Characterising the set of deterministic quantum correlations in prepare-and-measure scenarios

In quantum information science, distinguishing between classical and quantum correlations is fundamental. While Bell scenarios are the standard for this, they require complex resources like entanglement. This paper investigates the simpler 'prepare-and-measure' scenario, where a party prepares a quantum state and another measures it. The authors aim to create a general, systematic framework to certify when these correlations are 'non-classical' (i.e., cannot be explained by a deterministic model) under various communication restrictions, moving beyond simple dimension-based constraints.

The researchers adopt an adversarial perspective: they posit an adversary (Eve) who has full knowledge of the hidden variables governing the experiment. They define a correlation as 'classical' if Eve can perfectly predict the measurement outcomes of the receiver (Bob). By framing the problem this way, the authors show that certifying non-classicality is mathematically complementary to the task of quantum random number generation (QRNG). This allows them to adapt existing numerical techniques—specifically semidefinite programming (SDP) hierarchies—to test for the existence of deterministic models under three distinct constraints: fixed ensembles, bounded pairwise overlaps of states, and restricted observable expectation values.

The paper demonstrates that this adversarial approach is highly versatile. First, the authors derive an analytical bound for generic linear witnesses, showing that for a connected non-orthogonality graph, classical correlations are strictly bounded. Second, they provide numerical SDP-based feasibility tests for three representative scenarios. For example, in state-discrimination tasks, they show how the maximum success probability allowed by a classical model decreases as the overlap between prepared states changes, providing a clear threshold for certifying quantum behavior. These results show that the framework can be applied to diverse physical setups, including those with photon-number restrictions or specific state-overlap bounds, without needing to assume a fixed Hilbert space dimension.