We usually think of point coordinates as just numbers: first x, then y, and it doesn't matter in which order you multiply them. But in some theories at very small scales, where the very notion of a point blurs, coordinates turn into matrices. Then their product depends on the order: x·y is not equal to y·x. This is called noncommutativity. Previously, it was unclear whether such a structure could be detected at all if single particles behave as usual. It turns out that it can — if you look at two particles at once.
The authors considered a source that emits two particles in sequence. Each flies to its own observer — Alice and Bob. When the particles swap places, an extra rotation appears in their joint state, a phase that depends on their momenta. This phase is the trace of noncommutativity. If the momenta are chosen correctly, their differences are not parallel, the phase is nonzero. Then Alice and Bob, performing measurements, obtain correlations that violate Bell's inequality. This means the particles are entangled, and a classical explanation is impossible. The magnitude of the violation directly depends on the noncommutativity parameter.
Previously, it was thought that noncommutative geometry is a pure abstraction that cannot be tested. Now it is clear that it leaves a trace in measurement statistics. Admittedly, such an experiment is not yet technically feasible, but the criterion itself already exists. If it can ever be carried out, it will show whether space is truly twisted at the deepest level.
If the momentum differences at Alice's and Bob's are parallel, the phase χ_θ vanishes, and noncommutativity becomes invisible no matter how large it is. You need the two-dimensional geometry of momentum modes.