEB_B[\rho_AB]\right)\\ \leq\frack,roman_sup start_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT end_POSTSUBSCRIPT roman_Δ ( roman_Λ start_POSTSUPERSCRIPT Sym start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ( italic_k ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT [ italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ] , roman_Λ start_POSTSUPERSCRIPT EB end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT [ italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ] ) ≤ divide start_ARG | italic_B | end_ARG start_ARG italic_k end_ARG , (4)
where |B|𝐵|B|| italic_B | indicates the dimension of system B𝐵Bitalic_B, i.e., of ℋBsubscriptℋ𝐵\mathcalH_Bcaligraphic_H start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT. Furthermore, it is clear that any entanglement-breaking map is k𝑘kitalic_k-Bose-symmetric extendible, since for any entanglement-breaking map (3) we can consider ΛB→B+k[⋅]=∑yTr(MyB⋅)(|βy⟩⟨βy|⊗k)B+kfragmentssubscriptΛ→𝐵subscriptsuperscript𝐵𝑘fragments[⋅]subscript𝑦Trfragments(subscriptsuperscript𝑀𝐵𝑦⋅)fragments(|subscript𝛽𝑦⟩subscriptfragments⟨subscript𝛽𝑦superscript|tensor-productabsent𝑘)subscriptsuperscript𝐵𝑘\Lambda_B\rightarrow B^k_+[\cdot]=\sum_y\operatornameTr(M^B_y% \cdot)(|\beta_y\rangle\langle\beta_y|^\otimes k)_B^k_+roman_Λ start_POSTSUBSCRIPT italic_B → italic_B start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT end_POSTSUBSCRIPT [ ⋅ ] = ∑ start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT roman_Tr ( italic_M start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ⋅ ) ( | italic_β start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ⟩ ⟨ italic_β start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT ⊗ italic_k end_POSTSUPERSCRIPT ) start_POSTSUBSCRIPT italic_B start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT end_POSTSUBSCRIPT. Denote by ℒSym+(k)=ΛSym+(k)superscriptℒsubscriptSym𝑘superscriptΛsubscriptSym𝑘\mathcalL^\textrmSym_+(k)=\\Lambda^\textrmSym_+(k)\caligraphic_L start_POSTSUPERSCRIPT Sym start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ( italic_k ) end_POSTSUPERSCRIPT = roman_Λ start_POSTSUPERSCRIPT Sym start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ( italic_k ) end_POSTSUPERSCRIPT the class of channels with a k𝑘kitalic_k-Bose-symmetric extension, and by ℒEB=ΛEBsuperscriptℒEBsuperscriptΛEB\mathcalL^\textrmEB=\\Lambda^\textrmEB\caligraphic_L start_POSTSUPERSCRIPT EB end_POSTSUPERSCRIPT = roman_Λ start_POSTSUPERSCRIPT EB end_POSTSUPERSCRIPT the class of entanglement-breaking channels 111We are here considering maps with the same fixed input/output space, but varying k𝑘kitalic_k.. We can then write ℒEB⊆ℒSym+(k)superscriptℒEBsuperscriptℒsubscriptSym𝑘\mathcalL^\textrmEB\subseteq\mathcalL^\textrmSym_+(k)caligraphic_L start_POSTSUPERSCRIPT EB end_POSTSUPERSCRIPT ⊆ caligraphic_L start_POSTSUPERSCRIPT Sym start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ( italic_k ) end_POSTSUPERSCRIPT and ℒSym+(k)→ℒEB→superscriptℒsubscriptSym𝑘superscriptℒEB\mathcalL^\textrmSym_+(k)\rightarrow\mathcalL^\textrmEBcaligraphic_L start_POSTSUPERSCRIPT Sym start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ( italic_k ) end_POSTSUPERSCRIPT → caligraphic_L start_POSTSUPERSCRIPT EB end_POSTSUPERSCRIPT for k→∞→𝑘k\rightarrow\inftyitalic_k → ∞.
Mutual information, recoverability, and discord-The mutual information between A𝐴Aitalic_A and B𝐵Bitalic_B is defined as I(A:B)ρ=S(A)ρ+S(B)ρ-S(AB)ρfragmentsIsubscriptfragments(A:B)𝜌Ssubscriptfragments(A)𝜌Ssubscriptfragments(B)𝜌Ssubscriptfragments(AB)𝜌I(A:B)_\rho=S(A)_\rho+S(B)_\rho-S(AB)_\rhoitalic_I ( italic_A : italic_B ) start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT = italic_S ( italic_A ) start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT + italic_S ( italic_B ) start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT - italic_S ( italic_A italic_B ) start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT, and is a fundamental measure of the total correlations present between A𝐴Aitalic_A and B𝐵Bitalic_B Nielsen and Chuang (2010); Groisman et al. (2005); Wilde (2013). The conditional mutual information can be defined as Nielsen and Chuang (2010) I(A:B|C)ρ=I(A:BC)ρ-I(A:C)ρfragmentsIsubscriptfragments(A:B|C)𝜌Isubscriptfragments(A:BC)𝜌Isubscriptfragments(A:C)𝜌I(A:B|C)_\rho=I(A:BC)_\rho-I(A:C)_\rhoitalic_I ( italic_A : italic_B | italic_C ) start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT = italic_I ( italic_A : italic_B italic_C ) start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT - italic_I ( italic_A : italic_C ) start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT, i.e., it is equivalent to the decrease of correlations between A𝐴Aitalic_A and BC𝐵𝐶BCitalic_B italic_C due to the loss of system B𝐵Bitalic_B. The celebrated strong subadditivity of the von Neumann entropy Lieb and Ruskai (1973) is equivalent to
I(A:B|C)ρ≥0.fragmentsIsubscriptfragments(A:B|C)𝜌0.I(A:B|C)_\rho\geq 0.italic_I ( italic_A : italic_B | italic_C ) start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT ≥ 0 . (5)
