Constant Compositions in the Sphere Packing Bound for Classical-Quantum Channels

TL;DR

This paper extends the sphere packing bound for classical-quantum channels to constant composition codes, linking it to Lovász theta functions and proposing a bound extension for varying channels and codewords.

Contribution

It introduces the first sphere packing bound for constant composition codes in classical-quantum channels and connects it to Lovász theta functions, also extending to varying channels.

Findings

Extended sphere packing bound for constant composition codes.

Linked the bound to a variation of Lovász theta function.

Proposed a bound extension for varying channels and codewords.

Abstract

The sphere packing bound, in the form given by Shannon, Gallager and Berlekamp, was recently extended to classical-quantum channels, and it was shown that this creates a natural setting for combining probabilistic approaches with some combinatorial ones such as the Lov\'asz theta function. In this paper, we extend the study to the case of constant composition codes. We first extend the sphere packing bound for classical-quantum channels to this case, and we then show that the obtained result is related to a variation of the Lov\'asz theta function studied by Marton. We then propose a further extension to the case of varying channels and codewords with a constant conditional composition given a particular sequence. This extension is then applied to auxiliary channels to deduce a bound which can be interpreted as an extension of the Elias bound.