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arXiv:2212.11666 · IEEE Transactions on Information Theory

Channel Simulation: Finite Blocklengths and Broadcast Channels

Michael X. Cao, Navneeth Ramakrishnan, Mario Berta, Marco Tomamichel

Published Aug 19, 2024• 11 citations
arXivHTMLPDF

Abstract

We study channel simulation under common randomness assistance in the finite-blocklength regime and identify the smooth channel max-information as a linear program one-shot converse on the minimal simulation cost for fixed error tolerance. We show that this one-shot converse can be achieved exactly using no-signaling-assisted codes, and approximately achieved using common randomness-assisted codes. Our one-shot converse thus takes on an analogous role to the celebrated meta-converse in the complementary problem of channel coding, and we find tight relations between these two bounds. We asymptotically expand our bounds on the simulation cost for discrete memoryless channels, leading to the second-order as well as the moderate-deviation rate expansion, which can be expressed in terms of the channel capacity and channel dispersion known from noisy channel coding. Our bounds imply the well-known fact that the optimal asymptotic rate of one channel to simulate another under common randomness assistance is given by the ratio of their respective capacities. Additionally, our higher-order asymptotic expansion shows that this reversibility falls apart in the second order. Our techniques extend to discrete memoryless broadcast channels. In stark contrast to the elusive broadcast channel capacity problem, we show that the reverse problem of broadcast channel simulation under common randomness assistance allows for an efficiently computable single-letter characterization of the asymptotic rate region in terms of the broadcast channel’s multipartite mutual information.

DOI
10.1109/tit.2024.3445998

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We study channel simulation under common randomness assistance in the finite-blocklength regime and identify the smooth channel max-information as a linear program one-shot converse on the minimal simulation cost for fixed error tolerance. We show that this one-shot converse can be achieved exactly using no-signaling-assisted codes, and approximately achieved using common randomness-assisted codes. Our one-shot converse thus takes on an analogous role to the celebrated meta-converse in the complementary problem of channel coding, and we find tight relations between these two bounds. We asymptotically expand our bounds on the simulation cost for discrete memoryless channels, leading to the second-order as well as the moderate-deviation rate expansion, which can be expressed in terms of the channel capacity and channel dispersion known from noisy channel coding. Our bounds imply the well-known fact that the optimal asymptotic rate of one channel to simulate another under common randomness assistance is given by the ratio of their respective capacities. Additionally, our higher-order asymptotic expansion shows that this reversibility falls apart in the second order. Our techniques extend to discrete memoryless broadcast channels. In stark contrast to the elusive broadcast channel capacity problem, we show that the reverse problem of broadcast channel simulation under common randomness assistance allows for an efficiently computable single-letter characterization of the asymptotic rate region in terms of the broadcast channel’s multipartite mutual information.
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 R´enyi 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 [Caoet al., IEEE Trans. Inf. Theory (2024)], which corresponds to nonsignaling assisted codes. We prove this to be asymptotically tight by employing the approximation algorithms from [Bertaet 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.
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.
We consider the problem of shared randomness-assisted multiple access channel (MAC) simulation for product inputs and characterize the one-shot communication cost region via almost-matching inner and outer bounds in terms of the smooth max-information of the channel, featuring auxiliary random variables of bounded size. The achievability relies on a rejection-sampling algorithm to simulate an auxiliary channel between each sender and the decoder, and producing the final output based on the output of these intermediate channels. The converse follows via information-spectrum based arguments. To bound the cardinality of the auxiliary random variables, we employ the perturbation method from [Anantharamet al., IEEE Trans. Inf. Theory (2019)] in the one-shot setting. For the asymptotic setting and vanishing errors, our result expands to a tight single-letter rate characterization and consequently extends a special case of the simulation results of [Kurriet al., IEEE Trans. Inf. Theory (2022)] for fixed, independent and identically distributed (iid) product inputs to universal simulation for any product inputs. We broaden our discussion into the quantum realm by studying feedback simulation of quantum-to-classical (QC) MACs with product measurements [Atifet al., IEEE Trans. Inf. Theory (2022)]. For fixed product inputs and with shared randomness assistance, we give a quasi tight one-shot communication cost region with corresponding single-letter asymptotic iid expansion.
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.
We study the optimal rates of emulation (also called interconversion) between quantum channels, in three regimes: when the encoder and the decoder are unassisted, when they share classical randomness, and when they share an unlimited amount of entanglement. When the source and the target channels are idempotent, we completely characterise these three regimes, in terms of the shape vectors of the two channels, which describe the structure of their ranges. In the unassisted and shared randomness regimes, we give a single-letter expression for the emulation capacity, show that it coincides with the zero-error emulation capacity, and prove a finite-blocklength converse which implies a strong converse. In these two regimes, shared randomness does not increase the emulation capacity, and the emulation of idempotent channels is not reversible. Moreover, we provide an efficient algorithm to approximate this capacity to arbitrary precision. When an unlimited amount of shared entanglement is allowed, we determine the minimal emulation error exactly at every finite blocklength. It is governed by a single quantity, the ℓ2\ell_2ℓ2​-norm of the shape vector, so that the emulation becomes reversible, and exact emulation is achievable with finitely many maximally entangled states. Altogether, these results draw a comprehensive picture of the task of emulating idempotent channels with each other.