When (5) is satisfied with equality, ρABCsubscript𝜌𝐴𝐵𝐶\rho_ABCitalic_ρ start_POSTSUBSCRIPT italic_A italic_B italic_C end_POSTSUBSCRIPT is said to form a Markov chain: indeed, a strong result by Petz Petz (1986, 1988); Hayden et al. (2004) ensures that there exist a recovery channel ℛC→BCsubscriptℛ→𝐶𝐵𝐶\mathcalR_C\rightarrow BCcaligraphic_R start_POSTSUBSCRIPT italic_C → italic_B italic_C end_POSTSUBSCRIPT such that ρABC=ℛC→BC[ρAC]subscript𝜌𝐴𝐵𝐶subscriptℛ→𝐶𝐵𝐶delimited-[]subscript𝜌𝐴𝐶\rho_ABC=\mathcalR_C\rightarrow BC[\rho_AC]italic_ρ start_POSTSUBSCRIPT italic_A italic_B italic_C end_POSTSUBSCRIPT = caligraphic_R start_POSTSUBSCRIPT italic_C → italic_B italic_C end_POSTSUBSCRIPT [ italic_ρ start_POSTSUBSCRIPT italic_A italic_C end_POSTSUBSCRIPT ]. Fawzi and Renner recently generalized this by proving that, for any tripartite state ρABCsubscript𝜌𝐴𝐵𝐶\rho_ABCitalic_ρ start_POSTSUBSCRIPT italic_A italic_B italic_C end_POSTSUBSCRIPT, there always exists a recovery channel ℛC→BCsubscriptℛ→𝐶𝐵𝐶\mathcalR_C\rightarrow BCcaligraphic_R start_POSTSUBSCRIPT italic_C → italic_B italic_C end_POSTSUBSCRIPT such that Fawzi and Renner (2014) (see also Brandao et al. (2014))
F(ℛC→BC[ρAC],ρABC)≥2-12I(A:B|C)ρ,𝐹subscriptℛ→𝐶𝐵𝐶delimited-[]subscript𝜌𝐴𝐶subscript𝜌𝐴𝐵𝐶superscript2fragments12Isubscriptfragments(A:B|C)𝜌F(\mathcalR_C\rightarrow BC[\rho_AC],\rho_ABC)\geq 2^C)_\rho,italic_F ( caligraphic_R start_POSTSUBSCRIPT italic_C → italic_B italic_C end_POSTSUBSCRIPT [ italic_ρ start_POSTSUBSCRIPT italic_A italic_C end_POSTSUBSCRIPT ] , italic_ρ start_POSTSUBSCRIPT italic_A italic_B italic_C end_POSTSUBSCRIPT ) ≥ 2 start_POSTSUPERSCRIPT - divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_I ( italic_A : italic_B | italic_C ) start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT end_POSTSUPERSCRIPT , (6)
that is, roughly speaking, the smaller the decrease of correlations between A𝐴Aitalic_A and BC𝐵𝐶BCitalic_B italic_C due to the loss of B𝐵Bitalic_B, the better the original ABC𝐴𝐵𝐶ABCitalic_A italic_B italic_C state can be recovered from operating just on C𝐶Citalic_C alone.
Consider measurement maps ℳB→Y[⋅]=∑yTr(MyB⋅)|y⟩⟨y|Yfragmentssubscriptℳ→𝐵𝑌fragments[⋅]subscript𝑦Trfragments(subscriptsuperscript𝑀𝐵𝑦⋅)|y⟩⟨ysubscript|𝑌\mathcalM_B\rightarrow Y[\cdot]=\sum_y\operatornameTr(M^B_y\cdot)|% y\rangle\langle y|_Ycaligraphic_M start_POSTSUBSCRIPT italic_B → italic_Y end_POSTSUBSCRIPT [ ⋅ ] = ∑ start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT roman_Tr ( italic_M start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ⋅ ) | italic_y ⟩ ⟨ italic_y | start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT, where Mysubscript𝑀𝑦\M_y\ italic_M start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT is a POVM, and the |y⟩ket𝑦|y\rangle| italic_y ⟩s are orthogonal vector states. The discord of ρ𝜌\rhoitalic_ρ between A𝐴Aitalic_A and B𝐵Bitalic_B with measurement on B𝐵Bitalic_B can be defined as Piani (2012); Seshadreesan and Wilde (2014)
D(A:B¯)ρfragmentsDsubscriptfragments(A:¯𝐵)𝜌\displaystyle D(A:\underlineB)_\rhoitalic_D ( italic_A : under¯ start_ARG italic_B end_ARG ) start_POSTSUBSCRIPT italic_ρ end_POSTSUBSCRIPT =minℳB→Y(I(A:B)ρAB-I(A:Y)ℳB→Y[ρAB])fragmentssubscriptsubscriptℳ→𝐵𝑌fragments(Isubscriptfragments(A:B)subscript𝜌𝐴𝐵Isubscriptfragments(A:Y)subscriptℳ→𝐵𝑌delimited-[]subscript𝜌𝐴𝐵)\displaystyle=\min_\mathcalM_B\rightarrow Y\left(I(A:B)_\rho_AB-I(A:% Y)_\mathcalM_B\rightarrow Y[\rho_AB]\right)= roman_min start_POSTSUBSCRIPT caligraphic_M start_POSTSUBSCRIPT italic_B → italic_Y end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_I ( italic_A : italic_B ) start_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT end_POSTSUBSCRIPT - italic_I ( italic_A : italic_Y ) start_POSTSUBSCRIPT caligraphic_M start_POSTSUBSCRIPT italic_B → italic_Y end_POSTSUBSCRIPT [ italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ] end_POSTSUBSCRIPT ) (7)
=minVB→YE(I(A:YE)ρAYE-I(A:Y)ρAY)fragmentssubscriptsubscript𝑉→𝐵𝑌𝐸fragments(Isubscriptfragments(A:YE)subscript𝜌𝐴𝑌𝐸Isubscriptfragments(A:Y)subscript𝜌𝐴𝑌)\displaystyle=\min_V_B\rightarrow YE\left(I(A:YE)_\rho_AYE-I(A:Y)_% \rho_AY\right)= roman_min start_POSTSUBSCRIPT italic_V start_POSTSUBSCRIPT italic_B → italic_Y italic_E end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_I ( italic_A : italic_Y italic_E ) start_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_A italic_Y italic_E end_POSTSUBSCRIPT end_POSTSUBSCRIPT - italic_I ( italic_A : italic_Y ) start_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_A italic_Y end_POSTSUBSCRIPT end_POSTSUBSCRIPT )
=minVB→YEI(A:Y|E)ρAYE,fragmentssubscriptsubscript𝑉→𝐵𝑌𝐸Isubscriptfragments(A:Y|E)subscript𝜌𝐴𝑌𝐸,\displaystyle=\min_V_B\rightarrow YEI(A:Y|E)_\rho_AYE,= roman_min start_POSTSUBSCRIPT italic_V start_POSTSUBSCRIPT italic_B → italic_Y italic_E end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_I ( italic_A : italic_Y | italic_E ) start_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_A italic_Y italic_E end_POSTSUBSCRIPT end_POSTSUBSCRIPT ,
