Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Some Trace Inequalities of Quantum Information

Bo Liu

Published Jan 1, 2010• 0 citations
OpenAlex

Abstract

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.

Related papers

Yanagi Trace Inequality of Quantum Information Theory

Bo Liu

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.

•Jan 1, 2010•0 citations

Some trace inequalities for exponential and logarithmic functions

Eric A. Carlen, Élliott H. Lieb

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.

Bulletin of Mathematical Sciences•May 29, 2018•4 citations

More on a trace inequality in quantum information theory

Naresh Kumar Sharma

It is known that for a completely positive and trace preserving (cptp) map ${\cal N}$, $\text{Tr}$ $\exp$$\{ \log σ$ $+$ ${\cal N}^\dagger [\log {\cal N}(ρ)$ $-\log {\cal N}(σ)] \}$ $\leqslant$ $\text{Tr}$ $ρ$ when $ρ$, $σ$, ${\cal N}(ρ)$, and ${\cal N}(σ)$ 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.

arXiv:1512.00226•Dec 1, 2015•1 citation

A Determinantal Inequality Involving Partial Traces

Minghua Lin

Abstract Let A be a density matrix in . Audenaert [J. Math. Phys. 48(2007) 083507] proved an inequality for Schatten p-norms: where Tr1 and Tr2 stand for the first and second partial trace, respectively. As an analogue of his result, we prove a determinantal inequality

Canadian Mathematical Bulletin•Jan 4, 2016•18 citations

Schatten q-norms and determinantal inequalities for matrices with numerical ranges in a sector

Junjian Yang, Linzhang Lu, Zhen Chen

When A is a density matrix in , Audenaert [J Math Phys. 2007;48] proved an interesting inequality for Schatten q-norms where and stand for the first and second partial trace, respectively, a notion from quantum information theory. As an analogue of this result, Lin [Canad Math Bull. 2016;59] recently presented the following determinantal inequality In this note, we first correct a result of Kuai [Linear Multilinear Algebra. doi:10.1080/03081087.2017.1304521] and then generalize the above two inequalities to a larger class of matrices whose numerical ranges are contained in a sector.

Linear and Multilinear Algebra•Dec 22, 2017•5 citations

Interpolation functions and inequalities

Hiroyuki Osaka, Yoichi Udagawa

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)|).

Linear and Multilinear Algebra•Mar 18, 2021•0 citations
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.
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.
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.
When A is a density matrix in , Audenaert [J Math Phys. 2007;48] proved an interesting inequality for Schatten q-norms where and stand for the first and second partial trace, respectively, a notion from quantum information theory. As an analogue of this result, Lin [Canad Math Bull. 2016;59] recently presented the following determinantal inequality In this note, we first correct a result of Kuai [Linear Multilinear Algebra. doi:10.1080/03081087.2017.1304521] and then generalize the above two inequalities to a larger class of matrices whose numerical ranges are contained in a sector.