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 $O(\log n)$ 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.
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The paper studies how many uses of a finite classical target channel $W$ can be simulated from $n$ uses of a resource channel $T$, under non-signalling (NS) assistance and worst-input total-variation error $\epsilon$. It determines the fixed-error second-order correction to the capacity-ratio rate, rather than separately optimizing channel coding and simulation.
When each channel has a single positive capacity-achieving dispersion, the optimal target length is $m^*_{\mathrm{NS}}=rn+(d_\epsilon/C_W)\sqrt n+O(\log n)$, where $r=C_T/C_W$ and $d_\epsilon=\inf_{\epsilon<\eta<1}\{\sqrt{V_T}\Phi^{-1}(\eta)-\sqrt{rV_W}\Phi^{-1}(\eta-\epsilon)\}$. This is the inverse sesquinormal profile known from state-dichotomy conversion. The limiting error at the capacity-ratio length vanishes when $V_T/C_T=V_W/C_W$; otherwise, a positive threshold error cancels the second-order term. A generalization treats nonunique positive dispersions through an optimization over resource Gaussian mixtures and target variances.
The converse uses binary hypothesis-testing comparisons with uniform bounds covering correlated inputs and protocol-dependent reference distributions. Achievability combines type-class symmetrization with explicit completion of both NS marginals, paying $O(\log n)$ resource uses for auxiliary communication. Exact finite-blocklength formulas are also derived for additive channels and BEC-to-BSC conversion.
Shared randomness attains the NS second-order profile for packet-erasure-to-bit-erasure conversion and BSC self-conversion, but a dispersion-matched BEC-to-BSC example exhibits a strict separation. General shared-randomness and entanglement-assisted characterizations remain open, as do general nonadditive zero-dispersion cases and third-order refinements.
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Based on previous tight second-order results for the interconversion of specific channels, the general characterization given here was developed with ChatGPT (5.6 Sol and 6 Astra).
Error found: The abstract states the singleton-dispersion characterization without the strictly positive capacities and dispersions required by the main theorem, while the paper does not establish the claimed generality for nonadditive zero-dispersion channels.
Written by gpt-6.1-sol from the paper's full text. Automated: it can be wrong.
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(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.
The paper studies how many uses of a finite classical target channel W can be simulated from n uses of a resource channel T, under non-signalling (NS) assistance and worst-input total-variation error ϵ. It determines the fixed-error second-order correction to the capacity-ratio rate, rather than separately optimizing channel coding and simulation.
When each channel has a single positive capacity-achieving dispersion, the optimal target length is mNS∗=rn+(dϵ/CW)n+O(logn), where r=CT/CW and dϵ=infϵ<η<1{VTΦ−1(η)−rVWΦ−1(η−ϵ)}. This is the inverse sesquinormal profile known from state-dichotomy conversion. The limiting error at the capacity-ratio length vanishes when VT/CT=VW/CW; otherwise, a positive threshold error cancels the second-order term. A generalization treats nonunique positive dispersions through an optimization over resource Gaussian mixtures and target variances.
The converse uses binary hypothesis-testing comparisons with uniform bounds covering correlated inputs and protocol-dependent reference distributions. Achievability combines type-class symmetrization with explicit completion of both NS marginals, paying O(logn) resource uses for auxiliary communication. Exact finite-blocklength formulas are also derived for additive channels and BEC-to-BSC conversion.
Shared randomness attains the NS second-order profile for packet-erasure-to-bit-erasure conversion and BSC self-conversion, but a dispersion-matched BEC-to-BSC example exhibits a strict separation. General shared-randomness and entanglement-assisted characterizations remain open, as do general nonadditive zero-dispersion cases and third-order refinements.
Abstract In their seminal work, Bennett et al. [1] showed that, with sufficient shared randomness, one noisy channel can simulate another at a rate equal to the ratio of their capacities. We establish that when coding above this channel interconversion capacity, the exact strong converse exponent is characterized by a simple optimization involving the difference of the corresponding Rényi channel capacities with Hölder dual parameters. We further extend this result to the entanglement-assisted interconversion of classical–quantum channels, showing that the strong converse exponent is likewise determined by differences of sandwiched Rényi channel capacities. The converse bound is obtained by relaxing to non-signaling assisted codes and applying Hölder duality together with the data processing inequality for Rényi divergences. Achievability is proven by concatenating refined channel coding and simulation protocols that go beyond first-order capacities, attaining an exponentially small conversion error, remaining robust under small variations in the input distribution, and tolerating a sublinear gap between the conversion rates.
