Lower semicontinuity of the relative entropy disturbance and its corollaries

TL;DR

This paper proves that the decrease of quantum relative entropy under quantum operations is lower semicontinuous, leading to important implications for its discontinuity jumps and convexity properties, with various applications.

Contribution

It establishes the lower semicontinuity of the quantum relative entropy decrease and related functions, providing new insights into quantum information theory.

Findings

Quantum relative entropy decrease is lower semicontinuous.

Discontinuity jumps do not increase under quantum operations.

Lower semicontinuity of the joint convexity modulus of quantum relative entropy.

Abstract

It is proved that the decrease of the quantum relative entropy under action of a quantum operation is a lower semicontinuous function of a pair of its arguments. This property implies, in particular, that the local discontinuity jumps of the quantum relative entropy do not increase under action of quantum operations. It implies also the lower semicontinuity of the modulus of the joint convexity of the quantum relative entropy (as a function of ensembles of quantum states). Various corollaries and applications of these results are considered.