New monotonicity property of the quantum relative entropy

TL;DR

This work studies how local discontinuities in the quantum relative entropy $D( ho||\sigma)$ behave under quantum channels. It proves that the local jump cannot increase under any quantum operation, formalized as $dj(D(\Phi( ho_n)||\Phi(\sigma_n))) \le dj(D( ho_n||\sigma_n))$. The proof leverages Lindblad's extension of $D$, Stinespring dilations, and projector-based truncations together with Donald identity to control limits, improving upon earlier results on local continuity. The result strengthens the monotone and continuity properties of quantum relative entropy and has implications for information processing in quantum systems.

Abstract

It is proved that the local discontinuity jumps of the quantum relative entropy do not increase under action of quantum channels and operations.