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arXiv:2410.10770 · IEEE Transactions on Information Theory

Exponents for Classical-Quantum Channel Simulation in Purified Distance

Aadil Oufkir, Yongsheng Yao, Mario Berta

Published May 6, 2026• 1 citation
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Abstract

We determine the exact error and strong converse exponent for entanglement-assisted classical-quantum channel simulation in worst case input purified distance. The error exponent is expressed as a single-letter formula optimized over sandwiched Rényi divergences of order $α\in [1, \infty)$, notably without the need for a critical rate--a sharp contrast to the error exponent for classical-quantum channel coding. The strong converse exponent is expressed as a single-letter formula optimized over sandwiched Rényi divergences of order $α\in [\frac{1}{2},1]$. As in the classical work [Oufkir et al., arXiv:2410.07051], we start with the goal of asymptotically expanding the meta-converse for channel simulation in the relevant regimes. However, to deal with non-commutativity issues arising from classical-quantum channels and entanglement-assistance, we critically use various properties of the quantum fidelity, additional auxiliary channel techniques, approximations via Chebyshev inequalities, and entropic continuity bounds.

DOI
10.1109/tit.2026.3690906

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The Quantum Reverse Shannon Theorem has been a milestone in quantum information theory. It states that asymptotically reliable simulation of a quantum channel, assisted by unlimited shared entanglement, requires a rate of classical communication equal to the channel’s entanglement-assisted classical capacity. In this paper, we study the error exponent of quantum channel simulation, which characterizes the optimal speed of exponential convergence of the performance towards the perfect, as the blocklength increases. Based on channel purified distance, we derive lower and upper bounds for the error exponent. Then we show that the two bounds coincide when the classical communication rate is below a critical value, and hence, we have determined the exact formula of the error exponent in the low-rate case. This enables us to obtain an operational interpretation to the channel’s sandwiched Rényi information of order from 1 to 2, since our formula is expressed as a transform of this quantity. In the derivation, we have also obtained an achievability bound for quantum channel simulation in the finite-blocklength setting, which is of realistic significance.
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.
Quantum information decoupling is a fundamental primitive in quantum information theory, underlying various applications in quantum physics. We prove a novel one-shot decoupling theorem formulated in terms of quantum relative entropy distance, with the decoupling error bounded by two sandwiched Rényi conditional entropies. In the asymptotic i.i.d. setting of standard information decoupling via partial trace, we show that this bound is ensemble-tight in quantum relative entropy distance and thereby yields a characterization of the associated decoupling error exponent in the low-cost-rate regime. Leveraging this framework, we derive several operational applications formulated in terms of purified distance: (i) single-letter expressions for the exact error exponents of quantum state merging in the low-entanglement-cost and high-entanglement-distillation-rate regimes, in terms of Petz-Rényi conditional entropies, and (ii) regularized expressions for achievable error exponents of entanglement distillation and quantum channel coding in terms of Petz-Rényi coherent informations. We further prove that these achievable bounds are tight for maximally correlated states and generalized dephasing channels, respectively, for the high distillation-rate/coding-rate regimes.
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 determine the strong converse exponent for the entanglement cost of quantum state merging, showing that it is characterized by the optimized ααα-zzz conditional Rényi entropies with z=α/2∈[1/2,1]z=α/2\in[1/2,1]z=α/2∈[1/2,1]. This contrasts with the sandwiched conditional Rényi entropies that typically govern strong converse exponents in quantum information theory. As a consequence, we derive the strong converse exponent of the partially smoothed conditional min-entropy in purified distance. This exponent is governed by club-sandwiched conditional entropies, whereas global smoothing leads to a sandwiched expression.
We determine the exact strong converse exponent for entanglement-assisted classical communication of a quantum channel. Our main contribution is the derivation of an upper bound for the strong converse exponent which is characterized by the sandwiched Rényi divergence. It turns out that this upper bound coincides with the lower bound of Gupta and Wilde (Commun. Math. Phys. 334:867-887, 2015). Thus, the strong converse exponent follows from the combination of these two bounds. Our result has two implications. Firstly, it implies that the exponential bound for the strong converse property of quantum-feedback-assisted classical communication, derived by Cooney, Mosonyi and Wilde (Commun. Math. Phys. 344:797-829, 2016), is optimal. This answers their open question in the affirmative. Hence, we have determined the exact strong converse exponent for this problem as well. Secondly, due to an observation of Leung and Matthews, it can be easily extended to deal with the transmission of quantum information under the assistance of entanglement or quantum feedback, yielding similar results. The above findings provide, for the first time, a complete operational interpretation to the channel’s sandwiched Rényi information of order α > 1.