where in the second and third lines the minimization is over all isometries VB→YEsubscript𝑉→𝐵𝑌𝐸V_B\rightarrow YEitalic_V start_POSTSUBSCRIPT italic_B → italic_Y italic_E end_POSTSUBSCRIPT that realize measurement maps ℳB→Ysubscriptℳ→𝐵𝑌\mathcalM_B\rightarrow Ycaligraphic_M start_POSTSUBSCRIPT italic_B → italic_Y end_POSTSUBSCRIPT, with E𝐸Eitalic_E considered as the environment of the dilation Nielsen and Chuang (2010); Seshadreesan and Wilde (2014). That is, E𝐸Eitalic_E is the system that is traced out, or lost, in ℳB→Y[⋅]=TrE(VB→YE⋅VB→YE†)subscriptℳ→𝐵𝑌delimited-[]⋅subscriptTr𝐸⋅subscript𝑉→𝐵𝑌𝐸superscriptsubscript𝑉→𝐵𝑌𝐸†\mathcalM_B\rightarrow Y[\cdot]=\operatornameTr_E(V_B\rightarrow YE% \cdot V_B\rightarrow YE^\dagger)caligraphic_M start_POSTSUBSCRIPT italic_B → italic_Y end_POSTSUBSCRIPT [ ⋅ ] = roman_Tr start_POSTSUBSCRIPT italic_E end_POSTSUBSCRIPT ( italic_V start_POSTSUBSCRIPT italic_B → italic_Y italic_E end_POSTSUBSCRIPT ⋅ italic_V start_POSTSUBSCRIPT italic_B → italic_Y italic_E end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ), and ρAYE=VB→YEρABVB→YE†subscript𝜌𝐴𝑌𝐸subscript𝑉→𝐵𝑌𝐸subscript𝜌𝐴𝐵superscriptsubscript𝑉→𝐵𝑌𝐸†\rho_AYE=V_B\rightarrow YE\rho_ABV_B\rightarrow YE^\daggeritalic_ρ start_POSTSUBSCRIPT italic_A italic_Y italic_E end_POSTSUBSCRIPT = italic_V start_POSTSUBSCRIPT italic_B → italic_Y italic_E end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT italic_V start_POSTSUBSCRIPT italic_B → italic_Y italic_E end_POSTSUBSCRIPT start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT. Notice that I(A:B)ρAB=I(A:YE)ρAYEfragmentsIsubscriptfragments(A:B)subscript𝜌𝐴𝐵Isubscriptfragments(A:YE)subscript𝜌𝐴𝑌𝐸I(A:B)_\rho_AB=I(A:YE)_\rho_AYEitalic_I ( italic_A : italic_B ) start_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT end_POSTSUBSCRIPT = italic_I ( italic_A : italic_Y italic_E ) start_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_A italic_Y italic_E end_POSTSUBSCRIPT end_POSTSUBSCRIPT. It can be proven Hayashi (2006); Luo and Sun (2010) that the only states with vanishing discord are quantum-classical states of the form (2).
In the case of a (local) measurement, the recovery map (for our intentions, directly to B𝐵Bitalic_B, rather than YE𝑌𝐸YEitalic_Y italic_E) can be assumed to be of the form Seshadreesan and Wilde (2014) ℛY→B[⋅]=∑kTr(|y⟩⟨y|Y⋅)σByfragmentssubscriptℛ→𝑌𝐵fragments[⋅]subscript𝑘Trfragments(|y⟩fragments⟨ysubscript|𝑌⋅)subscriptsuperscript𝜎𝑦𝐵\mathcalR_Y\rightarrow B[\cdot]=\sum_k\operatornameTr(|y\rangle\langle y% |_Y\cdot)\sigma^y_Bcaligraphic_R start_POSTSUBSCRIPT italic_Y → italic_B end_POSTSUBSCRIPT [ ⋅ ] = ∑ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT roman_Tr ( | italic_y ⟩ ⟨ italic_y | start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT ⋅ ) italic_σ start_POSTSUPERSCRIPT italic_y end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT, with σBysubscriptsuperscript𝜎𝑦𝐵\sigma^y_Bitalic_σ start_POSTSUPERSCRIPT italic_y end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT states, so that the combination of measurement and recovery, ℛY→B∘ℳB→Ysubscriptℛ→𝑌𝐵subscriptℳ→𝐵𝑌\mathcalR_Y\rightarrow B\circ\mathcalM_B\rightarrow Ycaligraphic_R start_POSTSUBSCRIPT italic_Y → italic_B end_POSTSUBSCRIPT ∘ caligraphic_M start_POSTSUBSCRIPT italic_B → italic_Y end_POSTSUBSCRIPT, is an entanglement-breaking map (3) Horodecki et al. (2003). Then, combining (6) and (7), one has Seshadreesan and Wilde (2014)
supΛEB∈ℒEBF(ΛBEB[ρAB],ρAB)≥2-12D(A:B¯).subscriptsupremumsuperscriptΛEBsuperscriptℒEB𝐹subscriptsuperscriptΛEB𝐵delimited-[]subscript𝜌𝐴𝐵subscript𝜌𝐴𝐵superscript212𝐷:𝐴¯𝐵\sup_\Lambda^\textEB\in\mathcalL^\textEBF(\Lambda^\textEB_B[% \rho_AB],\rho_AB)\geq 2^-\frac12D(A:\underlineB).roman_sup start_POSTSUBSCRIPT roman_Λ start_POSTSUPERSCRIPT EB end_POSTSUPERSCRIPT ∈ caligraphic_L start_POSTSUPERSCRIPT EB end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_F ( roman_Λ start_POSTSUPERSCRIPT EB end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT [ italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ] , italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ) ≥ 2 start_POSTSUPERSCRIPT - divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_D ( italic_A : under¯ start_ARG italic_B end_ARG ) end_POSTSUPERSCRIPT . (8)
Introducing the surprisal of measurement recoverability Seshadreesan and Wilde (2014) DF(A:B¯):=-logsupΛEB∈ℒEBF2(ΛBEB[ρAB],ρAB)fragmentssubscript𝐷𝐹fragments(A:¯𝐵)assignsubscriptsupremumsuperscriptΛEBsuperscriptℒEBsuperscript𝐹2fragments(subscriptsuperscriptΛEB𝐵fragments[subscript𝜌𝐴𝐵],subscript𝜌𝐴𝐵)D_F(A:\underlineB):=-\log\sup_\Lambda^\textEB\in\mathcalL^\textEB% F^2(\Lambda^\textEB_B[\rho_AB],\rho_AB)italic_D start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ( italic_A : under¯ start_ARG italic_B end_ARG ) := - roman_log roman_sup start_POSTSUBSCRIPT roman_Λ start_POSTSUPERSCRIPT EB end_POSTSUPERSCRIPT ∈ caligraphic_L start_POSTSUPERSCRIPT EB end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_F start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_Λ start_POSTSUPERSCRIPT EB end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT [ italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ] , italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ), one can cast (8) as DF(A:B¯)≤D(A:B¯)fragmentssubscript𝐷𝐹fragments(A:¯𝐵)Dfragments(A:¯𝐵)D_F(A:\underlineB)\leq