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arXiv:2510.04552

Quantum Reverse Shannon Theorem Simplified

Gilad Gour

Published Oct 6, 2025• 0 citations
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Abstract

We revisit the quantum reverse Shannon theorem, a central result in quantum information theory that characterizes the resources needed to simulate quantum channels when entanglement is freely available. We derive a universal additive upper bound on the smoothed max-information in terms of the sandwiched Rényi mutual information. This bound yields tighter single-shot results, eliminates the need for the post-selection technique, and leads to a conceptually simpler proof of the quantum reverse Shannon theorem. By consolidating and streamlining earlier approaches, our result provides a clearer and more direct understanding of the resource costs of simulating quantum channels.

DOI
10.48550/arxiv.2510.04552

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The Quantum Reverse Shannon Theorem states that any quantum channel can be simulated by an unlimited amount of shared entanglement and an amount of classical communication equal to the channel’s entanglement assisted classical capacity. In this paper, we provide a new proof of this theorem, which has previously been proved by Bennett, Devetak, Harrow, Shor, and Winter. Our proof has a clear structure being based on two recent information-theoretic results: one-shot Quantum State Merging and the Post-Selection Technique for quantum channels.
We study the general framework of quantum channel simulation, that is, the ability of a quantum channel to simulate another one using different classes of codes. Our main results are as follows. First, we show that the minimum error of simulation under non-signalling assisted codes is efficiently computable via semidefinite programming. The cost of simulating a channel via noiseless quantum channels under non-signalling assisted codes can also be characterized as a semidefinite program. Second, we introduce the channel's smooth max-information, which can be seen as a one-shot generalization of the channel's mutual information. We show that the one-shot quantum simulation cost under non-signalling assisted codes is exactly equal to the channel's smooth max-information. Due to the quantum reverse Shannon theorem, the channel's smooth max-information converges to the channel's mutual information in the independent and identically distributed asymptotic limit. Together with earlier findings on the (activated) non-signalling assisted one-shot capacity of channels [Wang et al., arXiv:1709.05258], this suggest that the operational min- and max-type one-shot analogues of the channel's mutual information are the channel's hypothesis testing relative entropy and the channel's smooth max-information, respectively.
The entanglement cost of a quantum channel is the minimal rate at which entanglement (between sender and receiver) is needed in order to simulate many copies of a quantum channel in the presence of free classical communication. In this paper, we show how to express this quantity as a regularized optimization of the entanglement formation over states that can be generated between sender and receiver. Our formula is the channel analog of a well-known formula for the entanglement cost of quantum states in terms of the entanglement of formation and shares a similar relation to the recently shattered hope for additivity. The entanglement cost of a quantum channel can be seen as the analog of the quantum reverse Shannon theorem in the case where free classical communication is allowed. The techniques used in the proof of our result are then also inspired by a recent proof of the quantum reverse Shannon theorem and feature the one-shot formalism for quantum information theory, the postselection technique for quantum channels as well as Sion's minimax theorem. We discuss two applications of our result. First, we are able to link the security in the noisy-storage model to a problem of sending quantum rather than classical information through the adversary's storage device. This not only improves the range of parameters where security can be shown, but also allows us to prove security for storage devices for which no results were known before. Second, our result has consequences for the study of the strong converse quantum capacity. Here, we show that any coding scheme that sends quantum information through a quantum channel at a rate larger than the entanglement cost of the channel has an exponentially small fidelity.
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.
Characterizing the minimal communication needed for quantum channel simulation is a fundamental task in the quantum information theory. In this paper, we show that, in fidelity, the quantum channel simulation can be directly achieved via quantum state splitting without using a technique known as the de Finetti reduction, and thus provide a pair of tighter one-shot bounds. This opens up new potentials for higher-order analysis. Using the bounds, we also recover the quantum reverse Shannon theorem in a much simpler way.