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The most trivial way to simulate classically the communication of a quantumstate is to transmit the classical description of the quantum state itself.However, this requires an infinite amount of classical communication if thesimulation is exact. A more intriguing and potentially less demanding strategywould encode the full information about the quantum state into the probabilitydistribution of the communicated variables, so that this information is neversent in each single shot. This kind of simulation is called weak, as opposed tostrong simulations, where the quantum state is communicated in individualshots. In this paper, we introduce a bounded-error weak protocol for simulatingthe communication of an arbitrary number of qubits and a subsequent two-outcomemeasurement consisting of an arbitrary pure state projector and its complement.This protocol requires an amount of classical communication independent of thenumber of qubits and proportional to Delta^{-1}, where Delta is the error and afree parameter of the protocol. Conversely, a bounded-error strong protocolrequires an amount of classical communication growing exponentially with thenumber of qubits for a fixed error. Our result improves a previous protocol,based on the Johnson-Lindenstrauss lemma, with communication cost scaling asDelta^{-2} log Delta^{-1}.

Exponential communication gap between weak and strong classical simulations of quantum communication