D(A:\underlineB)italic_D start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ( italic_A : under¯ start_ARG italic_B end_ARG ) ≤ italic_D ( italic_A : under¯ start_ARG italic_B end_ARG ). The surprisal of measurement recoverability quantifies the necessary disturbance introduced by manipulating locally (on B𝐵Bitalic_B) the state ρABsubscript𝜌𝐴𝐵\rho_ABitalic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT, through measurement and preparation. Notice that this can be generalized to any class of maps that correspond to a non-trivial (local) manipulation (see Piani et al. (2014)), i.e., one can consider DF,ℒ(A:B¯):=-logsupΛ∈ℒF2(ΛB[ρAB],ρAB)fragmentssubscript𝐷𝐹ℒfragments(A:¯𝐵)assignsubscriptsupremumΛℒsuperscript𝐹2fragments(subscriptΛ𝐵fragments[subscript𝜌𝐴𝐵],subscript𝜌𝐴𝐵)D_F,\mathcalL(A:\underlineB):=-\log\sup_\Lambda\in\mathcalLF^2(% \Lambda_B[\rho_AB],\rho_AB)italic_D start_POSTSUBSCRIPT italic_F , caligraphic_L end_POSTSUBSCRIPT ( italic_A : under¯ start_ARG italic_B end_ARG ) := - roman_log roman_sup start_POSTSUBSCRIPT roman_Λ ∈ caligraphic_L end_POSTSUBSCRIPT italic_F start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( roman_Λ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT [ italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ] , italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ), for ℒℒ\mathcalLcaligraphic_L some class of channels. With this notation, we can write DF(A:B¯)=DF,ℒEB(A:B¯)fragmentssubscript𝐷𝐹fragments(A:¯𝐵)subscript𝐷𝐹superscriptℒEBfragments(A:¯𝐵)D_F(A:\underlineB)=D_F,\mathcalL^\textEB(A:\underlineB)italic_D start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ( italic_A : under¯ start_ARG italic_B end_ARG ) = italic_D start_POSTSUBSCRIPT italic_F , caligraphic_L start_POSTSUPERSCRIPT EB end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_A : under¯ start_ARG italic_B end_ARG ), where, we recall, ℒEBsuperscriptℒEB\mathcalL^\textEBcaligraphic_L start_POSTSUPERSCRIPT EB end_POSTSUPERSCRIPT indicates the set of entanglement-breaking channels. Notice that if ℒEB⊆ℒsuperscriptℒEBℒ\mathcalL^\textrmEB\subseteq\mathcalLcaligraphic_L start_POSTSUPERSCRIPT EB end_POSTSUPERSCRIPT ⊆ caligraphic_L, it necessarily holds
DF,ℒ(A:B¯)≤DF,ℒEB(A:B¯)≤D(A:B¯).fragmentssubscript𝐷𝐹ℒfragments(A:¯𝐵)subscript𝐷𝐹superscriptℒEBfragments(A:¯𝐵)Dfragments(A:¯𝐵).D_F,\mathcalL(A:\underlineB)\leq D_F,\mathcalL^\textEB(A:% \underlineB)\leq D(A:\underlineB).italic_D start_POSTSUBSCRIPT italic_F , caligraphic_L end_POSTSUBSCRIPT ( italic_A : under¯ start_ARG italic_B end_ARG ) ≤ italic_D start_POSTSUBSCRIPT italic_F , caligraphic_L start_POSTSUPERSCRIPT EB end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_A : under¯ start_ARG italic_B end_ARG ) ≤ italic_D ( italic_A : under¯ start_ARG italic_B end_ARG ) . (9)
In particular, we will consider ℒ=ℒSym+(k)ℒsuperscriptℒsubscriptSym𝑘\mathcalL=\mathcalL^\textrmSym_+(k)caligraphic_L = caligraphic_L start_POSTSUPERSCRIPT Sym start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ( italic_k ) end_POSTSUPERSCRIPT. Notice that, in the other direction, Eq. (4) implies (see Appendix) supΛEBF(ρAB,ΛBEB[ρAB])≥supΛSym+(k)F(ρAB,ΛBSym+(k)[ρAB])-(2|B|)/ksubscriptsupremumsuperscriptΛEB𝐹subscript𝜌𝐴𝐵subscriptsuperscriptΛEB𝐵delimited-[]subscript𝜌𝐴𝐵subscriptsupremumsuperscriptΛsubscriptSym𝑘𝐹subscript𝜌𝐴𝐵subscriptsuperscriptΛsubscriptSym𝑘𝐵delimited-[]subscript𝜌𝐴𝐵2𝐵𝑘\sup_\Lambda^\textEBF(\rho_AB,\Lambda^\textEB_B[\rho_AB])\geq% \sup_\Lambda^\textrmSym_+(k)F(\rho_AB,\Lambda^\textrmSym_+(k)_% B[\rho_AB])-\sqrt(2roman_sup start_POSTSUBSCRIPT roman_Λ start_POSTSUPERSCRIPT EB end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_F ( italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT , roman_Λ start_POSTSUPERSCRIPT EB end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT [ italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ] ) ≥ roman_sup start_POSTSUBSCRIPT roman_Λ start_POSTSUPERSCRIPT Sym start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ( italic_k ) end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_F ( italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT , roman_Λ start_POSTSUPERSCRIPT Sym start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ( italic_k ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT [ italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ] ) - square-root start_ARG ( 2 | italic_B | ) / italic_k end_ARG, so DF,ℒSym+(k)(A:B¯)→DF,ℒEB(A:B¯)fragmentssubscript𝐷𝐹superscriptℒsubscriptSym𝑘fragments(A:¯𝐵)→subscript𝐷𝐹superscriptℒEBfragments(A:¯𝐵)D_F,\mathcalL^\textrmSym_+(k)(A:\underlineB)\rightarrow D_F,% \mathcalL^\textEB(A:\underlineB)italic_D start_POSTSUBSCRIPT italic_F , caligraphic_L start_POSTSUPERSCRIPT Sym start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ( italic_k ) end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_A : under¯ start_ARG italic_B end_ARG ) → italic_D start_POSTSUBSCRIPT italic_F , caligraphic_L start_POSTSUPERSCRIPT EB end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_A : under¯ start_ARG italic_B end_ARG ) for k→∞→𝑘k\rightarrow\inftyitalic_k → ∞.
