A Unified Treatment of Convexity of Relative Entropy and Related Trace Functions, With Conditions for Equality

Abstract

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 * A p KB 1−p Lieb'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 Tr 1 (Tr 2 A p 12 ) 1/p . 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. a In [23], only concavity of the conditional entropy was proved explicitly, but the same argument [36, Sec. V.B] yields joint convexity of the relative entropy. Independently, Lindblad ([26]) observed that this follows directly from (2) by differentiating at p = 1. Rev. Math. Phys. 2010.22:1099-1121. Downloaded from www.worldscientific.com by MONASH UNIVERSITY on 09/17/13. For personal use only. Unified Treatment of Convexity of Relative Entropy and Related Trace Functions 1101even for A = B. Although this might seem unnecessary for convexity and concavity questions, it is crucial to a unified treatment. Lieb also considered Tr K * A p KB q with p, q > 0 and 0 ≤ p + q ≤ 1 and Ando considered 1 < q ≤ p ≤ 2. In Sec. 2.2, we extend our results to this situation. However, we also show that for q = 1−p, equality holds only under trivial conditions. Therefore, we concentrate on the case q = 1 − p.Next, we introduce our notation and conventions. In Sec. 2, we first describe our generalization of relative entropy and prove its convexity; then consider the extension to q = 1 − p mentioned above; and finally prove monotonicity under partial traces including a generalization of strong subadditivity to p = 1. In Sec. 3, we consider several formulations of equality conditions. In Sec. 4, we show how to use these results to obtain equality conditions in the results of Lieb and Carlen ([7, 8]). For completeness, we include an appendix which contains the proof of a basic convexity result from [37] that is key to our results. Rev. Math. Phys. 2010.22:1099-1121. Downloaded from www.worldscientific.com by MONASH UNIVERSITY on 09/17/13. For personal use only.

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