Lower bounds on the Probability of Error for Classical and Classical-Quantum Channels

TL;DR

This paper extends classical bounds on error probability to classical-quantum channels, introduces a new framework for low-rate error bounds, and reveals deep connections between various bounds and capacities in quantum information theory.

Contribution

It generalizes classical error bounds to quantum channels and proposes a new approach for low-rate error probability bounds, linking Lovász' bound with sphere packing in quantum contexts.

Findings

Lovász' bound emerges from sphere packing in quantum channels

A new lower bound framework for channels with zero-error capacity

Connections established between Lovász theta, Gallager's expurgated bound, and quantum sphere packing

Abstract

In this paper, lower bounds on error probability in coding for discrete classical and classical-quantum channels are studied. The contribution of the paper goes in two main directions: i) extending classical bounds of Shannon, Gallager and Berlekamp to classical-quantum channels, and ii) proposing a new framework for lower bounding the probability of error of channels with a zero-error capacity in the low rate region. The relation between these two problems is revealed by showing that Lov\'asz' bound on zero-error capacity emerges as a natural consequence of the sphere packing bound once we move to the more general context of classical-quantum channels. A variation of Lov\'asz' bound is then derived to lower bound the probability of error in the low rate region by means of auxiliary channels. As a result of this study, connections between the Lov\'asz theta function, the expurgated bound of Gallager, the cutoff rate of a classical channel and the sphere packing bound for classical-quantum channels are established.