科研之友
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摘要
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 TrK* 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(12)(p))(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.
- 出版日期2010-10
