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A unified treatment of convexity of relative entropy and related trace functions, with conditions for equality, Rev
 Math. Phys
"... We introduce a generalization of relative entropy derived from the WignerYanaseDyson entropy and give a simple, selfcontained proof that it is convex. Moreover, special cases yield the joint convexity of relative entropy, and for Tr K∗ApKB 1−p Lieb’s joint concavity in (A,B) for 0 < p < 1 a ..."
Abstract

Cited by 16 (2 self)
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We introduce a generalization of relative entropy derived from the WignerYanaseDyson entropy and give a simple, selfcontained proof that it is convex. Moreover, special cases yield the joint convexity of relative entropy, and for Tr K∗ApKB 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 Lieb and Carlen for Tr1(Tr2 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. Supported by the grants VEGA 2/0032/09 and APVV 007106
Rev. Math. Phys. 23, 691–747 (2011). Quantum fdivergences and error correction
"... Quantum fdivergences are a quantum generalization of the classical notion of fdivergences, and are a special case of Petz ’ quasientropies. Many wellknown distinguishability measures of quantum states are given by, or derived from, fdivergences; special examples include the quantum relative ent ..."
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Quantum fdivergences are a quantum generalization of the classical notion of fdivergences, and are a special case of Petz ’ quasientropies. Many wellknown distinguishability measures of quantum states are given by, or derived from, fdivergences; special examples include the quantum relative entropy, the Rényi relative entropies, and the Chernoff and Hoeffding measures. Here we show that the quantum fdivergences are monotonic under substochastic maps whenever the defining function is operator convex. This extends and unifies all previously known monotonicity results for this class of distinguishability measures. We also analyze the case where the monotonicity inequality holds with equality, and extend Petz ’ reversibility theorem for a large class of fdivergences and other distinguishability measures. We apply our findings to the problem of quantum error correction, and show that if a stochastic map preserves the pairwise distinguishability on a set of states, as measured by a suitable fdivergence, then its action can be reversed on that set by another stochastic map that can be constructed from the original one in a canonical way. We also provide an integral representation for operator convex