Randomness cost of symmetric twirling

TL;DR

This paper investigates the randomness cost of implementing symmetric twirling channels, deriving bounds that highlight the exponential resource requirements and discussing their implications for resource-efficient quantum protocols.

Contribution

It provides the first bounds on the randomness cost of symmetric twirling, connecting quantum channel implementation to Shannon theory and resource efficiency.

Findings

Randomness cost grows exponentially with the number of systems.

Symmetric twirling can be viewed as a Shannon theoretic protocol.

Protocols are resource-inefficient when randomness is costly.

Abstract

We study random unitary channels which reproduce the action of the twirling channel corresponding to the representation of the symmetric groupon an n-fold tensor product. We derive upper andlower bounds on the randomness cost of implementing such a map which depend exponentially on the number of systems. Consequently, symmetrictwirling can be regarded as a reasonable Shannon theoretic protocol. On the other hand, such protocols are disqualified by their resource-inefficiency in situations where randomness is a costly resource.