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arXiv:2601.09786 · IEEE International Symposium on Information Theory (ISIT)

Zero-Error List Decoding for Classical-Quantum Channels

Marco Dalai, Filippo Girardi, Ludovico Lami

quant-phcs.ITmath-phSubmitted Jan 14, 2026• Announced Jan 16, 2026• 0 citations• Revised Apr 16, 2026 (v2)
v1 · Jan 14, 2026v2 · Apr 16, 2026
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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.

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5+1 pages, 1 figure

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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.
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A fundamental quantity of interest in Shannon theory, classical or quantum, is the error exponent of a given channel W and rate R: the constantE(W,R)E(W,R)E(W,R)which governs the exponential decay of decoding error when using ever larger optimal codes of fixed rate R to communicate over ever more (memoryless) instances of a given channel W. Nearly matching lower and upper bounds are well-known for classical channels. Here I show a lower bound on the error exponent of communication over arbitrary classical-quantum (CQ) channels which matches Dalai’s sphere-packing upper bound for rates above a critical value, exactly analogous to the case of classical channels. This proves a conjecture made by Holevo in his investigation of the problem. Unlike the classical case, however, the argument does not proceed via a refined analysis of a suitable decoder, but instead by leveraging a bound by Hayashi on the error exponent of the cryptographic task of privacy amplification. This bound is then related to the coding problem via tight entropic uncertainty relations and Gallager’s method of constructing capacity-achieving parity-check codes for arbitrary channels. Along the way, I find a lower bound on the error exponent of the task of compression of classical information relative to quantum side information that matches the sphere-packing upper bound of Cheng et al. In turn, the polynomial prefactors to the sphere-packing bound found by Cheng et al. may be translated to the privacy amplification problem, sharpening a recent result by Li, Yao, and Hayashi, at least for linear randomness extractors.
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We study commitment scheme for classical-quantum channels. To accomplish this we define various notions of commitment capacity for these channels and prove matching upper and lower bound on it in terms of the conditional entropy. Our achievability (lower bound) proof is quantum generalisation of the work of one of the authors (arXiv:2103.11548) which studied the problem of secure list decoding and its application to bit-string commitment. The techniques we use in the proof of converse (upper bound) is similar in spirit to the techniques introduced by Winter, Nascimento and Imai (Cryptography and Coding 2003) to prove upper bound on the commitment capacity of classical channels. However, generalisation of this technique to the quantum case is not so straightforward and requires some new constructions, which can be of independent interest.