# Concavity of certain matrix trace and norm functions. II

@article{Hiai2012ConcavityOC,
title={Concavity of certain matrix trace and norm functions. II},
author={Fumio Hiai},
journal={arXiv: Functional Analysis},
year={2012}
}
• F. Hiai
• Published 28 October 2012
• Mathematics
• arXiv: Functional Analysis
Jointly convex mappings related to Lieb’s theorem and Minkowski type operator inequalities
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• 2021
Employing the notion of operator log-convexity, we study joint concavity/convexity of multivariable operator functions related to Lieb’s theorem: $$(A,B)\mapsto F(A,B)=\psi \left[ \Phi (f(A))\ \sigma Dimension reduction as an optimization problem over a set of generalized functions The problem is reduced to the minimization of a certain loss function I(q) measuring the distance from q to p_{\rm{emp}} over a pertinent set of generalized functions, denoted \mathcal{G}_k. Convexity of a certain operator trace functional • Mathematics Linear Algebra and its Applications • 2022 Inequalities for quantum divergences and the Audenaert–Datta conjecture • Computer Science Journal of Physics A: Mathematical and Theoretical • 2018 The problem, its context, and the methods that have been used to obtain the results that are known at present are reviewed, presenting a unified treatment of developments that have unfolded in a number of different papers. Quantum Markov chains, sufficiency of quantum channels, and Renyi information measures • Mathematics, Computer Science ArXiv • 2015 This paper provides an alternate characterization of short quantum Markov chains and sufficient quantum channels, and gives further support to these quantities as being legitimate Renyi generalizations of the conditional mutual information and the relative entropy difference. Data processing for the sandwiched Rényi divergence: a condition for equality • Computer Science • 2017 A necessary and sufficient algebraic condition for equality in the data processing inequality for the α-sandwiched Rényi divergence is derived and the known upper bound on the entanglement fidelity in terms of the usual fidelity is saturated only by pure states. Quantitative combinatorial geometry for concave functions • Mathematics J. Comb. Theory, Ser. A • 2021 ## References SHOWING 1-10 OF 26 REFERENCES Norm and anti-norm inequalities for positive semi-definite matrices • Mathematics • 2010 Some subadditivity results involving symmetric (unitarily invariant) norms are obtained. For instance, if g(t)=\sum_{k=0}^m a_kt^k is a polynomial of degree m with non-negative coefficients, A Minkowski Type Trace Inequality and Strong Subadditivity of Quantum Entropy II: Convexity and Concavity • Mathematics • 2008 AbstractWe revisit and prove some convexity inequalities for trace functions conjectured in this paper’s antecedent. The main functional considered is$$ \Phi_{p,q} (A_1,\, A_2, \ldots, A_m) =
A Minkowski Type Trace Inequality and Strong Subadditivity of Quantum Entropy
• Mathematics
• 2002
We consider the following trace function on n-tuples of positive operators: $${\Phi _P}({A_1},{A_2},...,{A_n}) = Tr({(\sum\limits_{j = 1}^n {A_j^P} )^{1/P}})$$ and prove that it is jointly
Convex Trace Functions and the Wigner-Yanase-Dyson Conjecture
• H.
• Mathematics
This paper is concerned with certain convex or concave mappings of linear operators on a Hilbert space into the reals. [f(A) is convex if f(ilA + (1 il)B) <; t..j(A) + (1 il)f(B) for 0 < il < 1 and
A UNIFIED TREATMENT OF CONVEXITY OF RELATIVE ENTROPY AND RELATED TRACE FUNCTIONS, WITH CONDITIONS FOR EQUALITY
• Mathematics
• 2010
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
CONCAVITY OF CERTAIN MATRIX TRACE FUNCTIONS
We demonstrate how Epstein's method using theory of Pick func-tions improves the existing results and also proves new ones on the joint con-cavity of trace functions of the form Tr (F(A1, . . .