Qualitative equivalence between incompatibility and Bell nonlocality

TL;DR

This paper proves that any non-trivial measurement incompatibility structure in quantum theory is both necessary and sufficient for Bell inequality violations, linking measurement incompatibility directly to nonlocality.

Contribution

It establishes a qualitative equivalence between measurement incompatibility structures and Bell nonlocality, showing sufficiency of incompatibility for Bell violations in finite cases.

Findings

Any non-trivial joint measurability structure admits a Bell violation.

Measurement incompatibility is both necessary and sufficient for Bell nonlocality.

The results connect measurement incompatibility directly to quantum nonlocality.

Abstract

Measurements in quantum theory can fail to be jointly measurable. Like entanglement, this incompatibility of measurements is necessary but not sufficient for violating Bell inequalities. The (in)compatibility relations among a set of measurements can be represented by a joint measurability structure, i.e., a hypergraph whose vertices denote measurements and hyperedges denote all and only compatible sets of measurements. Since incompatibility is necessary for a Bell violation, the joint measurability structure on each wing of a Bell experiment must necessarily be non-trivial, i.e., it must admit a subset of incompatible vertices. Here we show that for any non-trivial joint measurability structure with a finite set of vertices, there exists a quantum realization with a set of measurements that enables a Bell violation, i.e., given that Alice has access to this incompatible set of measurements, there exists a set of measurements for Bob and an entangled state shared between them such that they can jointly violate a Bell inequality. Hence, a non-trivial joint measurability structure is not only necessary for a Bell violation, but also sufficient.