The Classical-Communication Rate of Quantum Resources

Quantum theory is, in some way, “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 a noiseless quantum channel with some finite sets of quantum states and measurements. As a second application, we determine the classicalcommunication 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 nonmaximally entangled pure qubit pairs, where we find the required communication rate to behave monotonously 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.