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arXiv:1807.05354 · IEEE Transactions on Information Theory

Quantum Channel Simulation and the Channel’s Smooth Max-Information

Kun Fang, Xin Wang, Marco Tomamichel, Mario Berta

Published Sep 27, 2019• 62 citations
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Abstract

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. First, we show that the minimum error of simulation and the one-shot quantum simulation cost under no-signalling assisted codes are given by semidefinite programs. Second, we introduce the channel's smooth max-information, which can be seen as a one-shot generalization of the mutual information of a quantum channel. We provide an exact operational interpretation of the channel's smooth max-information as the one-shot quantum simulation cost under no-signalling assisted codes, which significantly simplifies the study of channel simulation and provides insights and bounds for the case under entanglement-assisted codes. Third, we derive the asymptotic equipartition property of the channel's smooth max-information; i.e., it converges to the quantum mutual information of the channel in the independent and identically distributed asymptotic limit. This implies the quantum reverse Shannon theorem in the presence of no-signalling correlations. Finally, we explore the simulation cost of various quantum channels.

DOI
10.1109/tit.2019.2943858

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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. First, we show that the minimum error of simulation and the one-shot quantum simulation cost under no-signalling assisted codes are given by semidefinite programs. Second, we introduce the channel's smooth max-information, which can be seen as a one-shot generalization of the mutual information of a quantum channel. We provide an exact operational interpretation of the channel's smooth max-information as the one-shot quantum simulation cost under no-signalling assisted codes, which significantly simplifies the study of channel simulation and provides insights and bounds for the case under entanglement-assisted codes. Third, we derive the asymptotic equipartition property of the channel's smooth max-information; i.e., it converges to the quantum mutual information of the channel in the independent and identically distributed asymptotic limit. This implies the quantum reverse Shannon theorem in the presence of no-signalling correlations. Finally, we explore the simulation cost of various 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.
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.
Understanding the classical communication cost of simulating a quantum channel is a fundamental problem in quantum information theory, which becomes even more intriguing when considering the role of non-locality in quantum information processing. This paper investigates the bidirectional classical communication cost of simulating a bipartite quantum channel assisted by non-signalling correlations, which are permitted across the spatial dimension between the two parties. By introducing non-signalling superchannels, we present lower and upper bounds on the one-shot εεε-error one-way classical communication cost of a bipartite channel via its smooth max-relative entropy of one-way classical communication, and establish that the asymptotic exact cost is given by its max-relative entropy of one-way classical communication. For the bidirectional scenario, we derive semidefinite programming (SDP) formulations for the one-shot exact bidirectional classical communication cost via non-signalling bipartite superchannels. We further introduce a channel's bipartite conditional min-entropy as an efficiently computable lower bound on the asymptotic cost of bidirectional classical communication. Our results in both one-shot and asymptotic settings provide lower bounds on the entanglement-assisted simulation cost in scenarios where entanglement is available to the two parties. Moreover, we propose a seesaw-based algorithm to compute an upper bound on the minimum simulation error via local operations and shared entanglement, which provides valuable insights into the relationship between non-signalling bipartite superchannels and more physically realizable protocols. Numerical experiments demonstrate the effectiveness of our bounds in estimating communication costs for various quantum channels, showing that our bounds can be tight in different scenarios.
We study entanglement-assisted quantum and classical communication over a single use of a quantum channel, which itself can correspond to a finite number of uses of a channel with arbitrarily correlated noise. We obtain characterizations of the corresponding one-shot capacities by establishing upper and lower bounds on them in terms of the difference of two smoothed entropic quantities. In the case of a memoryless channel, the upper and lower bounds converge to the known single-letter formulas for the corresponding capacities, in the limit of asymptotically many uses of it. Our results imply that the difference of two smoothed entropic quantities characterizing the one-shot entanglement-assisted capacities serves as a one-shot analog of the mutual information, since it reduces to the mutual information, between the output of the channel and a system purifying its input, in the asymptotic, memoryless scenario.
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.
Using one channel to simulate another exactly with the aid of quantum no-signalling correlations has been studied recently. The one-shot no-signalling assisted classical zero-error simulation cost of non-commutative bipartite graphs has been formulated as semidefinite programms [Duan and Winter, IEEE Trans. Inf. Theory 62, 891 (2016)]. Before our work, it was unknown whether the one-shot (or asymptotic) no-signalling assisted zero-error classical simulation cost for general non-commutative graphs is multiplicative (resp. additive) or not. In this paper we address these issues and give a general sufficient condition for the multiplicativity of the one-shot simulation cost and the additivity of the asymptotic simulation cost of non-commutative bipartite graphs, which include all known cases such as extremal graphs and classical-quantum graphs. Applying this condition, we exhibit a large class of so-called cheapest-full-rank graphs whose asymptotic zero-error simulation cost is given by the one-shot simulation cost. Finally, we disprove the multiplicativity of one-shot simulation cost by explicitly constructing a special class of qubit-qutrit non-commutative bipartite graphs.