Quantum violations of bipartite Bell facets
- Field
- Topic
Problem
Must every nontrivial facet Bell inequality in a finite bipartite scenario have a quantum violation? Let \(\mathcal L\) be the local polytope, and let \(\mathcal Q\) consist of behaviors realizable as
where \(\rho_{AB}\) is a finite-dimensional state and \(\{M_x^a\}_a\) and \(\{N_y^b\}_b\) are local POVMs. Equation (1) defines the quantum set used below.
For a linear functional \(F(p)=\sum_{a,b,x,y}c_{abxy}p(a,b\mid x,y)\), suppose \(F(p)\leq\beta_{\mathrm L}\) supports a facet of \(\mathcal L\) and is not a positivity facet within the normalization and no-signalling affine hull. Is it necessarily true that
Equation (2) asks whether the bipartite local and quantum sets can share a nontrivial facet-supporting hyperplane.
Source
Ramanathan explicitly asks whether every nontrivial bipartite facet Bell inequality, rather than only every such almost-quantum facet, admits a quantum violation [Ram21].
Progress
Escolà-Farràs, Calsamiglia, and Winter proved Eq. (2) for every tight bipartite correlation inequality, including every facet inequality of a two-player XOR game. Their theorem does not cover general Bell behaviors with arbitrary outcome probabilities [ECW20].
Ramanathan proved that every nontrivial bipartite facet is violated by the almost-quantum set and that the quantum statement holds for the correlation-polytope projection. Because the almost-quantum set strictly contains the quantum set, this does not establish Eq. (2) in general [Ram21].
Comment
The missing step is to replace the almost-quantum violation by a quantum behavior for every nontrivial bipartite facet. A solution may instead exhibit a facet for which \(\sup_{p\in\mathcal Q}F(p)=\beta_{\mathrm L}\), rather than the strict inequality in Eq. (2).