Choi-Jamiołkowski isomorphism and k𝑘kitalic_k-extendible maps.-The Choi-Jamiołkowski isomorphism Choi (1975); Jamiołkowski (1972) is a one-to-one correspondence between linear maps ΛX→YsubscriptΛ→𝑋𝑌\Lambda_X\rightarrow Yroman_Λ start_POSTSUBSCRIPT italic_X → italic_Y end_POSTSUBSCRIPT from L(ℋX)𝐿subscriptℋ𝑋L(\mathcalH_X)italic_L ( caligraphic_H start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ) to L(ℋY)𝐿subscriptℋ𝑌L(\mathcalH_Y)italic_L ( caligraphic_H start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT ) and linear operators WXYsubscript𝑊𝑋𝑌W_XYitalic_W start_POSTSUBSCRIPT italic_X italic_Y end_POSTSUBSCRIPT in L(ℋX⊗ℋY)𝐿tensor-productsubscriptℋ𝑋subscriptℋ𝑌L(\mathcalH_X\otimes\mathcalH_Y)italic_L ( caligraphic_H start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ⊗ caligraphic_H start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT ). It reads
J(Λ)XY=(idX⊗ΛX′→Y)[ψ~XX′+],𝐽subscriptΛ𝑋𝑌tensor-productsubscriptid𝑋subscriptΛ→superscript𝑋′𝑌delimited-[]subscriptsuperscript~𝜓𝑋superscript𝑋′J(\Lambda)_XY=(\rm id_X\otimes\Lambda_X^\prime\rightarrow Y)[\tilde% \psi^+_XX^\prime],italic_J ( roman_Λ ) start_POSTSUBSCRIPT italic_X italic_Y end_POSTSUBSCRIPT = ( roman_id start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ⊗ roman_Λ start_POSTSUBSCRIPT italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT → italic_Y end_POSTSUBSCRIPT ) [ over~ start_ARG italic_ψ end_ARG start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_X italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ] , (10)
with inverse
(J-1(WXY))X→Y[ρX]=TrX(WXYΓXρX).subscriptsuperscript𝐽1subscript𝑊𝑋𝑌→𝑋𝑌delimited-[]subscript𝜌𝑋subscriptTr𝑋subscriptsuperscript𝑊subscriptΓ𝑋𝑋𝑌subscript𝜌𝑋(J^-1(W_XY))_X\rightarrow Y[\rho_X]=\operatornameTr_X(W^\Gamma_X% _XY\rho_X).( italic_J start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_W start_POSTSUBSCRIPT italic_X italic_Y end_POSTSUBSCRIPT ) ) start_POSTSUBSCRIPT italic_X → italic_Y end_POSTSUBSCRIPT [ italic_ρ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ] = roman_Tr start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ( italic_W start_POSTSUPERSCRIPT roman_Γ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_X italic_Y end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ) . (11)
Here ψ~XX′+=|ψ~+⟩⟨ψ~+|XX′subscriptsuperscript~𝜓𝑋superscript𝑋′ketsuperscript~𝜓subscriptbrasuperscript~𝜓𝑋superscript𝑋′\tilde\psi^+_XX^\prime=|\tilde\psi^+\rangle\langle\tilde\psi^+% |_XX^\primeover~ start_ARG italic_ψ end_ARG start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_X italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = | over~ start_ARG italic_ψ end_ARG start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ⟩ ⟨ over~ start_ARG italic_ψ end_ARG start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT | start_POSTSUBSCRIPT italic_X italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT, with the unnormalized maximally entangled state |ψ~+⟩XX′=∑x|x⟩X|x⟩X′subscriptketsuperscript~𝜓𝑋superscript𝑋′subscript𝑥subscriptket𝑥𝑋subscriptket𝑥superscript𝑋′|\tilde\psi^+\rangle_XX^\prime=\sum_x|x\rangle_X|x\rangle_X^% \prime| over~ start_ARG italic_ψ end_ARG start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT ⟩ start_POSTSUBSCRIPT italic_X italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT | italic_x ⟩ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT | italic_x ⟩ start_POSTSUBSCRIPT italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT, for x⟩ket𝑥\ italic_x ⟩ an orthonormal basis of ℋXsubscriptℋ𝑋\mathcalH_Xcaligraphic_H start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT, and ΓXsubscriptΓ𝑋^\Gamma_Xstart_FLOATSUPERSCRIPT roman_Γ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT end_FLOATSUPERSCRIPT indicates partial transposition on X𝑋Xitalic_X. The operator J(Λ)𝐽ΛJ(\Lambda)italic_J ( roman_Λ ) encodes all the information about the map ΛΛ\Lambdaroman_Λ. In particular, the linear map (J-1(WXY))X→Ysubscriptsuperscript𝐽1subscript𝑊𝑋𝑌→𝑋𝑌(J^-1(W_XY))_X\rightarrow Y( italic_J start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_W start_POSTSUBSCRIPT italic_X italic_Y end_POSTSUBSCRIPT ) ) start_POSTSUBSCRIPT italic_X → italic_Y end_POSTSUBSCRIPT defined via (11) is a valid quantum channel from X𝑋Xitalic_X to Y𝑌Yitalic_Y if and only if WXYsubscript𝑊𝑋𝑌W_XYitalic_W start_POSTSUBSCRIPT italic_X italic_Y end_POSTSUBSCRIPT is positive semidefinite and WX=TrY(WXY)=𝟙𝕏subscript𝑊𝑋subscriptTr𝑌subscript𝑊𝑋𝑌subscript𝟙𝕏W_X=\operatornameTr_Y(W_XY)=\openone_Xitalic_W start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT = roman_Tr start_POSTSUBSCRIPT italic_Y end_POSTSUBSCRIPT ( italic_W start_POSTSUBSCRIPT italic_X italic_Y end_POSTSUBSCRIPT ) = blackboard_1 start_POSTSUBSCRIPT blackboard_X end_POSTSUBSCRIPT. Also, (J-1(WXY))X→Ysubscriptsuperscript𝐽1subscript𝑊𝑋𝑌→𝑋𝑌(J^-1(W_XY))_X\rightarrow Y( italic_J start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_W start_POSTSUBSCRIPT italic_X italic_Y end_POSTSUBSCRIPT ) ) start_POSTSUBSCRIPT italic_X → italic_Y end_POSTSUBSCRIPT is an entanglement breaking channel if and only if WXYsubscript𝑊𝑋𝑌W_XYitalic_W start_POSTSUBSCRIPT italic_X italic_Y end_POSTSUBSCRIPT satisfies the additional condition of being proportional to a separable state. Finally, it is easily checked that J-1(WXY)X→Ysuperscript𝐽1subscriptsubscript𝑊𝑋𝑌→𝑋𝑌J^-1(W_XY)_X\rightarrow Yitalic_J start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( italic_W start_POSTSUBSCRIPT italic_X italic_Y end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_X → italic_Y end_POSTSUBSCRIPT is a k𝑘kitalic_k-Bose-symmetric extendible channel if and only if, besides satisfying the conditions to be isomorphic to a channel, WXYsubscript𝑊𝑋𝑌W_XYitalic_W start_POSTSUBSCRIPT italic_X italic_Y end_POSTSUBSCRIPT admits k𝑘kitalic_k-Bose-symmetric extensions on Y𝑌Yitalic_Y.
