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International Journal of Quantum Information

LIEB'S SIMPLE PROOF OF CONCAVITY OF (A, B) ↦ Tr Ap K† B1-p K AND REMARKS ON RELATED INEQUALITIES

Mary Beth Ruskai

Published Sep 1, 2005• 15 citations
Publisher

Abstract

A simple, self-contained proof is presented for the concavity of the map (A, B) ↦ Tr Ap K† B1-p K. The author makes no claim to originality; this paper gives Lieb's original argument in its simplest, rather than its most general, form. A sketch of the chain of implications from this result to concavity of A ↦ Tr eK+ log A is then presented. An independent elementary proof is given for the joint convexity of the map [Formula: see text], which plays a key role in entropy inequalities.

DOI
10.1142/s0219749905001109

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A simple, self-contained proof is presented for the concavity of the map (A, B) ↦ Tr Ap K† B1-p K. The author makes no claim to originality; this paper gives Lieb's original argument in its simplest, rather than its most general, form. A sketch of the chain of implications from this result to concavity of A ↦ Tr eK+ log A is then presented. An independent elementary proof is given for the joint convexity of the map [Formula: see text], which plays a key role in entropy inequalities.
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 convexity of relative entropy, and for Tr K* ApK B1-pLieb's joint concavity in (A, B) for 0 < p < 1 and Ando's joint convexity for 1 < p ≤ 2. This approach allows us to obtain conditions for equality in these cases, as well as conditions for equality in a number of inequalities which follow from them. These include the monotonicity under partial traces, and some Minkowski type matrix inequalities proved by Carlen and Lieb for [Formula: see text]. In all cases, the equality conditions are independent of p; for extensions to three spaces they are identical to the conditions for equality in the strong subadditivity of relative entropy.
We derive the monotonicity of the quantum relative entropy by an elementary operational argument based on Stein's lemma in quantum hypothesis testing. For the latter we present an elementary and short proof that requires the law of large numbers only. Joint convexity of the quantum relative entropy is proven too, resulting in a self-contained elementary version of Tropp's approach to Lieb's concavity theorem, according to which the map tr(exp(h+log a)) is concave in a on positive operators for self-adjoint h.
Many trace inequalities can be expressed either as concavity/convexity theorems or as monotonicity theorems. A classic example is the joint convexity of the quantum relative entropy which is equivalent to the Data Processing Inequality. The latter says that quantum operations can never increase the relative entropy. The monotonicity versions often have many advantages, and often have direct physical application, as in the example just mentioned. Moreover, the monotonicity results are often valid for a larger class of maps than, say, quantum operations (which are completely positive). In this paper we prove several new monotonicity results, the first of which is a monotonicity theorem that has as a simple corollary a celebrated concavity theorem of Epstein. Our starting points are the monotonicity versions of the Lieb Concavity and the Lieb Convexity Theorems. We also give two new proofs of these in their general forms using interpolation. We then prove our new monotonicity theorems by several duality arguments.
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.