Correlations constrained by composite measurements

How to understand the set of correlations admissible in nature is one outstanding open problem in the core of the foundations of quantum theory. Here we take a complementary viewpoint to the device-independent approach, and explore the correlations that physical theories may feature when restricted by some particular constraints on their measurements. We show that demanding that a theory exhibits a composite measurement imposes a hierarchy of constraints on the structure of its sets of states and effects, which translate to a hierarchy of constraints on the allowed correlations themselves. We moreover focus on the particular case where one demands the existence of a correlated measurement that reads out the parity of local fiducial measurements. By formulating a non-linear Optimisation Problem, and semidefinite relaxations of it, we explore the consequences of the existence of such a parity reading measurement for violations of Bell inequalities. In particular, we show that in certain situations this assumption has surprisingly strong consequences, namely, that Tsirelson's bound can be recovered.

Automated summary

Understanding the permissible set of correlations in nature is a fundamental problem in quantum theory. This paper takes a complementary viewpoint and investigates the correlations that physical theories can exhibit under certain constraints on measurements. It is shown that demanding a theory to have a composite measurement imposes constraints on the structure of its sets of states and effects, leading to constraints on the allowed correlations. The existence of a correlated measurement that reads out the parity of local fiducial measurements is specifically studied, and it is found that this assumption has strong consequences for violations of Bell inequalities, including the recovery of Tsirelson's bound in certain situations.