We address the problem of exact and approximate transformation of quantum dichotomies in the asymptotic regime, i.e., the existence of a quantum channel E mapping ρ1⊗n into ρ2⊗Rnn with an error ϵn (measured by trace distance) and σ1⊗n into σ2⊗Rnn exactly, for a large number n . We derive second-order asymptotic expressions for the optimal transformation rate Rn in the small-, moderate-, and large-deviation error regimes, as well as the zero-error regime, for an arbitrary pair (ρ1,σ1) of initial states and a commuting pair (ρ2,σ2) of final states. We also prove that for σ1 and σ2 given by thermal Gibbs states, the derived optimal transformation rates in the first three regimes can be attained by thermal operations. This allows us, for the first time, to study the second-order asymptotics of thermodynamic state interconversion with fully general initial states that may have coherence between different energy eigenspaces. Thus, we discuss the optimal performance of thermodynamic protocols with coherent inputs and describe three novel resonance phenomena allowing one to significantly reduce transformation errors induced by finite-size effects. What is more, our result on quantum dichotomies can also be used to obtain, up to second-order asymptotic terms, optimal conversion rates between pure bipartite entangled states under local operations and classical communication. Published by the American Physical Society 2024
An important part of the information theory folklore had been about the output statistics of codes that achieve the capacity and how the empirical distributions compare to the output distributions induced by the optimal input in the channel capacity problem. Results for a variety of such empirical output distributions of good codes have been known in the literature, such as the comparison of the output distribution of the code to the optimal output distribution in vanishing and non-vanishing error probability cases. Motivated by these, we aim to achieve similar results for the quantum codes that are used for classical communication, that is the setting in which the classical messages are communicated through quantum codewords that pass through a noisy quantum channel. We first show the uniqueness of the optimal output distribution, to be able to talk more concretely about the optimal output distribution. Then, we extend the vanishing error probability results to the quantum case, by using techniques that are close in spirit to the classical case. We also extend non-vanishing error probability results to the quantum case on block codes, by using the second-order converses for such codes based on hypercontractivity results for the quantum generalized depolarizing semi-groups.
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 study relaxations of entanglement-assisted quantum channel coding and establish that non-signaling assistance and a natural semi-definite programming relaxation — termed meta-converse — are equivalent in terms of success probabilities. We then present a rounding procedure that transforms any non-signaling-assisted strategy into an entanglement-assisted one and prove an approximation ratio of (1–e−1) in success probabilities for the special case of measurement channels. For fully quantum channels, we give a weaker (dimension dependent) approximation ratio, that is nevertheless still tight to characterize the strong converse exponent of entanglement-assisted channel coding [Li and Yao, IEEE Tran. Inf. Theory (2024)]. Our derivations leverage ideas from position-based coding, quantum decoupling theorems, the matrix Chernoff inequality, and input flattening techniques.
The asymptotic rates of information-theoretic protocols - including error exponents, data-compression rates, and channel capacities - are traditionally derived under the idealised assumption that the underlying resources are independent and identically distributed (i.i.d.). Somewhat surprisingly, even slight departures from the exact i.i.d. structure can drastically alter the asymptotic behaviour predicted by the i.i.d. theory. If the precise nature of the perturbation is known, for instance in the case of a pointwise defect, one can design a bespoke protocol that compensates for it, e.g. by discarding the corrupted subsystem. In realistic physical settings, however, exact i.i.d. behaviour cannot be guaranteed, and deviations from the ideal regime cannot generally be identified precisely. This raises a fundamental question: which notions of almost i.i.d. structure are sufficiently robust to preserve the asymptotic predictions of quantum Shannon theory? We investigate this question for three central information-theoretic tasks: asymmetric hypothesis testing, classical and quantum data compression, and classical communication through quantum channels. Rather than designing protocols tailored to specific defects, we seek robust protocols that remain asymptotically optimal and that are universal within a broad class of almost i.i.d. resources whose precise deviations from the ideal regime are unknown. To this end, we study three inequivalent notions of almost i.i.d. structure, and determine which of them preserve the asymptotic rates and error exponents predicted by the i.i.d. theory. Along the way, we introduce the notion of an almost i.i.d. process and a new distance measure between quantum channels - the club distance - designed to capture stability under local perturbations. These notions may be of independent interest.