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Quantum

Quantum channel coding: Approximation algorithms and strong converse exponents

Aadil Oufkir, Mario Berta

Published Oct 6, 2025• 2 citations
Publisher

Abstract

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.

DOI
10.22331/q-2025-10-06-1877

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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.
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)(1 - e^{-1})(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.
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.
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.
A quantum channel can transmit quantum states, classical messages, or classical messages that remain secret from its environment. We prove exponential strong converses for all three tasks over arbitrary finite-dimensional memoryless quantum channels, at their respective regularized capacities and allowing arbitrary entangled inputs across channel uses and joint decoding. Above quantum or classical capacity, entanglement fidelity or decoding success probability, respectively, decays exponentially with the number of channel uses. Above private capacity, fidelity to an ideal shared secret key decays exponentially; in particular, vanishing secrecy error forces decoding success to vanish. The proofs combine one-shot quantum blowing-up lemmas, which convert codes with low success into reliable codes with a controlled loss in code size, with a low-degree polynomial approximation of the channel's Stinespring image projector. The loss is governed by the projective tensor norm of the approximation across the receiver--environment bipartition. For memoryless channels, the polynomial construction gives exponential accuracy while keeping the rate loss arbitrarily small, turning the corresponding weak converses into strong converses.
We study the effect of shared non-signaling correlations for the problem of simulating a channel using noiseless communication in the one-shot setting. For classical channels, we show how to round any non-signaling-assisted simulation strategy - which exactly corresponds to the meta-converse for channel simulation - to a strategy that only uses shared randomness. For quantum channels, we round any non-signaling-assisted simulation strategy to a strategy that only uses shared entanglement. As our main result, we prove a guarantee on the ratio of success probabilities of at least(1−−1e)(1-\frac{-1}{\mathbf{e}})(1−e−1​), for both the classical and the quantum setting. We further - show this ratio to be optimal. It can be improved to(1−1t)(1-\frac{1}{t})(1−t1​)usingooo(ln(t)(t)(t)) additional bits (qubits) of communication.
The entanglement-assisted capacity of a quantum channel admits an additive single-letter characterization, implying that joint encodings across channel uses cannot increase the ultimate communication rate. Here, we show that this additive picture does not extend to communication reliability. Specifically, we prove that the Petz-Rényi channel information can be strictly superadditive for every α∈[1/2,1)α\in[1/2,1)α∈[1/2,1), yielding a genuine multi-copy enhancement of the entanglement-assisted random-coding error exponent, even though the entanglement-assisted capacity remains additive. We establish this phenomenon analytically already for measurement channels, which are entanglement-breaking and have additive unassisted capacity. Remarkably, this strict superadditivity is witnessed by a separable, classically correlated two-copy channel-input marginal, demonstrating that no entanglement between the transmitted systems is required. Our results show that, although correlations across channel uses cannot increase the ultimate rate of entanglement-assisted communication, they can enhance its reliability.