Eur. Phys. J. Plus (2024) 139: 828
https://doi.org/10.1140/epjp/s13360-024-05612-2

Regular Article

On two classes of Rényi entropy functions of a quantum channel

¹ School of Mathematics and Statistics, Henan University, 475004, Kaifeng, Henan, China
² Laboratory of Quantum Science and Engineering, South China University of Technology, 510641, Guangzhou, Guangdong, China
³ Department of Mathematics, South China University of Technology, 510641, Guangzhou, Guangdong, China

^b huangfang@henu.edu.cn

Received: 5 September 2023
Accepted: 1 September 2024
Published online: 18 September 2024

Abstract

In this paper, we define the Rényi entropy $H_{\text{QC}}^p(\mathrm{\Phi })$ of a quantum channel $\mathrm{\Phi }$, leveraging the isomorphic correspondence between quantum channels and their Choi states. We demonstrate that it satisfies all axioms of an entropy function. Additionally, we introduce the Rényi relative entorpy $H_{\text{Re}}^p(\mathrm{\Phi })$ of the quantum channel $\mathrm{\Phi }$ and establish its relationship with $H_{\text{QC}}^p(\mathrm{\Phi })$. Furthermore, by the Rényi entropy $H_p(\mathrm{\Phi })$ of the quantum channel $\mathrm{\Phi }$ defined by Gour and Wilde, we proved that $H_{\text{QC}}^p(\mathrm{\Phi })$ serves as an upper bound for $H_p(\mathrm{\Phi })$ a general quantum channel $\mathrm{\Phi }$. As an example, we examined a specific type of Werner-Holevo channels and showed that these two Rényi entropies are equal for the parameter $p\in [\frac{1}{2},+\infty )$ and are a fixed constant number independent on the parameter p. Based on these results, we proved that the two classes of Rényi channel entropies are equal for finite-dimensional covariant channels and $p\in [\frac{1}{2},+\infty )$.