Abstract
The characterization of the set of quantum correlations is a problem of fundamental importance in quantum information. The question whether every proper (tight) Bell inequality is violated in quantum theory is an intriguing one in this regard. Here we make significant progress in answering this question, by showing that every tight Bell inequality is violated by ``almost-quantum'' correlations, a semidefinite programming relaxation of the set of quantum correlations. As a consequence, we show that many (classes of) Bell inequalities including two-party correlation Bell inequalities and multioutcome nonlocal computation games, which do not admit quantum violations, are not facets of the classical Bell polytope. To do this, we make use of the intriguing connections between Bell correlations and the graph-theoretic Lov\'asz-theta set, discovered by Cabello-Severini-Winter (CSW). We also exploit connections between the cut polytope of graph theory and the classical correlation Bell polytope, to show that correlation Bell inequalities that define facets of the lower dimensional correlation polytope are violated in quantum theory.
Highlights
Quantum correlations, i.e., the correlations between quantum systems in a Bell-type experiment, are of central interest in quantum information theory
A problem of fundamental importance in the characterization of quantum correlations was raised by Gill in [11], namely whether every tight Bell inequality is violated in quantum theory
The corresponding question for multiparty tight Bell inequalities had been previously answered in a fundamental breakthrough result by Fritz et al [14] who identified a class of nontrivial tight Bell inequalities that are not violated in quantum theory following results in [15,16], when three or more parties are involved in the Bell experiment
Summary
The boxes P that satisfy the above constraints form the no-signaling polytope of the Bell scenario NS[B(2; mA, dA; mB, dB)]. The convex hull of these LDBs forms the classical or Bell polytope denoted by C[B(2; mA, dA; mB, dB)] This is the set of all correlations obtainable from local hidden variable theories. Q[B(2; mA, dA; mB, dB)] lies within the no-signaling polytope This set consists of boxes P where each component POA,OB|IA,IB (oA, oB|iA, iB ) is obtained as POA,OB|IA,IB (oA, oB|iA, iB ) = Tr ρ EiAA,oA ⊗ EiBB,oB (4). A Bell inequality BGi · |P ωc(Gi ) supports a facet of the classical polytope if and only if D affinely independent boxes of C[B(2; mA, dA; mB, dB)] satisfy it with equality. A number of Bell inequalities achieve their optimal quantum violations already at this level, including the aforementioned correlation Bell inequalities
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