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arXiv:2102.12543 · Physical Review Research

Certifying the classical simulation cost of a quantum channel

Brian Doolittle, Eric Chitambar

Published Oct 26, 2021• 9 citations
arXivHTMLPDF

Abstract

A fundamental objective in quantum information science is to determine the cost in classical resources of simulating a particular quantum system. The classical simulation cost is quantified by the signaling dimension which specifies the minimum amount of classical communication needed to perfectly simulate a channel's input-output correlations when unlimited shared randomness is held between encoder and decoder. This paper provides a collection of device-independent tests that place lower and upper bounds on the signaling dimension of a channel. Among them, a single family of tests is shown to determine when a noisy classical channel can be simulated using an amount of communication strictly less than either its input or its output alphabet size. In addition, a family of eight signaling dimension witnesses is presented that completely characterize when any four-outcome measurement channel, such as a Bell measurement, can be simulated using one communication bit and shared randomness. Finally, we bound the signaling dimension for all partial replacer channels in $d$ dimensions. The bounds are found to be tight for the special case of the erasure channel.

DOI
10.1103/physrevresearch.3.043073

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IEEE International Symposium on Information Theory (ISIT)•Jun 26, 2022•1 citation
A fundamental objective in quantum information science is to determine the cost in classical resources of simulating a particular quantum system. The classical simulation cost is quantified by the signaling dimension which specifies the minimum amount of classical communication needed to perfectly simulate a channel's input-output correlations when unlimited shared randomness is held between encoder and decoder. This paper provides a collection of device-independent tests that place lower and upper bounds on the signaling dimension of a channel. Among them, a single family of tests is shown to determine when a noisy classical channel can be simulated using an amount of communication strictly less than either its input or its output alphabet size. In addition, a family of eight signaling dimension witnesses is presented that completely characterize when any four-outcome measurement channel, such as a Bell measurement, can be simulated using one communication bit and shared randomness. Finally, we bound the signaling dimension for all partial replacer channels in ddd dimensions. The bounds are found to be tight for the special case of the erasure channel.
A quantum system of any fixed dimension can be prepared in a continuum of states, yet it cannot be used to transmit an unlimited amount of classical information. Similarly, the correlations observed between measurement outcomes on separate parts of a shared quantum system can be stronger than classical correlations, but they cannot transmit information. These fundamental limitations suggest that the statistics observed from quantum communication and quantum correlations may admit a simulation using a finite amount of classical communication. This expectation is confirmed in the smallest nontrivial quantum dimension, with two classical bits being necessary and sufficient to exactly simulate qubit communication and all correlations between qubits. Despite significant efforts during the previous decades, this remained the only solved case. Here we resolve both problems for every quantum dimension. The solution reveals an unexpected qualitative transition starting at dimension four: no finite amount of classical communication can exactly simulate ququart communication nor all quantum correlations of two entangled ququarts, even with unlimited shared randomness. One might have expected this transition, if it existed, to appear already for qutrits. Instead, we construct an explicit protocol that exactly simulates qutrit communication using 357357357 classical bits, and consequently, all correlations of two entangled qutrits.
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.
In the general probabilistic model of information theory, the signaling dimension of a given communication protocol (or, equivalently, of its state space) is the smallest possible complexity of a classical communication protocol capable of perfectly simulating the given one in terms of input-output correlations. The signaling dimensions of three-, four-, and five-dimensional cross-polytopes have been known to be 3. We give logarithmic lower and upper bounds for the signaling dimensions of higher dimensional cross-polytopes, pinning down the value up to an additive gap of 2. The lower bound shows that the signaling dimension goes to infinity, in contrast to the case of balls and hypercubes.
Bell nonlocal correlations cannot be reproduced by local hidden-variable models without communication, making classical communication cost a natural quantitative measure of nonlocality. Although finite communication always suffices to simulate any correlation in a fixed finite Bell scenario, determining the minimum amount needed to (exactly) simulate any given nonlocal correlation remains challenging, especially in the average-cost setting. Here, we derive lower bounds on input-average communication cost from nonlocal games, allowing variable-length, fully interactive two-way protocols. When applied to parallel games, these bounds yield explicit finite quantum correlations for which the exact simulation cost exceeds any prescribed finite input-average communication budget. For nnn parallel Magic-Square-winning correlations, we prove a lower bound of nlog⁡2(3/2)n\log_2(3/2)nlog2​(3/2) bits and give a one-way communication protocol that attains this rate asymptotically. For the quantum correlation maximally winning the nnn-copy of the Clauser--Horne--Shimony--Holt nonlocal game, we obtain an input-average lower bound of approximately 0.04627n0.04627n0.04627n bits for exact simulation. Finally, we formulate a hierarchy of lower bounds and complementary upper bounds using deterministic correlations up to chosen communication costs. Both recover the exact input-average cost at their final levels, and if the bounds agree at lower levels, their common value yields the exact input-average cost without needing to consider all communication strategies.
Abstract Can quantum processes be simulated using only classical resources? This question delineates the boundary between classical and quantum models and clarifies the origin of quantum advantage in information processing. We address this question through the task of quantum channel simulation, where a sender (Alice) holds the classical description of a quantum state and wishes to transmit it to a receiver (Bob) for measurement. Prior work has shown that, for qubit channels, 2 bits of forward communication with shared randomness suffice to reproduce the statistics of any single-qubit measurement. We argue, however, that true channel simulation requires reproducing statistics of joint measurements—including entangled effects—on Alice’s state and an auxiliary system held by Bob. Such scenarios naturally arise in network communication, where some nodes know the state, while others do not. We prove that a perfect qubit channel cannot be simulated with any finite amount of classical communication, even using the most general multi-round, bidirectional protocols. We further show that this no-go result is rooted in the necessity of reproducing statistics associated with entangled effects. On the other hand, we show that noisy qubit channels, such as depolarizing channels, admit classical simulation, though the required communication diverges as noise decreases.
There are various ways to quantify the communication capabilities of a quantum channel. In this work we study the communication value (cv) of channel, which describes the optimal success probability of transmitting a randomly selected classical message over the channel. The cv also offers a dual interpretation as the classical communication cost for zero-error channel simulation using non-signaling resources. We first provide an entropic characterization of the cv as a generalized conditional min-entropy over the cone of separable operators. We evaluate the cv exactly for all qubit channels and the Werner-Holevo family of channels. The latter is shown to have non-multiplicative cv when d > 2. On the other hand, we prove that any pair of qubit channels have multiplicative cv when used in parallel. Even stronger, all entanglement-breaking channels and the partially depolarizing channel are shown to have multiplicative cv when used in parallel with any channel. We then turn to the entanglement-assisted cv and prove that it is equivalent to the conditional min-entropy of the Choi matrix of the channel. Combining with previous work on zero-error channel simulation, this implies that the entanglement-assisted cv is the classical communication cost for perfectly simulating a channel using quantum non-signaling resources. A final component of this work investigates relaxations of the channel cv to other cones such as the set of operators having a positive partial transpose (PPT).