2009/03/31 by Anna Jencova, Anna Jenčová, Mary Beth Ruskai · 62 citations
Computer Science · Mathematics · Physics and Astronomy · #Convexity #Economics #Entropy (arrow of time) #Kullback–Leibler divergence #Mathematical Inequalities and Applications #Mathematical functions and polynomials #Mathematics #Matrix Theory and Algorithms #Philosophy #Physics #Pure mathematics #Quantum mechanics #Statistics #TRACE (psycholinguistics) #math-ph #math.MP #msc:15A39 #msc:81P45 #quant-ph
paper · pdf · doi:10.1142/s0129055x10004144
published in Reviews in Mathematical Physics 22(09), 1099-1121 (World Scientific) · Final version to appear in Rev. Math. Phys. with many typos and minor errors corrected
arxiv created 2010/07/29 · openalex publication_date 2010/10/01 · arxiv updated 2015/05/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 K B 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 [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.