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arXiv:1301.4441 · Physical Review Letters

Communication Complexity of Channels in General Probabilistic Theories

A. Montina, Marcell Pfaffhauser, Stefan Wolf

Published Oct 18, 2013• 10 citations
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Abstract

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.

DOI
10.1103/physrevlett.111.160502

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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.
The communication complexity of a quantum channel is the minimal amount of classical communication required for classically simulating a process of state preparation, transmission through the channel and subsequent measurement. It establishes a limit on the power of quantum communication in terms of classical resources. We show that classical simulations employing a finite amount of communication can be derived from a special class of hidden variable theories where quantum states represent statistical knowledge about the classical state and not an element of reality. This special class has attracted strong interest very recently. The communication cost of each derived simulation is given by the mutual information between the quantum state and the classical state of the parent hidden variable theory. Finally, we find that the communication complexity for single qubits is smaller than 1.28 bits. The previous known upper bound was 1.85 bits.
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.
In order to perform Quantum Cryptography procedures it is often essencial to ensure that the parties of the communication are authentic. Such task is accomplished by quantum authentication protocols which are distributed algorithms based on the intrinsic properties of Quantum Mechanics. The choice of an authentication protocol must consider that quantum states are very delicate and that the channel is subject to eavesdropping. However, even in face of the various existing definitions of quantum authentication protocols in the literature, little is known about them in this perspective, and this lack of knowledge may unfavor comparisons and wise choices. In the attempt to overcome this limitation, in the present work we aim at showing an approach to evaluate quantum authentication protocols based on the determination of their quantum communication complexity. Based on our investigation, no similar methods to analyze quantum authentication protocols were found in the literature. Pursuing this further, our approach has advantages that need to be highlighted: it characterizes a systematic procedure to evaluate quantum authentication protocols; its evaluation is intuitive, based only on the protocol execution; the resulting measure is a concise notation of what resources a quantum authentication protocol demands and how many communications are performed; it allows comparisons between protocols; it makes possible to analyze the communication effort when an eavesdropping occurs; and, lastly, it is likely to be applied in almost any quantum authentication protocol. To illustrate the proposed approach, we also bring results about its application in ten existing quantum authentication protocols (data origin authentication and identity authentication). Such evaluations increase the knowledge about the existing protocols, presenting its advantages, limitations and contrasts.
A two-party quantum communication process with classical inputs and outcomes can be simulated by replacing the quantum channel with a classical one. The minimal amount of classical communication required to reproduce the statistics of the quantum process is called its communication complexity. In the case of many instances simulated in parallel, the minimal communication cost per instance is called the asymptotic communication complexity. Previously, we reduced the computation of the asymptotic communication complexity to a convex minimization problem. In most cases, the objective function does not have an explicit analytic form, as the function is defined as the maximum over an infinite set of convex functions. Therefore, the overall problem takes the form of a minimax problem and cannot directly be solved by standard optimization methods. In this paper, we introduce a simple algorithm to compute the asymptotic communication complexity. For some special cases with an analytic objective function one can employ available convex-optimization libraries. In the tested cases our method turned out to be notably faster. Finally, using our method we obtain 1.2381.2381.238 bits as a lower bound on the asymptotic communication complexity of a noiseless quantum channel with the capacity of 1 qubit. This improves the previous bound of 1.2081.2081.208 bits.
We consider the process consisting of preparation, transmission through a quantum channel, and subsequent measurement of quantum states. The communication complexity of the channel is the minimal amount of classical communication required for classically simulating it. Recently, we reduced the computation of this quantity to a convex minimization problem with linear constraints. Every solution of the constraints provides an upper bound on the communication complexity. In this paper, we derive the dual maximization problem of the original one. The feasible points of the dual constraints, which are inequalities, give lower bounds on the communication complexity, as illustrated with an example. The optimal values of the two problems turn out to be equal (zero duality gap). By this property, we provide necessary and sufficient conditions for optimality in terms of a set of equalities and inequalities. We use these conditions and two reasonable but unproven hypotheses to derive the lower bound $n\ifmmode\times\else\texttimes\fi{}{2}^{n\ensuremath{-}1}$ for a noiseless quantum channel with capacity equal to nnn qubits. This lower bound can have interesting consequences in the context of the recent debate on the reality of the quantum state.