  The most trivial way to simulate classically the communication of a quantum
state is to transmit the classical description of the quantum state itself.
However, this requires an infinite amount of classical communication if the
simulation is exact. A more intriguing and potentially less demanding strategy
would encode the full information about the quantum state into the probability
distribution of the communicated variables, so that this information is never
sent in each single shot. This kind of simulation is called weak, as opposed to
strong simulations, where the quantum state is communicated in individual
shots. In this paper, we introduce a bounded-error weak protocol for simulating
the communication of an arbitrary number of qubits and a subsequent two-outcome
measurement consisting of an arbitrary pure state projector and its complement.
This protocol requires an amount of classical communication independent of the
number of qubits and proportional to Delta^{-1}, where Delta is the error and a
free parameter of the protocol. Conversely, a bounded-error strong protocol
requires an amount of classical communication growing exponentially with the
number of qubits for a fixed error. Our result improves a previous protocol,
based on the Johnson-Lindenstrauss lemma, with communication cost scaling as
Delta^{-2} log Delta^{-1}.