A faithful SDP lower bound to quantum discord-The major obstacle in the computation of the surprisal of measurement recoverability is the fact that it requires an optimization over entanglement breaking channels, i.e., via the Choi-Jamiołkowski isomorphism, over separable states, which cannot be easily parametrized.
In our case, relaxing the problem, we choose to maximize the fidelity between ρ=ρAB𝜌subscript𝜌𝐴𝐵\rho=\rho_ABitalic_ρ = italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT and σ=(idA⊗ΛBSym+(k))[ρAB])fragmentsσfragments(subscriptid𝐴tensor-productsuperscriptsubscriptΛ𝐵subscriptSym𝑘)fragments[subscript𝜌𝐴𝐵])\sigma=(\rm id_A\otimes\Lambda_B^\textrmSym_+(k))[\rho_AB])italic_σ = ( roman_id start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⊗ roman_Λ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT Sym start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ) [ italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ] ), optimizing over ΛBSym+(k)∈ℒSym+(k)superscriptsubscriptΛ𝐵subscriptSym𝑘superscriptℒsubscriptSym𝑘\Lambda_B^\textrmSym_+(k)\in\mathcalL^\textrmSym_+(k)roman_Λ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT Sym start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ( italic_k ) end_POSTSUPERSCRIPT ∈ caligraphic_L start_POSTSUPERSCRIPT Sym start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ( italic_k ) end_POSTSUPERSCRIPT. GAMING NEWS -Jamiołkoski isomorphism allows us to write this as an optimization over positive semidefinite operators WBB′subscript𝑊𝐵superscript𝐵′W_BB^\primeitalic_W start_POSTSUBSCRIPT italic_B italic_B start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT that satisfy WB=𝟙𝔹subscript𝑊𝐵subscript𝟙𝔹W_B=\openone_Bitalic_W start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT = blackboard_1 start_POSTSUBSCRIPT blackboard_B end_POSTSUBSCRIPT and admit k𝑘kitalic_k-Bose-symmetric extensions. Hence we can write this as an optimization over extended operators WBBksubscript𝑊𝐵superscript𝐵𝑘W_BB^kitalic_W start_POSTSUBSCRIPT italic_B italic_B start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_POSTSUBSCRIPT isomorphic to k𝑘kitalic_k-Bose-symmetric broadcasting channels. Putting everything together, we find that supΛ∈ℒSym(k)F(ρAB,ΛB[ρAB])subscriptsupremumΛsuperscriptℒSym(k)𝐹subscript𝜌𝐴𝐵subscriptΛ𝐵delimited-[]subscript𝜌𝐴𝐵\sup_\Lambda\in\mathcalL^\textrmSym(k)F(\rho_AB,\Lambda_B[\rho_AB% ])roman_sup start_POSTSUBSCRIPT roman_Λ ∈ caligraphic_L start_POSTSUPERSCRIPT Sym(k) end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_F ( italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT , roman_Λ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT [ italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ] ), from which DF,ℒSym+(k)(A:B¯)fragmentssubscript𝐷𝐹superscriptℒsubscriptSym𝑘fragments(A:¯𝐵)D_F,\mathcalL^\textrmSym_+(k)(A:\underlineB)italic_D start_POSTSUBSCRIPT italic_F , caligraphic_L start_POSTSUPERSCRIPT Sym start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ( italic_k ) end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_A : under¯ start_ARG italic_B end_ARG ) can be derived, corresponds to the solution of the following SDP optimization problem:
maximize 12(Tr(X)+Tr(X†))12Tr𝑋Trsuperscript𝑋†\displaystyle\frac12(\operatornameTr(X)+\operatornameTr(X^\dagger))divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( roman_Tr ( italic_X ) + roman_Tr ( italic_X start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT ) ) (12a)
subject to (ρABXX†Tr\AB1(WBBkΓBρAB))≥0matrixsubscript𝜌𝐴𝐵𝑋superscript𝑋†subscriptTr\absent𝐴subscript𝐵1subscriptsuperscript𝑊subscriptΓ𝐵𝐵superscript𝐵𝑘subscript𝜌𝐴𝐵0\displaystyle\beginpmatrix\rho_AB&X\\ X^\dagger&\operatornameTr_\backslash AB_1(W^\Gamma_B_BB^k\rho_% AB)\endpmatrix\geq 0( start_ARG start_ROW start_CELL italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT end_CELL start_CELL italic_X end_CELL end_ROW start_ROW start_CELL italic_X start_POSTSUPERSCRIPT † end_POSTSUPERSCRIPT end_CELL start_CELL roman_Tr start_POSTSUBSCRIPT \ italic_A italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_W start_POSTSUPERSCRIPT roman_Γ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B italic_B start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ) end_CELL end_ROW end_ARG ) ≥ 0 (12d)
WBBk≥0subscript𝑊𝐵superscript𝐵𝑘0\displaystyle W_BB^k\geq 0italic_W start_POSTSUBSCRIPT italic_B italic_B start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ≥ 0 (12e)
WB=𝟙𝔹subscript𝑊𝐵subscript𝟙𝔹\displaystyle W_B=\openone_Bitalic_W start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT = blackboard_1 start_POSTSUBSCRIPT blackboard_B end_POSTSUBSCRIPT (12f)
