Von Neumann entropy and majorization

TL;DR

This paper explores the properties of Shannon entropy under majorization, extends Uhlmann's theorem to infinite dimensions, and characterizes quantum channels that preserve entropy as isometric operations.

Contribution

It generalizes Uhlmann's theorem to infinite-dimensional Hilbert spaces and characterizes entropy-preserving quantum channels as isometries.

Findings

Shannon entropy respects majorization relations between distributions.

Uhlmann's theorem is extended to infinite-dimensional spaces.

Quantum channels preserving entropy are exactly isometric operations.

Abstract

We consider the properties of the Shannon entropy for two probability distributions which stand in the relationship of majorization. Then we give a generalization of a theorem due to Uhlmann, extending it to infinite dimensional Hilbert spaces. Finally we show that for any quantum channel $\Phi$, one has $S(\Phi(\rho))=S(\rho)$ for all quantum states $\rho$ if and only if there exists an isometric operator $V$ such that $\Phi(\rho)=V\rho V^*$.