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arXiv:2610.15877

Fine asymptotics for approximate simulation of symmetric channels

Valentinian Lungu, Aaron B. Wagner, Ioannis Kontoyiannis

Published Oct 11, 2026• 0 citations
arXivHTMLPDF

Abstract

We study the approximate channel simulation problem in the normal approximation regime. Our starting point is the fact established by Cao et al.~\cite{Cao_channel_simulation}, that the minimal codebook size $M_ε^*(W)$ for a fixed error tolerance $ε>0$ is closely related to the $ε$-smooth channel max-information, $I_{\max}^ε(W)$. Using this, we refine earlier large-blocklength approximations to $M_ε^*(W^{\otimes n})$ for discrete memoryless channels. Specifically, we provide tight third-order expansions for Gallager-symmetric channels and demonstrate that the third-order term is channel-dependent. As in corresponding channel coding expansions, we identify a substantial rate gap between singular and nonsingular channels. Notably, the direction of this gap is reversed relative to channel coding: The third-order term is $-\frac{1}{2}\log n$ for singular channels but $O(\log\log n)$ for nonsingular channels.

DOI
10.48550/arxiv.2610.15877

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We study the approximate channel simulation problem in the normal approximation regime. Our starting point is the fact established by Cao et al.~\cite{Cao_channel_simulation}, that the minimal codebook size Mε∗(W)M_ε^*(W)Mε∗​(W) for a fixed error tolerance ε>0ε>0ε>0 is closely related to the εεε-smooth channel max-information, Imax⁡ε(W)I_{\max}^ε(W)Imaxε​(W). Using this, we refine earlier large-blocklength approximations to Mε∗(W⊗n)M_ε^*(W^{\otimes n})Mε∗​(W⊗n) for discrete memoryless channels. Specifically, we provide tight third-order expansions for Gallager-symmetric channels and demonstrate that the third-order term is channel-dependent. As in corresponding channel coding expansions, we identify a substantial rate gap between singular and nonsingular channels. Notably, the direction of this gap is reversed relative to channel coding: The third-order term is −12log⁡n-\frac{1}{2}\log n−21​logn for singular channels but O(log⁡log⁡n)O(\log\log n)O(loglogn) for nonsingular channels.
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We determine the exact error and strong converse exponents of shared randomness-assisted channel simulation in worst case total-variation distance. Namely, we find that these exponents can be written as simple optimizations over the Rényi channel mutual information. Strikingly, and in stark contrast to channel coding, there are no critical rates, allowing a tight characterization for arbitrary rates below and above the simulation capacity. We derive our results by asymptotically expanding the meta-converse for channel simulation [Cao {\it et al.}, IEEE Trans.~Inf.~Theory (2024)], which corresponds to non-signaling assisted codes. We prove this to be asymptotically tight by employing the approximation algorithms from [Berta {\it et al.}, Proc.~IEEE ISIT (2024)], which show how to round any non-signaling assisted strategy to a strategy that only uses shared randomness. Notably, this implies that any additional quantum entanglement-assistance does not change the error or the strong converse exponents.
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.
We study channel interconversion under non-signalling superchannels, where the asymptotic conversion rate is given by the ratio of the channel capacities. Our main result determines the second-order expansion of the optimal target length. Namely, when each channel has a single capacity-achieving dispersion, the coefficient is given by the inverse sesquinormal distribution, with parameters determined by the capacities and dispersions. The result extends state-dichotomy theory to channels: one superchannel approximates the target uniformly over inputs and maps every reference channel whose output is independent of its input to another such channel. As in state dichotomies, there are finite-error resonances where the second-order term vanishes, leaving an O(log⁡n)O(\log n)O(logn) remainder. Crucially, our new characterization cannot be obtained by concatenating channel coding and simulation protocols [Cao {\it et al.}~IEEE Trans.~Inf.~Theory (2024)]. Instead, the proof combines comparisons of binary hypothesis testing experiments with uniform Gaussian estimates and a construction that uses symmetry within type classes to satisfy the non-signalling constraints. Shared randomness assistance already suffices at first order, but we show that its second-order behaviour can differ strictly from the non-signalling case.
A central challenge in quantum information theory is to determine whether entanglement provides an advantage for information processing when coherent operations across multiple channel uses are allowed. For depolarizing channels, King's proof of the additivity of the minimum output entropy rules out such an advantage for classical communication at leading asymptotic order, while leaving open the possibility of an advantage for higher orders. In this work, we conjecture a majorization relation for the outputs of depolarizing channels and prove an approximate version of this conjecture. Our proof combines an exact ordering of shifted log-determinants with an anti-concentration bound for the spectrum of the product-state output, refining King's analysis based on Schatten norms. As a direct application, we obtain a tight third-order expansion for the classical capacity of quantum depolarizing channels. Consequently, entangled codewords and collective decoding can improve the message size over the product-coding strategy by at most O(log⁡log⁡n)O(\log \log n)O(loglogn) bits, sharpening our understanding of the role of entanglement in the finite-blocklength regime.
Shannon’s channel coding theorem establishes the upper bound of reliable communication rate as the channel capacity C, while the rate R of practical codingschemes is usually strictly lower than C, and the difference between them constitutes the rate gap. Within the framework of constraint network informationgeometry, this paper encodes a discrete memoryless channel and a linear blockcode as a channel constraint network and a coding constraint network respectively,and proposes to strictly define the information dimension by the volume growthrate of the rigid kernel under the Fisher information metric. The core conclusionshows that the Shannon gap C−R is asymptotically exactly equal to the logarithmnormalized limit of the ratio of Fisher volumes of the rigid kernels of the channelconstraint network and the coding constraint network, i.e., the difference of thevolume growth rates of the two networks. Under the local asymptotic normalitycondition, this gap satisfies L2 = 2(C −R)+O(n−1/2logn), where L is the squareof the Fisher geodesic distance. For the repetition code (3,1) over the binary symmetric channel, the analytical probability distributions of all eight output patternsare given, the finite-length deviation of the KL divergence from the asymptoticgap is computed, and its sources are analyzed. For the Hamming (7,4) code overthe binary erasure channel, by combinatorially enumerating all one hundred andtwenty-eight erasure patterns over finite fields, the average residual non-recoverabledimension and its deviation from the asymptotic prediction n(C −R) are preciselygiven, verifying the error bound O(1/√n). Finally, a roadmap is provided to extendthe volume compression framework to LDPC codes via the peeling algorithm andto Polar codes via the polarization transform, and the applicability boundaries ofthe current framework in the cases of asymmetric channels, continuous alphabets,and non-linear codes are honestly declared.