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The general form of γfamily of quantum relative entropies
, 2011
"... We use the Falcone–Takesaki noncommutative flow of weights and the resulting theory of noncommutative Lp spaces in order to define the family of relative entropy functionals that naturally generalise the quantum relative entropies of Jenčová–Ojima and the classical relative entropies of Zhu–Rohwer ..."
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We use the Falcone–Takesaki noncommutative flow of weights and the resulting theory of noncommutative Lp spaces in order to define the family of relative entropy functionals that naturally generalise the quantum relative entropies of Jenčová–Ojima and the classical relative entropies of Zhu–Rohwer, and belong to an intersection of families of Petz relative entropies with Bregman relative entropies. For the purpose of this task, we generalise the notion of Bregman entropy to the infinitedimensional noncommutative case using the Legendre–Fenchel duality. In addition, we use the Falcone–Takesaki duality to extend the duality between coarse–grainings and Markov maps to the infinitedimensional noncommutative case. Following the recent result of Amari for the Zhu–Rohwer entropies, we conjecture that the proposed family of relative entropies is uniquely characterised by the Markov monotonicity and the Legendre–Fenchel duality. The role of these results in the foundations and applications of quantum information theory is discussed.
From quasientropy
"... The subject is the overview of the use of quasientropy in finite dimensional spaces. Operator monotone functions and relative modular operators are used. The origin is the relative entropy, and the fdivergence, monotone metrics, covariance and the χ2divergence are the most important particular ca ..."
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The subject is the overview of the use of quasientropy in finite dimensional spaces. Operator monotone functions and relative modular operators are used. The origin is the relative entropy, and the fdivergence, monotone metrics, covariance and the χ2divergence are the most important particular cases. The extension of monotone metrics to those with two parameters is a new concept. Monotone metrics are also characterized by their joint convexity property.
Publ. RIMS Kyoto Univ. 48(2012), 525–542. From quasientropy to various quantum information quantities
"... The subject is the applications of the use of quasientropy in finite dimensional spaces to many important quantities in quantum information. Operator monotone functions and relative modular operators are used. The origin is the relative entropy, and the fdivergence, monotone metrics, covariance an ..."
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The subject is the applications of the use of quasientropy in finite dimensional spaces to many important quantities in quantum information. Operator monotone functions and relative modular operators are used. The origin is the relative entropy, and the fdivergence, monotone metrics, covariance and the χ 2divergence are the most important particular cases. The extension of monotone metrics to those with two parameters is a new concept. Monotone metrics are also characterized by their joint convexity property.
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