Exact Classical Simulation of the GHZ Distribution

Abstract

John Bell has shown that the correlations entailed by quantum
mechanics cannot be reproduced by a classical process involving
non-communicating parties. But can they be simulated with the help
of bounded communication? This problem has been studied for more
than twenty years and it is now well understood in the case of
bipartite entanglement. However, the issue was still widely open for
multipartite entanglement, even for the simplest case, which is
the tripartite Greenberger-Horne-Zeilinger (GHZ) state.
We give an exact simulation of arbitrary independent von Neumann
measurements on general n-partite GHZ states.  Our protocol
requires O(n^2) bits of expected communication between the
parties, and O(n*log(n)) expected time is sufficient to carry it
out in parallel. Furthermore, we need only an expectation of
O(n) independent unbiased random bits, with no need for the
generation of continuous real random variables nor prior shared
random variables. In the case of equatorial measurements, we
improve earlier results with a protocol that needs only O(n*log(n)) bits of communication and O(log^2(n)) parallel time. At the
cost of a slight increase in the number of bits communicated, these
tasks can be accomplished with a constant expected number of rounds.