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

TL;DR

This paper develops a hierarchy of tests to characterize quantum correlations in prepare-and-measure chain-shaped networks, enabling analysis of quantum information tasks like QRACs and randomness certification with improved precision.

Contribution

It introduces an adapted NPA-hierarchy incorporating sequential measurement constraints for P extbar M quantum networks, advancing the analysis of quantum correlations and randomness certification.

Findings

Derived the optimal trade-off in 2→1 sequential QRACs.

Analyzed semi-device-independent randomness certification.

Quantified local and global randomness from complete probabilities.

Abstract

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 chain-shaped networks, assuming only the inner-product information of the non-orthogonal quantum states. The P\&M quantum chain-shaped networks involve multiple measurement parties, each measurement party potentially having multiple sequential receivers. Specifically, we adapt the original NPA-hierarchy by incorporating a finite number of linear and positive semi-definite constraints to characterize the quantum correlations in P\&M quantum chain-shaped networks. These constraints in each hierarchy are derived from sequential measurement operators and the inner-product matrix of the non-orthogonal quantum states. We apply the adapted NPA-hierarchy to tackle some quantum information tasks, including sequential quantum random access codes (QRACs) and randomness certification. First, we derive the optimal trade-off between the two sequential receivers in the $2 \to 1$ sequential QRACs. Furthermore, we have investigated semi-device-independent randomness certification in the double violation region of $2 \to 1$ sequential QRACs. Second, considering the presence of eavesdropper (Eve) in actual communication, we show how much global and local randomness can be certified using the optimal trade-off of $2 \to 1$ sequential QRACs. Additionally, we quantify the amount of local and global randomness that can be certified from the complete probabilities generated by the two sequential receivers. We conclude that utilizing the complete set of probabilities certifies more local and global randomness than relying solely on the optimal trade-off relationship.