WBBk=ΠBk+WBBkΠBk+.subscript𝑊𝐵superscript𝐵𝑘subscriptsuperscriptΠsuperscript𝐵𝑘subscript𝑊𝐵superscript𝐵𝑘subscriptsuperscriptΠsuperscript𝐵𝑘\displaystyle W_BB^k=\Pi^+_B^kW_BB^k\Pi^+_B^k.italic_W start_POSTSUBSCRIPT italic_B italic_B start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = roman_Π start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_W start_POSTSUBSCRIPT italic_B italic_B start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_POSTSUBSCRIPT roman_Π start_POSTSUPERSCRIPT + end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_POSTSUBSCRIPT . (12g)
We already argued that DF,ℒSym+(k)(A:B¯)fragmentssubscript𝐷𝐹superscriptℒsubscriptSym𝑘fragments(A:¯𝐵)D_F,\mathcalL^\textrmSym_+(k)(A:\underlineB)italic_D start_POSTSUBSCRIPT italic_F , caligraphic_L start_POSTSUPERSCRIPT Sym start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ( italic_k ) end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_A : under¯ start_ARG italic_B end_ARG ) converges to DF,ℒEB(A:B¯)fragmentssubscript𝐷𝐹superscriptℒEBfragments(A:¯𝐵)D_F,\mathcalL^\textrmEB(A:\underlineB)italic_D start_POSTSUBSCRIPT italic_F , caligraphic_L start_POSTSUPERSCRIPT EB end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_A : under¯ start_ARG italic_B end_ARG ). To see that it does so monotonically, i.e., that DF,ℒSym(k+1)(A:B¯)≥DF,ℒSym+(k)(A:B¯)fragmentssubscript𝐷𝐹superscriptℒSym𝑘1fragments(A:¯𝐵)subscript𝐷𝐹superscriptℒsubscriptSym𝑘fragments(A:¯𝐵)D_F,\mathcalL^\textrmSym(k+1)(A:\underlineB)\geq D_F,\mathcalL^% \textrmSym_+(k)(A:\underlineB)italic_D start_POSTSUBSCRIPT italic_F , caligraphic_L start_POSTSUPERSCRIPT Sym ( italic_k + 1 ) end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_A : under¯ start_ARG italic_B end_ARG ) ≥ italic_D start_POSTSUBSCRIPT italic_F , caligraphic_L start_POSTSUPERSCRIPT Sym start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ( italic_k ) end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_A : under¯ start_ARG italic_B end_ARG ), it is enough to notice that, if WBBk+1subscript𝑊𝐵superscript𝐵𝑘1W_BB^k+1italic_W start_POSTSUBSCRIPT italic_B italic_B start_POSTSUPERSCRIPT italic_k + 1 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT is Bose-symmetric on Bk+1superscript𝐵𝑘1B^k+1italic_B start_POSTSUPERSCRIPT italic_k + 1 end_POSTSUPERSCRIPT, then TrBk+1(WBBk+1)subscriptTrsubscript𝐵𝑘1subscript𝑊𝐵superscript𝐵𝑘1\operatornameTr_B_k+1(W_BB^k+1)roman_Tr start_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT italic_k + 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_W start_POSTSUBSCRIPT italic_B italic_B start_POSTSUPERSCRIPT italic_k + 1 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) is Bose-symmetric on Bksuperscript𝐵𝑘B^kitalic_B start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT. We also remark again that DF,ℒSym(2)(A:B¯)fragmentssubscript𝐷𝐹superscriptℒSym2fragments(A:¯𝐵)D_F,\mathcalL^\textrmSym(2)(A:\underlineB)italic_D start_POSTSUBSCRIPT italic_F , caligraphic_L start_POSTSUPERSCRIPT Sym ( 2 ) end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_A : under¯ start_ARG italic_B end_ARG ) is already a faithful quantifier of discord, in the sense that, thanks to the no-local-broadcasting theorem, we know it is strictly positive for any state that is not classical on B𝐵Bitalic_B. Finally, thanks to the properties of the fidelity F𝐹Fitalic_F, in particular its monotonicity under quantum operations, i.e., F(Λ[σ],Λ[ρ])≥F(σ,ρ)𝐹Λdelimited-[]𝜎Λdelimited-[]𝜌𝐹𝜎𝜌F(\Lambda[\sigma],\Lambda[\rho])\geq F(\sigma,\rho)italic_F ( roman_Λ [ italic_σ ] , roman_Λ [ italic_ρ ] ) ≥ italic_F ( italic_σ , italic_ρ ) Nielsen and Chuang (2010), it is immediate to check that each DF,ℒSym+(k)(A:B¯)fragmentssubscript𝐷𝐹superscriptℒsubscriptSym𝑘fragments(A:¯𝐵)D_F,\mathcalL^\textrmSym_+(k)(A:\underlineB)italic_D start_POSTSUBSCRIPT italic_F , caligraphic_L start_POSTSUPERSCRIPT Sym start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ( italic_k ) end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( italic_A : under¯ start_ARG italic_B end_ARG ) is invariant under local unitaries on B𝐵Bitalic_B, and monotonically decreasing under general local operations on A𝐴Aitalic_A 222A detailed proof for the case of DF,ℒEBsubscript𝐷𝐹superscriptℒEBD_F,\mathcalL^\textrmEBitalic_D start_POSTSUBSCRIPT italic_F , caligraphic_L start_POSTSUPERSCRIPT EB end_POSTSUPERSCRIPT end_POSTSUBSCRIPT, which can be immediately adapted to DF,ℒSym+(k)subscript𝐷𝐹superscriptℒsubscriptSym𝑘D_F,\mathcalL^\textrmSym_+(k)italic_D start_POSTSUBSCRIPT italic_F , caligraphic_L start_POSTSUPERSCRIPT Sym start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ( italic_k ) end_POSTSUPERSCRIPT end_POSTSUBSCRIPT, is presented in Seshadreesan and Wilde (2014).. Thus, each DF,ℒSym+(k)subscript𝐷𝐹superscriptℒsubscriptSym𝑘D_F,\mathcalL^\textrmSym_+(k)italic_D start_POSTSUBSCRIPT italic_F , caligraphic_L start_POSTSUPERSCRIPT Sym start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ( italic_k ) end_POSTSUPERSCRIPT end_POSTSUBSCRIPT, in particular in the case k=2𝑘2k=2italic_k = 2, constitutes in itself a well-behaved measure of the general quantumness of correlations Brodutch and Modi (2012); Piani (2012).
