Distributed instrument simulation with quantum side information in the one-shot regime

Three distributed parties, two transmitters (Txs) and a receiver (Rx), hold one component each of a tripartite quantum state (rho^). The goal is to simulate the action of a separable instrument acting on the (A_1) and (A_2) components, with the Rx recovering the classical outcome. To enable this, each Tx (k) can transfer bits on a noiseless bit pipe and share randomness at rates (R_k) and (C_k), respectively, with the Rx. Undertaking a Shannon-theoretic study, we characterize two new sets of inner bounds. The first set, derived for the one-shot regime, is based on instrument simulation protocols built using unstructured IID codes, while the second set, derived for the asymptotic regime, relies on coset codes and new decoding POVMs. The first set of bounds recovers current known inner bounds for instrument and measurement simulation in all previously studied scenarios. Our protocols are based on likelihood POVMs, and our analysis leverages Sen's smooth multiparty covering and simultaneous decoding, while handling the distributed-component scenario via a compatible operator sliding trick.