On two classes of Rényi entropy functions of a quantum channel

In this paper, we define the Rényi entropy H QC p ( Φ ) of a quantum channel Φ , 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 Re p ( Φ ) of the quantum channel Φ and establish its relationship with H QC p ( Φ ) . Furthermore, by the Rényi entropy H p ( Φ ) of the quantum channel Φ defined by Gour and Wilde, we proved that H QC p ( Φ ) serves as an upper bound for H p ( Φ ) a general quantum channel Φ . 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 ∈ [ 1 2 , + ∞ ) 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 ∈ [ 1 2 , + ∞ ) .