# Maps on positive definite operators preserving the quantum $\chi_\alpha^2$-divergence

## TL;DR

This paper characterizes all bijective maps on positive definite operators that preserve the quantum hi_lpha^2-divergence, showing they are implemented by unitary or antiunitary operators, extending to semidefinite and density operators.

## Contribution

It provides a complete description of structure-preserving maps for the quantum hi_lpha^2-divergence, including on various operator cones, revealing their unitary or antiunitary nature.

## Key findings

- Preservers are implemented by unitary or antiunitary operators.
- Results extend to positive semidefinite and density operators.
- Characterizes structure-preserving transformations for quantum divergence.

## Abstract

We describe the structure of all bijective maps on the cone of positive definite operators acting on a finite and at least two-dimensional complex Hilbert space which preserve the quantum $\chi_\alpha^2$-divergence for some $\alpha \in [0,1]$. We prove that any such transformation is necessarily implemented by either a unitary or an antiunitary operator. Similar results concerning maps on the cone of positive semidefinite operators as well as on the set of all density operators are also derived.