Holevo barycenter does not multiply under combined quantum channels

Non-Multiplicativity of the Holevo Barycenter of Quantum Channels

Information Theory

Summary

The Holevo barycenter is a way to describe the average output state from a quantum communication channel. People wondered whether combining two channels side-by-side would make their barycenters simply multiply or combine in a straightforward way. The authors show that this is not always true—the barycenter for the combined channel can be different from the product of the individual barycenters. They also find that the entropy of these barycenters does not reliably add up or break down in simple ways when channels are combined. This answers an open question about how quantum information capacity behaves when channels are used together.

What this means in practice

A theory result. No direct application yet.

Authors

Sayantan Chakraborty, Stefano Mancini, Leonardo Rossetti, Andreas Winter

Abstract

The Holevo barycenter of a quantum channel is the unique output state obtained as the average output of any ensemble achieving the Holevo capacity. Given two quantum channels, the multiplicativity problem asks whether this barycenter tensorizes under parallel composition, namely whether the barycenter of the product channel coincides with the tensor product of the individual barycenters. This question is closely related to the additivity problem for the Holevo capacity: additivity implies tensorization of the Holevo barycenter, while tensorization alone is not sufficient for additivity. Although a construction is known that demonstrates the existence of channels with non-additive Holevo capacity, their corresponding Holevo barycenters still tensorize, leaving open whether multiplicativity might ultimately hold universally. Here, we answer this question in the negative by exhibiting channels for which the Holevo barycenter is not multiplicative under the tensor product of the channel with itself. Moreover, we show that the entropy of the Holevo barycenter is neither universally subadditive nor universally superadditive under tensor product.
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