We investigate whether certain non-classical communication channels can be simulated by a classical channel with a given number of states and a given `amount' of noise. It is proved that any noisy quantum channel can be simulated by a corresponding classical channel with `the same amount' of noise. Classical simulations of general probabilistic channels are also studied.
We investigate whether certain non-classical communication channels can be simulated by a classical channel with a given number of states and a given amount' of noise. It is proved that any noisy quantum channel can be simulated by a corresponding classical channel with the same amount' of noise. Classical simulations of general probabilistic channels are also studied.
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.
We study the problem of simulating protocols in a quantum communication setting over noisy channels. This problem falls at the intersection of quantum information theory and quantum communication complexity, and it will be of importance for eventual real-world applications of interactive quantum protocols, which can be proved to have exponentially lower communication costs than their classical counterparts for some problems. These are the first results concerning the quantum version of this problem, originally studied by Schulman in a classical setting [L. J. Schulman, Communication on noisy channels: A coding theorem for computation, in Proceedings of the 33rd Annual IEEE Symposium on Foundations of Computer Science, IEEE, 1992, pp. 724--733], [L. J. Schulman, Deterministic coding for interactive communication, in Proceedings of the 25th Annual ACM Symposium on Theory of Computing, ACM, 1993, pp. 747--756]. We simulate a length N quantum communication protocol by a length O(N) protocol with arbitrarily small error. Under adversarial noise, our strategy can withstand, for arbitrarily small ε>0, error rates as high as 1/2−ε when parties preshare perfect entanglement, but the classical channel is noisy. We show that this is optimal. We provide extension of these results in several other models of communication, including when also the entanglement is noisy, and when there is no preshared entanglement but communication is quantum and noisy. We also study the case of random noise, for which we provide simulation protocols with positive communication rates and no preshared entanglement over some quantum channels with quantum capacity CQ=0, proving that CQ is in general not the right characterization of a channel's capacity for interactive quantum communication. Our results are stated for a general quantum communication protocol in which Alice and Bob collaborate, and these results hold in particular in the quantum communication complexity settings of the Yao and Cleve--Buhrman models.
We study the problem of simulating protocols in a quantum communication setting over noisy channels. This problem falls at the intersection of quantum information theory and quantum communication complexity, and will be of importance for eventual real-world applications of interactive quantum protocols, which can be proved to have exponentially lower communication costs than their classical counterparts for some problems. These are the first results concerning the quantum version of this problem, originally studied by Schulman in a classical setting (FOCS '92, STOC '93). We simulate a length N quantum communication protocol by a length O(N) protocol with arbitrarily small error. Our simulation strategy has a far higher communication rate than a naive one that encodes separately each particular round of communication to achieve comparable success. Such a strategy would have a communication rate going to 0 in the worst interaction case as the length of the protocols increases, in contrast to our strategy, which has a communication rate proportional to the capacity of the channel used. Under adversarial noise, our strategy can withstand, for arbitrarily small ε > 0, error rates as high as 1/2 -- ε when parties preshare perfect entanglement, but the classical channel is noisy. We show that this is optimal. Note that in this model, the naive strategy would not work for any constant fraction of errors. We provide extension of these results in several other models of communication, including when also the entanglement is noisy, and when there is no pre-shared entanglement but communication is quantum and noisy. We also study the case of random noise, for which we provide simulation protocols with positive communication rates and no pre-shared entanglement over some quantum channels with quantum capacity Q = 0, proving that Q is in general not the right characterization of a channel's capacity for interactive quantum communication. Our results are stated for a general quantum communication protocol in which Alice and Bob collaborate, and hold in particular in the quantum communication complexity settings of the Yao and Cleve-Buhrman models.
We quantify in a unified approach the nonlocal resources of bipartite quantum channels and identify their component resisting the effects of dephasing noise. Despite its classical nature, we demonstrate that the latter plays a relevant role in performing quantum protocols, such as state transformations and quantum coding for noisy communication. When comparing different resource types, we show that simulating certain stochastic processes with quantum channels has a communication advantage with respect to their classical simulation, even when dephasing noise occurs. This approach also allows us to lower bound the amount of shared entanglement and communication needed to simulate bipartite quantum channels and to inner approximate the set of n-round local operations, assisted with classical communication.
The communication complexity of a quantum channel is the minimal amount of classical communication required for classically simulating the process of preparation, transmission through the channel, and subsequent measurement of a quantum state. At present, only little is known about this quantity. In this Letter, we present a procedure for systematically evaluating the communication complexity of channels in any general probabilistic theory, in particular, quantum theory. The procedure is constructive and provides the most efficient classical protocols. We illustrate this procedure by evaluating the communication complexity of a quantum depolarizing channel with some finite sets of quantum states and measurements.
Quantum theory is, in some sense, “non-classical.” For instance, the behavior of entangled systems (i.e., shared quantum information) under measurements cannot, in general, be explained by shared classical information. With classical communication, on the other hand, both the correlations entanglement leads to as well as quantum channels can be reproduced in principle. Here, crucial questions are whether the required communications is finite; if so, then its exact amount is related to the “degree of non-classicality” of the quantum primitive. We apply information-theoretic results such as the reverse Shannon theorem for determining the required communication in the asymptotic limit. The communication complexity of a quantum channel is the minimal amount of classical communication required for classically simulating the process of preparation, transmission through the channel, and subsequent measurement of a quantum state. At present, only little is known about this quantity. Our generic procedure allows for systematically evaluating the communication complexity of channels in any general probabilistic theory, in particular quantum theory. The procedure is constructive and provides the most efficient classical protocols. We illustrate it by evaluating the communication complexity of sending single qubits over a noiseless quantum channel with some finite sets of quantum states and measurements. As a second application, we determine the classical-communication rate required for the simulation of the behavior under measurements of entangled states. Here, the communication cost can be directly interpreted as the “non-classicality” of the correlation. A particular example is the simulation of non-maximally entangled pure qubit pairs, where we find the required communication rate to behave monotonically with the strength of the entanglement. For different measures of non-locality, such as the number of required non-local (PR) boxes, another behavior had been reported for the single-shot scenario.