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arXiv:1101.1070 · Proceedings of the American Mathematical Society

From joint convexity of quantum relative entropy to a concavity theorem of Lieb

Joel Tropp

Published Aug 4, 2011• 26 citations
arXivHTMLPDF

Abstract

This paper provides a succinct proof of a 1973 theorem of Lieb that establishes the concavity of a certain trace function. The development relies on a deep result from quantum information theory, the joint convexity of quantum relative entropy, as well as a recent argument due to Carlen and Lieb.
DOI
10.1090/s0002-9939-2011-11141-9

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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.
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.
We consider both known and not previously studied trace functions with applications in quantum physics. By using perspectives we obtain convexity statements for different notions of residual entropy, including the entropy gain of a quantum channel as studied by Holevo and others. We give new and simplified proofs of the Carlen-Lieb theorems concerning concavity or convexity of certain trace functions by making use of the theory of operator monotone functions. We then apply these methods in a study of new types of trace functions. Keywords: Trace function, convexity, entropy gain, residual entropy, operator monotone function.
Carlen-Frank-Lieb conjecture and monotonicity of α−z\alpha-zα−z R\'enyi relative entropy
In this paper we study the joint convexity/concavity of the trace functions Ψp,q,s(A,B)=Tr(Bq2K∗ApKBq2)s,  p,q,s∈R, \Psi_{p,q,s}(A,B)=\text{Tr}(B^{\frac{q}{2}}K^*A^{p}KB^{\frac{q}{2}})^s,~~p,q,s\in \mathbb{R}, Ψp,q,s​(A,B)=Tr(B2q​K∗ApKB2q​)s,  p,q,s∈R, where AAA and BBB are positive definite matrices and KKK is any fixed invertible matrix. We will give full range of (p,q,s)∈R3(p,q,s)\in\mathbb{R}^3(p,q,s)∈R3 for Ψp,q,s\Psi_{p,q,s}Ψp,q,s​ to be jointly convex/concave for all KKK. As a consequence, we confirm a conjecture of Carlen, Frank and Lieb. In particular, we confirm a weaker conjecture of Audenaert and Datta and obtain the full range of (α,z)(\alpha,z)(α,z) for α\alphaα-zzz Renyi relative entropies to be monotone under completely positive trace preserving maps. We also give simpler proofs of many known results, including the concavity of Ψp,0,1/p\Psi_{p,0,1/p}Ψp,0,1/p​ for 0<p<10<p<10<p<1 which was first proved by Epstein using complex analysis. The key is to reduce the problem to the joint convexity/concavity of the trace functions Ψp,1−p,1(A,B)=TrK∗ApKB1−p,  −1≤p≤1, \Psi_{p,1-p,1}(A,B)=\text{Tr} K^*A^{p}KB^{1-p},~~-1\le p\le 1, Ψp,1−p,1​(A,B)=TrK∗ApKB1−p,  −1≤p≤1, using a variational method.