Relating entropies of quantum channels

TL;DR

This paper compares two definitions of quantum channel entropy, establishing bounds and conditions for their equality, with implications for understanding quantum information processing.

Contribution

It introduces and compares two entropy measures for quantum channels, proving bounds and saturation conditions, including for unital qubit channels and conjectures for high-dimensional channels.

Findings

The von Neumann entropy of the Choi state bounds the extended channel output entropy.

For unital qubit channels, the bound is tight and saturated.

Numerical evidence suggests the bound may be saturated for large random channels.

Abstract

In this work, we study two different approaches to defining the entropy of a quantum channel. One of these is based on the von Neumann entropy of the corresponding Choi-Jamio{\l}kowski state. The second one is based on the relative entropy of the output of the extended channel relative to the output of the extended completely depolarizing channel. This entropy then needs to be optimized over all possible input states. Our results first show that the former entropy provides an upper bound on the latter. Next, we show that for unital qubit channels, this bound is saturated. Finally, we conjecture and provide numerical intuitions that the bound can also be saturated for random channels as their dimension tends to infinity.