Fully Quantum Arbitrarily Varying Channels: Random Coding Capacity and Capacity Dichotomy

TL;DR

This paper studies the capacity of a fully quantum arbitrarily varying channel with a quantum jammer, establishing a reduction to a simpler model and revealing a capacity dichotomy based on shared randomness and channel properties.

Contribution

It introduces a reduction of the quantum arbitrarily varying channel capacity to a related compound channel and proves a capacity dichotomy theorem.

Findings

Capacity reduces to a related compound channel with i.i.d. jammer states

Shared randomness needed is logarithmic in block length

Either both classical and quantum capacities are zero or equal their random coding capacities

Abstract

We consider a model of communication via a fully quantum jammer channel with quantum jammer, quantum sender and quantum receiver, which we dub quantum arbitrarily varying channel (QAVC). Restricting to finite dimensional user and jammer systems, we show, using permutation symmetry and a de Finetti reduction, how the random coding capacity (classical and quantum) of the QAVC is reduced to the capacity of a naturally associated compound channel, which is obtained by restricting the jammer to i.i.d. input states. Furthermore, we demonstrate that the shared randomness required is at most logarithmic in the block length, using a random matrix tail bound. This implies a dichotomy theorem: either the classical capacity of the QAVC is zero, and then also the quantum capacity is zero, or each capacity equals its random coding variant.