On quantum quasi-relative entropy

TL;DR

This paper introduces bounds and error terms for quantum quasi-relative entropy and related inequalities, enhancing understanding of quantum information measures like relative entropy and skew information.

Contribution

It provides new error bounds for quantum quasi-relative entropy and generalizes operator inequalities, including strong subadditivity, with explicit bounds for key functions.

Findings

Error bounds for monotonicity and joint convexity of $S_f^K$

Explicit bounds for logarithmic and power functions related to quantum entropy

Remainder terms for strong subadditivity and Cauchy-Schwartz inequalities

Abstract

We consider a quantum quasi-relative entropy $S_f^K$ for an operator $K$ and an operator convex function $f$. We show how to obtain the error bounds for the monotonicity and joint convexity inequalities from the recent results for the $f$-divergences (i.e. $K=I$). We also provide an error term for a class of operator inequalities, that generalize 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.