  We consider the task of faithfully simulating a distributed quantum
measurement, wherein we provide a protocol for the three parties, Alice, Bob
and Eve, to simulate a repeated action of a distributed quantum measurement
using a pair of non-product approximating measurements by Alice and Bob,
followed by a stochastic mapping at Eve. The objective of the protocol is to
utilize minimum resources, in terms of classical bits needed by Alice and Bob
to communicate their measurement outcomes to Eve, and the common randomness
shared among the three parties, while faithfully simulating independent
repeated instances of the original measurement. To achieve this, we develop a
mutual covering lemma and a technique for random binning of distributed quantum
measurements, and, in turn, characterize a set of sufficient communication and
common randomness rates required for asymptotic simulatability in terms of
single-letter quantum information quantities. Furthermore, using these results
we address a distributed quantum rate-distortion problem, where we characterize
the achievable rate-distortion region through a single-letter inner bound.
Finally, via a technique of single-letterization of multi-letter quantum
information quantities, we provide an outer bound for the rate-distortion
region.
