Equality cases in monotonicity of quasi-entropies, Lieb's concavity and Ando's convexity

We revisit and improve joint concavity/convexity and monotonicity properties of quasi-entropies due to Petz in a new fashion. Then we characterize equality cases in the monotonicity inequalities (the data-processing inequalities) of quasi-entropies in several ways as follows: Let phi:B(H) -> B(K) be a trace-preserving map such that phi* is a Schwarz map. When f is an operator monotone or operator convex function on [0, infinity), we present several equivalent conditions for the equality S-f(K)(phi(rho)||phi(sigma))=S-f(phi*(K))(rho||sigma) to hold for given positive operators rho, sigma on H and K is an element of B(K). The conditions include equality cases in the monotonicity versions of Lieb's concavity and Ando's convexity theorems. Specializing the map phi we have equivalent conditions for equality cases in Lieb's concavity and Ando's convexity. Similar equality conditions are discussed also for monotone metrics and chi(2)-divergences. We further consider some types of linear preserver problems for those quantum information quantities.

Automated summary

This paper revisits and improves the joint concavity/convexity and monotonicity properties of quasi-entropies proposed by Petz. The authors characterize the equality cases in the monotonicity inequalities of quasi-entropies and provide equivalent conditions for the equality to hold. They also discuss the equality conditions for monotone metrics and chi(2)-divergences, and consider linear preserver problems for these quantum information quantities.