Another short and elementary proof of strong subadditivity of quantum entropy

Ruskai, Mary Beth

doi:10.1016/s0034-4877(07)00019-5

Cited by 30 publications

(32 citation statements)

References 28 publications

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“…It was subsequently used by Narnhofer and Thirring ( [29]) to give a new proof of SSA. The argument given here is similar to that in [18,31,37]; however, the unified treatment for 0 < p < 2 leading to equality conditions, is new. Moreover, a dual treatment can be given for −1 < p < 1 allowing extension to the full range (−1, 2).…”

Section: Introductionmentioning

confidence: 75%

A Unified Treatment of Convexity of Relative Entropy and Related Trace Functions, With Conditions for Equality

Jenčová

Ruskai

2010

Rev. Math. Phys.

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We consider a generalization of relative entropy derived from the Wigner-Yanase-Dyson entropy and give a simple, self-contained proof that it is convex. Moreover, special cases yield the joint convexity of relative entropy, and for Tr K * A p KB 1−p Lieb's joint concavity in (A, B) for 0 < p < 1 and Ando's joint convexity for 1 < p ≤ 2. This approach allows us to obtain conditions for equality in these cases, as well as conditions for equality in a number of inequalities which follow from them. These include the monotonicity under partial traces, and some Minkowski type matrix inequalities proved by Carlen and Lieb for Tr 1 (Tr 2 A p 12 ) 1/p . In all cases, the equality conditions are independent of p; for extensions to three spaces they are identical to the conditions for equality in the strong subadditivity of relative entropy. a In [23], only concavity of the conditional entropy was proved explicitly, but the same argument [36, Sec. V.B] yields joint convexity of the relative entropy. Independently, Lindblad ([26]) observed that this follows directly from (2) by differentiating at p = 1. Rev. Math. Phys. 2010.22:1099-1121. Downloaded from www.worldscientific.com by MONASH UNIVERSITY on 09/17/13. For personal use only. Unified Treatment of Convexity of Relative Entropy and Related Trace Functions 1101even for A = B. Although this might seem unnecessary for convexity and concavity questions, it is crucial to a unified treatment. Lieb also considered Tr K * A p KB q with p, q > 0 and 0 ≤ p + q ≤ 1 and Ando considered 1 < q ≤ p ≤ 2. In Sec. 2.2, we extend our results to this situation. However, we also show that for q = 1−p, equality holds only under trivial conditions. Therefore, we concentrate on the case q = 1 − p.Next, we introduce our notation and conventions. In Sec. 2, we first describe our generalization of relative entropy and prove its convexity; then consider the extension to q = 1 − p mentioned above; and finally prove monotonicity under partial traces including a generalization of strong subadditivity to p = 1. In Sec. 3, we consider several formulations of equality conditions. In Sec. 4, we show how to use these results to obtain equality conditions in the results of Lieb and Carlen ([7, 8]). For completeness, we include an appendix which contains the proof of a basic convexity result from [37] that is key to our results. Rev. Math. Phys. 2010.22:1099-1121. Downloaded from www.worldscientific.com by MONASH UNIVERSITY on 09/17/13. For personal use only.

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Section: Introductionmentioning

confidence: 75%

A Unified Treatment of Convexity of Relative Entropy and Related Trace Functions, With Conditions for Equality

Jenčová

Ruskai

2010

Rev. Math. Phys.

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“…The original proof is based on Lieb's theorem [106]. Simpler proofs were subsequently presented by Nielsen and Petz [126] and Ruskai [143], amongst others. In this book we proved this statement indirectly via the data-processing inequality for the relative entropy, which in turns follows by continuity from the data-processing inequality for the Rényi divergence in Chapter 4.…”

Section: Background and Further Readingmentioning

confidence: 99%

Quantum Information Processing with Finite Resources

Tomamichel¹

2016

SpringerBriefs in Mathematical Physics

284

398

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“…According to eq. (20), write |η k AB = (1 1 A ⊗ R k ) |φ + AA and |θ k CD = (1 1 C ⊗ S k ) |φ + CC . Then we have…”

Section: New Rank Inequalitiesmentioning

confidence: 99%

Inequalities for the ranks of multipartite quantum states

Cadney¹,

Huber

Linden³

et al. 2014

Linear Algebra and its Applications

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We investigate relations between the ranks of marginals of multipartite quantum states. These are the Schmidt ranks across all possible bipartitions and constitute a natural quantification of multipartite entanglement dimensionality. We show that there exist inequalities constraining the possible distribution of ranks. This is analogous to the case of von Neumann entropy (α-Rényi entropy for α = 1), where nontrivial inequalities constraining the distribution of entropies (such as e.g. strong subadditivity) are known. It was also recently discovered that all other α-Rényi entropies for α ∈ (0, 1) ∪ (1, ∞) satisfy only one trivial linear inequality (non-negativity) and the distribution of entropies for α ∈ (0, 1) is completely unconstrained beyond non-negativity. Our result resolves an important open question by showing that also the case of α = 0 (logarithm of the rank) is restricted by nontrivial linear relations and thus the cases of von Neumann entropy (i.e., α = 1) and 0-Rényi entropy are exceptionally interesting measures of entanglement in the multipartite setting.

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Another short and elementary proof of strong subadditivity of quantum entropy

Cited by 30 publications

References 28 publications

A Unified Treatment of Convexity of Relative Entropy and Related Trace Functions, With Conditions for Equality

A Unified Treatment of Convexity of Relative Entropy and Related Trace Functions, With Conditions for Equality

Quantum Information Processing with Finite Resources

Inequalities for the ranks of multipartite quantum states

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