Zero-Error List Decoding for Classical-Quantum Channels

2026-01-14

AI Summary

This study investigates the zero-error list-decoding capacity of pure-state classical-quantum channels. By integrating the information-theoretic framework of list decoding with analyses of quantum state overlaps and properties of positive semidefinite matrices, the authors establish an achievable bound for list size two and a general converse bound for any fixed list size. A key contribution is demonstrating that, even with arbitrarily large list sizes, the divergence rate suggested by the sphere-packing bound may not be attainable via zero-error list codes. Furthermore, when the pairwise overlap matrix of the quantum states is positive semidefinite, the achievable and converse bounds coincide, thereby precisely characterizing a fundamental distinction between classical and quantum settings in zero-error list decoding.

Technology Category

Machine Learning: Quantum Machine LearningCognitive Modeling & Cognitive Systems: Neural Spike CodingSearch and Optimization: Non-convex Optimization

Application Category

Security and Privacy: Large-scale security measurementsEconomics, Online Markets and Human Computation: LLM based quality controls for crowd workSearch and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• zero-error capacity
• classical-quantum channels
• list decoding
• pure-state
• sphere-packing bound

Innovation

• zero-error list decoding
• classical-quantum channels
• achievability bound
• converse bound
• sphere-packing bound