Minimal Entropy of States Emerging from Noisy Quantum Channels

TL;DR

This paper investigates the minimal entropy of qubit states transmitted through noisy quantum channels, providing evidence that this entropy is additive and exploring implications for channel capacity and the role of unitality.

Contribution

It offers a proof supporting the additivity of minimal entropy for certain quantum channels and explores the geometric and capacity implications for unital and non-unital maps.

Findings

Minimal entropy is additive for tensor products of unital stochastic maps.

Maximal output norm is multiplicative for most product maps, including unital ones.

Additivity of minimal entropy implies additivity of Holevo capacity for unital channels.

Abstract

In this paper, we consider the minimal entropy of qubit states transmitted through two uses of a noisy quantum channel, which is modeled by the action of a completely positive trace-preserving (or stochastic) map. We provide strong support for the conjecture that this minimal entropy is additive, namely that the minimum entropy can be achieved when product states are transmitted. Explicitly, we prove that for a tensor product of two unital stochastic maps on qubit states, using an entanglement that involves only states which emerge with minimal entropy cannot decrease the entropy below the minimum achievable using product states. We give a separate argument, based on the geometry of the image of the set of density matrices under stochastic maps, which suggests that the minimal entropy conjecture holds for non-unital as well as for unital maps. We also show that the maximal norm of the output states is multiplicative for most product maps on $n$-qubit states, including all those for which at least one map is unital. For the class of {\it unital} channels on ${\bf C}^2$, we show that additivity of minimal entropy implies that the Holevo capacity of the channel is {\it additive} over two inputs, achievable with orthogonal states, and equal to the Shannon capacity. This implies that superadditivity of the capacity is possible only for non-unital channels.