Transformations on density operators and on positive definite operators preserving the quantum R\'enyi divergence

TL;DR

This paper generalizes the quantum Rényi divergence, characterizes all transformations that preserve it on density operators and positive definite operators, and describes their structural properties.

Contribution

It introduces a generalized form of quantum Rényi divergence and characterizes all symmetry transformations that preserve this divergence.

Findings

Identifies all transformations on density operators preserving the generalized divergence

Determines the structure of bijections on positive definite operators that preserve the divergence

Provides a comprehensive description of symmetry groups related to the quantum Rényi divergence

Abstract

In a certain sense we generalize the recently introduced and extensively studied notion called quantum R\'enyi divergence (in another name, sandwiched R\'enyi relative entropy) and describe the structures of corresponding symmetries. More precisely, we characterize all transformations on the set of density operators which leave our new general quantity invariant and also determine the structure of all bijective transformations on the cone of positive definite operators which preserve the quantum R\'enyi divergence.