On quantum quasi-relative entropy

We consider a quantum quasi-relative entropy [Formula: see text] for an operator [Formula: see text] and an operator convex function [Formula: see text]. We show how to obtain the error bounds for the monotonicity and joint convexity inequalities from the recent results for the [Formula: see text]-divergences (i.e. [Formula: see text]). We also provide an error term for a class of operator inequalities, that generalizes operator strong subadditivity inequality. We apply those results to demonstrate explicit bounds for the logarithmic function, that leads to the quantum relative entropy, and the power function, which gives, in particular, a Wigner–Yanase–Dyson skew information. In particular, we provide the remainder terms for the strong subadditivity inequality, operator strong subadditivity inequality, WYD-type inequalities, and the Cauchy–Schwartz inequality.

Singular value inequalities for Hilbert space operators

In this paper we show that if Ai,Bi,Xi are Hilbert space operators such that Xi is compact i = 1,2,..., n and f,g are non-negative continuous functions on [0,?) with f (t)g(t)=t for all t?[0,?), also h is non-negative increasing operator convex function on [0,?), then h(Sj(?n,i=1 ?iAi*Xi*Bi)) ? Sj(?n,i=1 ?ih(Ai* f(|Xi*|)2Ai) ? ?n,i=1 ?ih(Bi* g(|Xi|)2Bi)) for j = 1,2,... and ?n,i=1 ?i = 1. Also, applications of some inequalities are given.

Contraction of Generalized Relative Entropy Under Stochastic Mappings on Matrices

The contraction coefficient of stochastic mappings between full matrix algebras is introduced with respect to a generalized relative entropy labeled by an operator convex function g. It is proved that the coefficient is actually independent of g, in particular, it can be most conveniently computed by means of the square function.