From Symmetry to Secrecy: Covariant Classical--Quantum Wiretap Channels

2026-10-02

AI Summary

This study addresses the computational difficulty of determining the secrecy capacity of covariant classical-quantum wiretap channels under finite blocklengths, as well as the absence of a rigorous proof for the non-additivity of private information. To overcome these challenges, this work proposes a group-convolution-based preprocessing framework for finite blocklengths, leveraging unitary group orbit structures to design correlated coding schemes. By integrating constant-composition coding, universal_2 hashing, and sandwiched Rényi divergence techniques, the transmission rate is optimized through entropy difference analysis. A primary contribution of this research is providing the first rigorous proof of private information non-additivity within this model. Numerical experiments demonstrate that mixed binary channels require only three to four channel uses to surpass the noisy repetition benchmark, while also establishing conditions for uniform input bounds.

Abstract

For classical--quantum wiretap channels with a transitive covariant input action, both output ensembles admit a description as unitary group orbits of fixed seed states. We use this structure to construct finite-block preprocessing schemes whose private information is an entropy difference evaluated at two distributions related by group convolution. Applying the construction to product groups allows correlations across any fixed number of channel uses. We give a sufficient condition for the associated uniform-input bound to equal the single-use private information. Constant-composition channel codes and universal$_2$ hashing achieve the resulting rates, with trace-distance leakage controlled by sandwiched Rényi information. For uniform outer randomization, the representation structure reduces the comparison-state optimization to Eve's invariant states. For the hybrid binary channel considered by Tikku, Berta and Renes, numerical optimization at three and four uses gives rates above our numerical estimate of the optimized noisy-repetition benchmark wherever that estimate is positive on the tested grid. The improvement comes from noise distributions beyond the family of independent physical flips and a symmetric logical flip. We also prove nonadditivity at an explicit channel parameter using a two-use encoder, an analytic single-use converse, and certified entropy bounds. Finally, we relate the hybrid model to binary phase-shift keying with a fixed receiver measurement.

Problem

• classical-quantum wiretap channel
• private information
• covariant channels
• nonadditivity
• secrecy capacity

Innovation

• classical-quantum wiretap channels
• covariant structure
• finite-block preprocessing
• sandwiched Rényi information
• nonadditivity