Convergence Properties of Good Quantum Codes for Classical Communication

2026-01-16

AI Summary

This work investigates the convergence of the empirical output distribution of classical information transmitted through quantum codes over noisy quantum channels toward the capacity-achieving output distribution. For both vanishing and non-vanishing error probability regimes, the study establishes the first theoretical framework characterizing the convergence behavior of quantum code output distributions and proves the uniqueness of the capacity-achieving output distribution. Methodologically, it extends techniques from classical information theory to the quantum setting by leveraging the hypercontractivity of quantum generalized depolarizing semigroups and a second-order converse theorem. The results not only generalize foundational aspects of classical coding theory to the quantum domain but also provide a new theoretical foundation for quantum channel coding.

Technology Category

• Machine Learning: Quantum Machine Learning
• Search and Optimization: Non-convex Optimization
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

An important part of the information theory folklore had been about the output statistics of codes that achieve the capacity and how the empirical distributions compare to the output distributions induced by the optimal input in the channel capacity problem. Results for a variety of such empirical output distributions of good codes have been known in the literature, such as the comparison of the output distribution of the code to the optimal output distribution in vanishing and non-vanishing error probability cases. Motivated by these, we aim to achieve similar results for the quantum codes that are used for classical communication, that is the setting in which the classical messages are communicated through quantum codewords that pass through a noisy quantum channel. We first show the uniqueness of the optimal output distribution, to be able to talk more concretely about the optimal output distribution. Then, we extend the vanishing error probability results to the quantum case, by using techniques that are close in spirit to the classical case. We also extend non-vanishing error probability results to the quantum case on block codes, by using the second-order converses for such codes based on hypercontractivity results for the quantum generalized depolarizing semi-groups.

Problem

• quantum codes
• classical communication
• output distribution
• channel capacity
• convergence properties

Innovation

• quantum codes
• output distribution
• vanishing error probability
• second-order converse
• hypercontractivity