Additivity and Distinguishability of Random Unitary Channels

TL;DR

This paper investigates the properties of random unitary channels, showing their relevance to key conjectures in quantum information theory and their computational complexity in distinguishing quantum circuits.

Contribution

It introduces an efficient approximation of general quantum channels by random unitary channels and analyzes the complexity of distinguishing such circuits.

Findings

Random unitary channels can approximate general quantum channels efficiently.

Conjectures on additivity and multiplicativity are equivalent when expressed with random unitary channels.

Distinguishing random unitary circuits is as hard as distinguishing general mixed state circuits.

Abstract

A random unitary channel is one that is given by a convex combination of unitary channels. It is shown that the conjectures on the additivity of the minimum output entropy and the multiplicativity of the maximum output $p$-norm can be equivalently restated in terms of random unitary channels. This is done by constructing a random unitary approximation to a general quantum channel. This approximation can be constructed efficiently, and so it is also applied to the computational problem of distinguishing quantum circuits. It is shown that the problem of distinguishing random unitary circuits is as hard as the problem of distinguishing general mixed state circuits, which is complete for the class of problems that have quantum interactive proof systems.