Bilocality

Bilocality is a concept in quantum networks that describes a constraint on correlations when two independent sources distribute quantum systems to a small network of three parties arranged in a line. In the canonical bilocal scenario, two sources emit systems to three parties: the left party A receives from source 1, the central party B receives from both sources, and the right party C receives from source 2. The key assumption is that the two sources operate independently of each other.

In a bilocal hidden variable model, the observed correlations P(a,b,c|x,y,z) are generated from two independent hidden variables, λ1 and λ2, with independent distributions p(λ1) and p(λ2). The joint distribution factors as P(a,b,c|x,y,z) = ∫ dλ1 dλ2 p(λ1)p(λ2) P(a|x,λ1) P(b|y,λ1,λ2) P(c|z,λ2). The independence of the sources is encoded by the factorization p(λ1,λ2) = p(λ1)p(λ2). Bilocality thus imposes a stronger constraint than standard locality, because it requires independent sources in addition to locality at each node.

To test bilocality, researchers use bilocality inequalities—derived under the bilocal model—that bound the correlations achievable with independent sources. Violations of these inequalities by experimental data indicate non-bilocal quantum correlations, often observed in entanglement-swapping experiments that involve two independent sources.

Bilocality is part of a broader study of network nonlocality, which extends beyond the three-node, two-source paradigm to more complex network topologies such as star or triangle networks. These investigations explore how independence of sources and network structure shape the space of possible correlations in quantum theory.