Additive bounds of minimum output entropies for unital channels and an exact qubit formula

TL;DR

This paper derives an exact formula for the minimum output entropy of qubit channels and provides bounds for unital channels, with implications for channel capacity and entropy additivity.

Contribution

It introduces an exact formula for qubit channels' minimum output entropy and bounds for unital channels based on operator norms, extending understanding of quantum channel capacities.

Findings

Exact formula for qubit channels' minimum output entropy

Bounds for unital quantum channels based on operator norms

Examples of channels with additive minimum output Re9nyi 2-entropy

Abstract

We investigate minimum output (R\'enyi) entropy of qubit channels and unital quantum channels. We obtain an exact formula for the minimum output entropy of qubit channels, and bounds for unital quantum channels. Interestingly, our bounds depend only on the operator norm of the matrix representation of the channels on the space of trace-less Hermitian operators. Moreover, since these bounds respect tensor products, we get bounds for the capacity of unital quantum channels, which is saturated by the Werner-Holevo channel. Furthermore, we construct an orthonormal basis, besides the Gell-Mann basis, for the space of trace-less Hermitian operators by using discrete Weyl operators. We apply our bounds to discrete Weyl covariant channels with this basis, and find new examples in which the minimum output R\'enyi $2$-entropy is additive.