On the Second-Order Asymptotics for Entanglement-Assisted Communication

TL;DR

This paper explores the second-order asymptotics of entanglement-assisted classical communication over quantum channels, providing a detailed analysis of how coding rates approach capacity with increasing channel uses.

Contribution

It introduces a quantum mutual information variance for entanglement-assisted channels and characterizes convergence rates for covariant channels.

Findings

Quantum mutual information variance generalizes classical dispersion.

For covariant channels, this variance equals channel dispersion.

Results apply to both classical and quantum entanglement-assisted communication.

Abstract

The entanglement-assisted classical capacity of a quantum channel is known to provide the formal quantum generalization of Shannon's classical channel capacity theorem, in the sense that it admits a single-letter characterization in terms of the quantum mutual information and does not increase in the presence of a noiseless quantum feedback channel from receiver to sender. In this work, we investigate second-order asymptotics of the entanglement-assisted classical communication task. That is, we consider how quickly the rates of entanglement-assisted codes converge to the entanglement-assisted classical capacity of a channel as a function of the number of channel uses and the error tolerance. We define a quantum generalization of the mutual information variance of a channel in the entanglement-assisted setting. For covariant channels, we show that this quantity is equal to the channel dispersion, and thus completely characterize the convergence towards the entanglement-assisted classical capacity when the number of channel uses increases. Our results also apply to entanglement-assisted quantum communication, due to the equivalence between entanglement-assisted classical and quantum communication established by the teleportation and super-dense coding protocols.