Structure of States Which Satisfy Strong Subadditivity of Quantum Entropy with Equality

Hayden, Patrick; Jozsa, Richard; Petz, Dénes; Winter, Andreas

doi:10.1007/s00220-004-1049-z

Cited by 382 publications

(610 citation statements)

References 24 publications

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“…Moreover, the operator D R n is in the fixed point space of α * 2,n if and only if it commutes with the Kraus operators L i,n for all i, [7].…”

Section: Be the Decomposition Of H Given By (B(h) S) There Is A Decmentioning

confidence: 99%

“…The case of equality was studied in several papers recently but always restricted to finite dimensional Hilbert spaces [7,10]. Our aim now is to allow infinite dimensional spaces.…”

Section: Exponential Familiesmentioning

confidence: 99%

“…Equality in the strong subadditivity is equivalent to sufficiency of the subalgebra B(H A ⊗ H B ) ⊗ C1 C for (B(H), S) where S := {ω ABC , ω A ⊗ ω BC }, and the latter is equivalent to The structure (29) of the density matrix ω ABC is similar to the finite dimensional situation discussed in [7,10], however the direct sum decomposition may be infinite.…”

Section: Theoremmentioning

confidence: 99%

See 2 more Smart Citations

Sufficiency in Quantum Statistical Inference

Jenčová

Petz

2006

Commun. Math. Phys.

129

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This paper attempts to develop a theory of sufficiency in the setting of non-commutative algebras parallel to the ideas in classical mathematical statistics. Sufficiency of a coarse-graining means that all information is extracted about the mutual relation of a given family of states. In the paper sufficient coarse-grainings are characterized in several equivalent ways and the non-commutative analogue of the factorization theorem is obtained. Among the applications the equality case for the strong subadditivity of the von Neumann entropy, the Imoto-Koashi theorem and exponential families are treated. The setting of the paper allows the underlying Hilbert space to be infinite dimensional.MSC: 46L53, 81R15, 62B05.

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“…Moreover, the operator D R n is in the fixed point space of α * 2,n if and only if it commutes with the Kraus operators L i,n for all i, [7].…”

Section: Be the Decomposition Of H Given By (B(h) S) There Is A Decmentioning

confidence: 99%

“…The case of equality was studied in several papers recently but always restricted to finite dimensional Hilbert spaces [7,10]. Our aim now is to allow infinite dimensional spaces.…”

Section: Exponential Familiesmentioning

confidence: 99%

Section: Theoremmentioning

confidence: 99%

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Sufficiency in Quantum Statistical Inference

Jenčová

Petz

2006

Commun. Math. Phys.

129

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“…The latter is fully analogous to the classical case of a Markov chain, where A and C are independent conditional on B (which we call the "pivot" of the chain), as explained in [4].…”

Section: Introductionmentioning

confidence: 99%

A New Inequality for the von Neumann Entropy

Linden

Winter

2005

Commun. Math. Phys.

Self Cite

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Strong subadditivity of von Neumann entropy, proved in 1973 by Lieb and Ruskai, is a cornerstone of quantum coding theory. All other known inequalities for entropies of quantum systems may be derived from it. Here we prove a new inequality for the von Neumann entropy which we prove is independent of strong subadditivity: it is an inequality which is true for any four party quantum state, provided that it satisfies three linear relations (constraints) on the entropies of certain reduced states.

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“…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 [33][34][35] 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 [27,28,36], 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 [32,[37][38][39].…”

mentioning

confidence: 99%

Hierarchy of Efficiently Computable and Faithful Lower Bounds to Quantum Discord

Piani

2016

Phys. Rev. Lett.

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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. PACS numbers:Introduction.-Correlations in quantum mechanics exhibit non-classical features that include non-locality [1], steering [2], entanglement [3], and quantum discord [4]. Quantum correlations play a fundamental role in quantum information processing and quantum technologies [5], which go from quantum cryptography [6] to quantum metrology [7]. 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 [8]. 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. [4,[9][10][11][12][13][14][15][16][17][18][19][20][21][22] 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.

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Structure of States Which Satisfy Strong Subadditivity of Quantum Entropy with Equality

Cited by 382 publications

References 24 publications

Sufficiency in Quantum Statistical Inference

Sufficiency in Quantum Statistical Inference

A New Inequality for the von Neumann Entropy

Hierarchy of Efficiently Computable and Faithful Lower Bounds to Quantum Discord

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