  We consider the problem of quantum measurement compression with side
information in the one-shot setting with shared randomness. In this problem,
Alice shares a pure state with Reference and Bob and she performs a measurement
on her registers. She wishes to communicate the outcome of this measurement to
Bob using shared randomness and classical communication, in such a way that the
outcome that Bob receives is correctly correlated with Reference and Bob's own
registers. Our goal is to simultaneously minimize the classical communication
and randomness cost. We provide a protocol based on convex-split and position
based decoding with its communication upper bounded in terms of smooth max and
hypothesis testing relative entropies.
  We also study the randomness cost of our protocol in both one-shot and
asymptotic and i.i.d. setting. By generalizing the convex-split technique to
incorporate pair-wise independent random variables, we show that our one shot
protocol requires small number of bits of shared randomness. This allows us to
construct a new protocol in the asymptotic and i.i.d. setting, which is optimal
in both the number of bits of communication and the number of bits of shared
randomness required.
  We construct a new protocol for the task of strong randomness extraction in
the presence of quantum side information. Our protocol achieves error guarantee
in terms of relative entropy (as opposed to trace distance) and extracts close
to optimal number of uniform bits. As an application, we provide new
achievability result for the task of quantum measurement compression without
feedback, in which Alice does not need to know the outcome of the measurement.
This leads to the optimal number of bits communicated and number of bits of
shared randomness required, for this task in the asymptotic and i.i.d. setting.
