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Linear and Multilinear Algebra

Interpolation functions and inequalities

Hiroyuki Osaka, Yoichi Udagawa

Published Mar 18, 2021• 0 citations
Publisher

Abstract

Using interpolation classes and the generalized Petz's trace inequality we give a new matrix inequality which might play an important role in the quantum information theory: For any pair of positive definite matrices A,B∈Mn(C) we have Tr⁡((A−B)(exp⁡(A)−exp⁡(B))≤Tr⁡(|A(exp⁡(A)−1)−B(exp⁡(B)−1)|).
DOI
10.1080/03081087.2021.1901840

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Aim: Supposebe Si be 2×2 is density matrix,π={πi}ni=1 is any a probability distribution,and A(s)≡∑i=1 n πiS11+si is invertible,for any 0≤s≤1,H(x)=-xlogx,and then Tr[A(s)s{∑j=1 n πjS11+sj(logS11+sj)2}-A(s)-1+s{∑j=1 n πjH(S11+sj)}2]≥0 which is a generalization of an inequality proved by K.Yanagi and others.Method: These problems are settled by applying Caushy-schwarz inequality,Jensen's inequality and some property of trace.Results: These inequalities related matrix logarithm give partial answer of the open problem posed by Yanagi.Conclusion: Some further studies on this base will been done because the results of those trace inequalities is just one case.
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