Measurement compression with quantum side information using shared randomness

We consider the problem of measurement compression with side information in the one-shot setting with shared randomness. In this problem, Alice shares a pure state with Referee and Bob and she performs a measurement on her registers. She wishes to communicate the outcome of this measurement to Bob using shared randomness only, in such a way that the outcome that Bob receives is correctly correlated with Referee and Bob’s own registers. We provide a protocol based on convex split and position based decodingwith its communication upper bounded in terms of smooth-max and hypothesis testing divergences.