  It is known that quantum correlations exhibited by a maximally entangled
qubit pair can be simulated with the help of shared randomness, supplemented
with additional resources, such as communication, post-selection or non-local
boxes. For instance, in the case of projective measurements, it is possible to
solve this problem with protocols using one bit of communication or making one
use of a non-local box. We show that this problem reduces to a distributed
sampling problem. We give a new method to obtain samples from a biased
distribution, starting with shared random variables following a uniform
distribution, and use it to build distributed sampling protocols. This approach
allows us to derive, in a simpler and unified way, many existing protocols for
projective measurements, and extend them to positive operator value
measurements. Moreover, this approach naturally leads to a local hidden
variable model for Werner states.
