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Yanagi Trace Inequality of Quantum Information Theory

Bo Liu

Published Jan 1, 2010• 0 citations
OpenAlex

Abstract

For selfadjoint matrices ρ and X,a skew information of Lieb's type Sf,g(ρ,X) is positive or negative if some conditions for realvalued(continuous) f and g is given.As another important application,yanagi trace inequality can be settled.That is,for positive matrices A and B,if every real number s,then Tr{(A+B)s[A(logA)2+B(logB)2]-(A+B)s-1(AlogA+BlogB)2}≥0 where 0≤s≤1.

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For selfadjoint matrices ρ and X,a skew information of Lieb's type Sf,g(ρ,X) is positive or negative if some conditions for realvalued(continuous) f and g is given.As another important application,yanagi trace inequality can be settled.That is,for positive matrices A and B,if every real number s,then Tr{(A+B)s[A(logA)2+B(logB)2]-(A+B)s-1(AlogA+BlogB)2}≥0 where 0≤s≤1.
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.
It is known that for a completely positive and trace preserving (cptp) map N{\cal N}N, Tr\text{Tr}Tr exp⁡\expexp{log⁡σ\{ \log σ{logσ +++ N†[log⁡N(ρ){\cal N}^\dagger [\log {\cal N}(ρ)N†[logN(ρ) −log⁡N(σ)]}-\log {\cal N}(σ)] \}−logN(σ)]} ⩽\leqslant⩽ Tr\text{Tr}Tr ρρρ when ρρρ, σσσ, N(ρ){\cal N}(ρ)N(ρ), and N(σ){\cal N}(σ)N(σ) are strictly positive. We state and prove a relevant version of this inequality for the hitherto unaddressed case of these matrices being nonnegative. Our treatment also provides an alternate proof for the strictly positive case.
Quantum entropy and skew information play important roles in quantum information science. They are defined by traces of positive operators, hence trace inequalities often have important roles to develop the mathematical theory in quantum information science. In this article, we study some properties for information quantities in quantum systems through trace inequalities. Especially, we give upper bounds and lower bounds on the Tsallis relative entropy, which is a one-parameter extension of the relative entropy in quantum systems. In addition, we compare the known bounds and the new bounds, for both upper and lower bounds, respectively. We also give an inequality for generalized skew information by introducing a generalized correlation measure.
Quantum entropy and skew information play important roles in quantum information science. They are defined by the trace of the positive operators so that the trace inequalities often have important roles to develop the mathematical theory in quantum information science. In this paper, we study some properties for information quantities in quantum system through trace inequalities. Especially, we give upper bounds and lower bounds of Tsallis relative entropy, which is a one-parameter extension of the relative entropy in quantum system. In addition, we compare the known bounds and the new bounds, for both upper and lower bounds, respectively. We also give an inequality for generalized skew information by introducing a generalized correlation measure.
Consider a function F(X,Y) of pairs of positive matrices with values in the positive matrices such that whenever X and Y commute F(X,Y)=XpYq. Our first main result gives conditions on F such that Tr[X log(F(Z,Y))]≤Tr[X(p log X+q log Y)] for all X,Y,Z such that TrZ=TrX. (Note that Z is absent from the right side of the inequality.) We give several examples of functions F to which the theorem applies. Our theorem allows us to give simple proofs of the well-known logarithmic inequalities of Hiai and Petz and several new generalizations of them which involve three variables X,Y,Z instead of just X,Y alone. The investigation of these logarithmic inequalities is closely connected with three quantum relative entropy functionals: The standard Umegaki quantum relative entropy D(X∥Y)=Tr[X(log X− log Y]), and two others, the Donald relative entropy DD(X∥Y), and the Belavkin–Stasewski relative entropy DBS(X∥Y). They are known to satisfy DD(X∥Y)≤D(X∥Y)≤DBS(X∥Y). We prove that the Donald relative entropy provides the sharp upper bound, independent of Z on Tr[X log(F(Z,Y))] in a number of cases in which F(Z,Y) is homogeneous of degree 1 in Z and −1 in Y. We also investigate the Legendre transforms in X of DD(X∥Y) and DBS(X∥Y), and show how our results for these lead to new refinements of the Golden–Thompson inequality.