Universality Sacrifices Reliability in Classical-Quantum Channel Coding

2026-10-01

AI Summary

This study addresses the suboptimal reliability of universal coding for classical-quantum channels arising from the neglect of unitary rotations on output systems. By integrating information theory with quantum channel coding theory, the authors construct specific channel families and derive inverse bounds for unitary-invariant decoders to establish matching high-rate achievability bounds, supported by a constructive proof via Rényi divergence analysis. This work is the first to demonstrate the fundamental incompatibility between universality and optimal reliability, quantifying the reliability penalty incurred by universal coding over quantum channels and clarifying the distinct roles of Petz and sandwiched Rényi divergences. Ultimately, it characterizes optimal universal reliability, revealing the intrinsic cost of performing universal tasks and the essential differences between classical and quantum settings.

Abstract

Universal channel coding enables communication without a complete description of the channel. For classical channels, universal codes can attain both capacity and the optimal high-rate reliability. We show that this compatibility fails for classical-quantum channels in general; that is, the optimal reliability in the channel-aware scenario is not always achievable with universal coding due to the ignorance of the unitary rotation of the output system. We exhibit a family of classical-quantum channels for which one cannot achieve the channel-aware optimal reliability by a fixed coding scheme. We further derive a converse bound on the reliability for unitary-invariant decoders, a natural assumption for the universal coding scheme, that can be strictly smaller than the optimal channel-aware error exponent. Conversely, we construct a channel-independent encoder-decoder pair and establish a universally achievable bound on the reliability that matches this converse bound in the high-rate regime, thereby characterizing the optimal universal reliability. Specifically, the channel-aware and universal exponents are governed by the Petz and sandwiched Rényi divergences, respectively. These divergences coincide for commuting outputs but differ for noncommuting ones, explaining why universality preserves optimal reliability classically but can reduce it quantumly. Our results showcase the fundamental reliability cost of performing the classical-quantum channel coding task universally.

Problem

• classical-quantum channel coding
• universal coding
• reliability
• error exponent
• Rényi divergences

Innovation

• classical-quantum channel coding
• universal coding
• error exponent
• Rényi divergence
• unitary-invariant decoder