Characterizing the set of quantum correlations in prepare-and-measure quantum multichain-shaped networks

We introduce a hierarchy of tests satisfied by any probability distribution P that represents the quantum correlations generated in prepare-and-measure (P&M) quantum multichain-shaped networks, assuming only the inner-product information of prepared states. The P&M quantum multichain-shaped networks involve multiple measurement parties, with each measurement party potentially having multiple sequential receivers. We adapt the original Navascués, Pironio, and Acín (NPA) hierarchy by incorporating a finite number of linear and positive semi-definite constraints to characterize the quantum correlations in P&M quantum multichain-shaped networks. These constraints in each hierarchy are derived from sequential measurements and the inner-product matrix of prepared states. The adapted NPA hierarchy is further applied to tackle some quantum information tasks, including sequential quantum random access codes (QRACs) and semi-device-independent randomness certification. First, we derive the optimal trade-off between the two sequential receivers in the 2→1 sequential QRACs, and investigate randomness certification in its double violation region. Second, considering the presence of an eavesdropper in actual communication, we show how much local and global randomness can be certified using the optimal trade-off of 2→1 sequential QRACs. We also quantify the amount of local and global randomness that can be certified from the complete set of probabilities generated by the two sequential receivers. Our conclusion is that utilizing the full set of probabilities certifies more randomness than relying solely on the optimal trade-off relationship.