Inflated graph states refuting communication-assisted local-hidden-variable models

Standard Bell inequalities hold when distant parties are not allowed to communicate. Barrett et al. [Phys. Rev. A 75, 012103 (2007)1050-294710.1103/PhysRevA.75.012103] found correlations from Pauli measurements on certain network graphs to refute a local-hidden-variable (LHV) description, even allowing some communication along the graph. This has recently found applications in proving separation between classical and quantum computing, in terms of shallow circuits, and distributed computing. The correlations presented by Barrett et al. can be understood as coming from an extension of three party Greenberger-Horne-Zeilinger state correlations which can be embedded on a graph state. In this paper, we propose systematic extensions of any graph state, which we dub inflated graph states, such that they exhibit correlations which refute any communication-assisted LHV model. We further show the smallest possible such example, with a seven-qubit linear graph state, as well as specially crafted smaller examples with five and four qubits. The latter is the smallest possible violation using binary inputs and outputs. © 2023 American Physical Society.