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Journal of Physics A Mathematical and General

Convexity inequalities for estimating free energy and relative entropy

Mary Beth Ruskai, Frank H. Stillinger

Published Jun 21, 1990• 38 citations
Publisher

Abstract

For quantum systems, it is shown that the relative entropy S(P,Q)=-Tr P log P+Tr P log Q of two positive semi-definite operators P and Q satisfies gamma -1 Tr(P-P 1+ gamma Q - gamma )<or=S(P,Q)<or= gamma -1 Tr(P 1- gamma Q gamma -P) for 0( gamma <or=1, and that these bounds become exact in the limit gamma to Phi . Analogous inequalities hold, for states in classical statistical mechanics or information theory, with trace replaced by integration or summation. Furthermore, the average of these bounds is, in general a better approximation to S(P,Q) than either bound alone, and the average is amenable to further improvement via repeated Richardson extrapolation. IF P and Q are Gibbs equilibrium states, then these inequalities can also be used to obtain bounds on the free energy of a perturbed system in terms of the free energy of the unperturbed system and expectations of the perturbation.

DOI
10.1088/0305-4470/23/12/023

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For quantum systems, it is shown that the relative entropy S(P,Q)=-Tr P log P+Tr P log Q of two positive semi-definite operators P and Q satisfies gamma -1 Tr(P-P 1+ gamma Q - gamma )<or=S(P,Q)<or= gamma -1 Tr(P 1- gamma Q gamma -P) for 0( gamma <or=1, and that these bounds become exact in the limit gamma to Phi . Analogous inequalities hold, for states in classical statistical mechanics or information theory, with trace replaced by integration or summation. Furthermore, the average of these bounds is, in general a better approximation to S(P,Q) than either bound alone, and the average is amenable to further improvement via repeated Richardson extrapolation. IF P and Q are Gibbs equilibrium states, then these inequalities can also be used to obtain bounds on the free energy of a perturbed system in terms of the free energy of the unperturbed system and expectations of the perturbation.
This article presents self-contained proofs of the strong subadditivity inequality for von Neumann’s quantum entropy, S(ρ), and some related inequalities for the quantum relative entropy, most notably its convexity and its monotonicity under stochastic maps. Moreover, the approach presented here, which is based on Klein’s inequality and Lieb’s theorem that the function A→Tr eK+log A is concave, allows one to obtain conditions for equality. In the case of strong subadditivity, which states that S(ρ123)+S(ρ2)⩽S(ρ12)+S(ρ23) where the subscripts denote subsystems of a composite system, equality holds if and only if log ρ123=log ρ12−log ρ2+log ρ23. Using the fact that the Holevo bound on the accessible information in a quantum ensemble can be obtained as a consequence of the monotonicity of relative entropy, we show that equality can be attained for that bound only when the states in the ensemble commute. The article concludes with an Appendix giving a short description of Epstein’s elegant proof of Lieb’s theorem.
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.
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.
The positivity conditions of the relative entropy between two thermal equilibrium states ρ[over ̂]_{1} and ρ[over ̂]_{2} are used to obtain upper and lower bounds for the subtraction of their entropies, the Helmholtz potential and the Gibbs potential of the two systems. These limits are expressed in terms of the mean values of the Hamiltonians, number operator, and temperature of the different systems. In particular, we discuss these limits for molecules that can be represented in terms of the Franck-Condon coefficients. We emphasize the case where the Hamiltonians belong to the same system at two different times t and t^{'}. Finally, these bounds are obtained for a general qubit system and for the harmonic oscillator with a time-dependent frequency at two different times.
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.
We consider a quantum quasi-relative entropy [Formula: see text] for an operator [Formula: see text] and an operator convex function [Formula: see text]. We show how to obtain the error bounds for the monotonicity and joint convexity inequalities from the recent results for the [Formula: see text]-divergences (i.e. [Formula: see text]). We also provide an error term for a class of operator inequalities, that generalizes operator strong subadditivity inequality. We apply those results to demonstrate explicit bounds for the logarithmic function, that leads to the quantum relative entropy, and the power function, which gives, in particular, a Wigner–Yanase–Dyson skew information. In particular, we provide the remainder terms for the strong subadditivity inequality, operator strong subadditivity inequality, WYD-type inequalities, and the Cauchy–Schwartz inequality.