Quantum $f$-divergence preserving maps on positive semidefinite operators acting on finite dimensional Hilbert spaces

TL;DR

This paper characterizes all bijections on positive semidefinite operators that preserve quantum $f$-divergence, showing they are implemented by unitary or antiunitary operators, thus revealing their structure.

Contribution

It provides a complete description of quantum $f$-divergence preserving maps on positive semidefinite operators for any strictly convex function.

Findings

Preserving maps are implemented by unitary or antiunitary operators.

The structure holds for arbitrary strictly convex functions.

The results apply to finite-dimensional Hilbert spaces.

Abstract

We determine the structure of all bijections on the cone of positive semidefinite operators which preserve the quantum $f$-divergence for an arbitrary strictly convex function $f$ defined on the positive halfline. It turns out that any such transformation is implemented by either a unitary or an antiunitary operator.