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From quasientropy to various quantum information quantities
 PUBL. RIMS KYOTO UNIV. 48(2012), 525–542.
, 2012
"... 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.
Concavity of certain matrix trace and norm functions. Linear Algebra and its
 Applications
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Remarks on Kim’s strong subadditivity matrix inequality: extensions and equality conditions
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Quantum fdivergences and error correction
 REV. MATH. PHYS. 23, 691–747 (2011).
, 2011
"... 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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Cited by 3 (0 self)
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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
Convexity of quasientropy type functions: Lieb’s and Ando’s convexity theorems revisited
, 2012
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Convexity of quasientropy type functions: Lieb’s and Ando’s convexity theorems revisited
 J. MATH. PHYS. 54, 062201 (2013)
, 2013
"... Given a positive function f on (0, ∞) and a nonzero real parameter θ, we consider a function I θ f (A, B, X) = Tr X ∗ (f(LAR −1 B)RB) −θ (X) in three matrices A, B> 0 and X. This generalizes the notion of monotone metrics on positive definite matrices, and in the literature θ = ±1 has been typi ..."
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Given a positive function f on (0, ∞) and a nonzero real parameter θ, we consider a function I θ f (A, B, X) = Tr X ∗ (f(LAR −1 B)RB) −θ (X) in three matrices A, B> 0 and X. This generalizes the notion of monotone metrics on positive definite matrices, and in the literature θ = ±1 has been typical. We investigate how operator monotony of f is sufficient and/or necessary for joint convexity/concavity of Iθ f (A, B, X). Similar discussions are given for quasientropies and quantum skew informations.