Convergence conditions for the quantum relative entropy and other applications of the deneralized quantum Dini lemma

TL;DR

This paper generalizes the quantum Dini lemma to analyze convergence of quantum information measures, providing new theorems and criteria for quantum relative entropy, mutual information, and von Neumann entropy.

Contribution

It introduces a generalized quantum Dini lemma considering sequences of functions, expanding the applicability of convergence analysis in quantum information theory.

Findings

Established new convergence conditions for quantum relative entropy.

Derived criteria for the mutual information of quantum channels.

Provided a simple convergence criterion for von Neumann entropy.

Abstract

We describe a generalized version of the result called quantum Dini lemma that was used previously for analysis of local continuity of basic correlation and entanglement measures. The generalization consists in considering sequences of functions instead of a single function. It allows us to expand the scope of possible applications of the method. We prove two general dominated convergence theorems and the theorem about preserving local continuity under convex mixtures. By using these theorems we obtain several convergence conditions for the quantum relative entropy and for the mutual information of a quantum channel considered as a function of a pair (channel, input state). A simple convergence criterion for the von Neumann entropy is also obtained.