Notice that, if the goal is that of lower-bounding the surprisal of measurement recoverability-and in turn standard discord-rather than just considering a class of physical channels like Bose-symmetric extendible ones, we can impose additional ‘unphysical’ properties that nonetheless make the considered class more closely approximate the class of entanglement-breaking channels. Correspondingly, the SDP optimization (Hierarchy of efficiently computable and faithful lower bounds to quantum discord) can be modified to include additional constraints, in particular asking for WBBksubscript𝑊𝐵superscript𝐵𝑘W_BB^kitalic_W start_POSTSUBSCRIPT italic_B italic_B start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_POSTSUBSCRIPT to be positive under partial transposition (PPT) in any bipartite cut. In particular, simply by asking that it is PPT with respect to the B:Bk:𝐵superscript𝐵𝑘B:B^kitalic_B : italic_B start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT partition, e.g., by adding to (Hierarchy of efficiently computable and faithful lower bounds to quantum discord) the condition WBBkΓB≥0superscriptsubscript𝑊𝐵superscript𝐵𝑘subscriptΓ𝐵0W_BB^k^\Gamma_B\geq 0italic_W start_POSTSUBSCRIPT italic_B italic_B start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_Γ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ≥ 0, we make the corresponding k𝑘kitalic_k-Bose-extendible channel PPT binding Horodecki et al. (2000), i.e., such that the state (idA⊗ΛB)[σAB]tensor-productsubscriptid𝐴subscriptΛ𝐵delimited-[]subscript𝜎𝐴𝐵(\rm id_A\otimes\Lambda_B)[\sigma_AB]( roman_id start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⊗ roman_Λ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ) [ italic_σ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT ] is PPT for all σABsubscript𝜎𝐴𝐵\sigma_ABitalic_σ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT. This is a non-trivial constraint also for the case k=1𝑘1k=1italic_k = 1, and, in the case |B|=2𝐵2|B|=2| italic_B | = 2, enough to make the channel entanglement breaking Horodecki et al. (1996) so that in this case the solution to the SDP provides exactly the surprisal of measurement recoverability. We implemented (Hierarchy of efficiently computable and faithful lower bounds to quantum discord) in MATLAB MATLAB (2014), making use of CVX Grant and Boyd (2014, 2008) and other tools publicly available Johnston (2015); Cubitt (2015). An example of the results is presented in Figure 2.
Discord, entanglement, and symmetric extensions.-Our approach, based on an SDP hierarchy dealing with symmetric extensions, is inspired by and very similar to the one used to verify entanglement Doherty et al. (2002, 2004) (see also Nowakowski and Horodecki (2009) for applications to the extendability of channels). In turn, the fact that fidelity can be expressed as an SDP program, which we exploited here, could also be adopted for the study and quantification of entanglement, providing a hierarchy of SDP programs that allows to calculate the largest fidelity of the given state ρABsubscript𝜌𝐴𝐵\rho_ABitalic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT with any state σABSym(k)subscriptsuperscript𝜎Sym(k)𝐴𝐵\sigma^\textrmSym(k)_ABitalic_σ start_POSTSUPERSCRIPT Sym(k) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT admitting a (Bose-)k𝑘kitalic_k-symmetric extension on B𝐵Bitalic_B, and converging to the fidelity of separability Streltsov et al. (2010). Our approach points to a illuminating conceptual relation between entanglement and discord, in terms of symmetric extensions and how they are generated: entanglement limits how well a state can be approximated by a state admitting a k𝑘kitalic_k-symmetric extension, and only separable states can be perfectly approximated for all k≥2𝑘2k\geq 2italic_k ≥ 2; on the other hand, discord limits how well a state can be locally transformed into a (Bose-)k𝑘kitalic_k-symmetric extension of itself, with only discord-free states that can be perfectly locally broadcast, for any k≥2𝑘2k\geq 2italic_k ≥ 2. Remarkably, while entanglement can be exactly characterized only in the limit k→∞→𝑘k\rightarrow\inftyitalic_k → ∞, discord can be pinned down already by considering the case k=2𝑘2k=2italic_k = 2-this is the content of the no-local-broadcasting theorem. This explains why, while entanglement verification is hard Gurvits (2003); Ioannou (2007); gha (2010), our hierarchy provides a faithful, reliable, and efficiently computable lower bound to discord already at the lowest level.
Conclusions.-We have introduced a hierarchy of discord-like quantifiers. They are defined in terms of how well a given quantum state ρABsubscript𝜌𝐴𝐵\rho_ABitalic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT can be locally broadcast. More precisely, in the lowest non-trivial level of the hierarchy, our quantifier answers the following question: Consider any mapping from B𝐵Bitalic_B to the symmetric subspace of two copies B1B2subscript𝐵1subscript𝐵2B_1B_2italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT of B𝐵Bitalic_B; how well can the resulting ρAB1subscript𝜌𝐴subscript𝐵1\rho_AB_1italic_ρ start_POSTSUBSCRIPT italic_A italic_B start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT (equivalently, ρAB2subscript𝜌𝐴subscript𝐵2\rho_AB_2italic_ρ start_POSTSUBSCRIPT italic_A italic_B start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT) approximate the original ρABsubscript𝜌𝐴𝐵\rho_ABitalic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT? In the limit where we consider infinite copies of B𝐵Bitalic_B, instead of just two, the question becomes that of how well the information about B𝐵Bitalic_B contained in ρABsubscript𝜌𝐴𝐵\rho_ABitalic_ρ start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT can be transmitted (equivalently, stored) in the form of classical information, through a measure, transmit (store), and re-prepare process. Our hierarchy is faithful at all non-trivial levels, i.e., the quantifiers are non-vanishing for states that are not classical on B𝐵Bitalic_B. Each element in the hierarchy corresponds to an SDP optimization problem; hence, it can be reliably and efficiently (in the dimensions of the systems) computed numerically Doherty et al. (2002, 2004). Furthermore, while each element has a clear physical meaning in itself and satisfies the basic properties to be expected for a meaningful quantifier of the quantumness, it also constitutes a lower bound to the standard quantum discord. Remarkably, in the case in which we are interested in the discord features of a qubit-qudit system, with measurement on the qudit, a tailored SDP program can provide exactly, i.e., up to numerical error, the surprisal of measurement recoverability defined by Seshadreesan and Wilde Seshadreesan and Wilde (2014), and thus the best possible lower bound to standard quantum discord based on the breakthrough result about quantum Markov chains of Fawzi and Renner Fawzi and Renner (2014).
Acknowledgements-I acknowledge support from NSERC. I would like to thank K. P. Seshadreesan and M. M. Wilde for discussions, and G. Adesso for useful correspondence.