Zero-Error List Decoding for Classical-Quantum Channels

TL;DR

The paper addresses zero-error communication over pure-state classical-quantum channels under list decoding. It derives an achievability bound for list size $L=2$ and a converse bound for fixed $L$, and identifies conditions (positive semi-definite absolute overlaps, pairwise non-obtuse geometry) under which these bounds match, yielding exact characterizations of $C_{0,L}$. A key finding is that, unlike the fully classical case, the sphere-packing rate $R_\infty$ may not be attainable by zero-error list codes in the quantum setting, as demonstrated by the Trine Channel. The results provide a nuanced understanding of zero-error list decodability in CQ channels and highlight fundamental differences from classical channels, with implications for quantum communications and related coding strategies.

Abstract

The aim of this work is to study the zero-error capacity of pure-state classical-quantum channels in the setting of list decoding. We provide an achievability bound for list-size two and a converse bound holding for every fixed list size. The two bounds coincide for channels whose pairwise absolute state overlaps form a positive semi-definite matrix. Finally, we discuss a remarkable peculiarity of the classical-quantum case: differently from the fully classical setting, the rate at which the sphere-packing bound diverges might not be achievable by zero-error list codes, even when we take the limit of fixed but arbitrarily large list size.

Figures (1)

• Figure 1: Trine channel. All the vectors lie in a plane; $\psi_1$, $\psi_2$ and $\psi_3$ form angles of $120^\circ$, and each vector $\varphi_i$ is orthogonal to $\psi_i$.

Theorems & Definitions (13)

• Theorem 1
• proof
• Theorem 2
• Lemma 3
• proof
• proof : Proof of Theorem \ref{['thm:main']}.
• Corollary 4
• Remark 5
• Theorem 6
• proof