Bell inequalities for the simplest exclusivity graph

Abstract

Which is the simplest logical structure for which there is quantum nonlocality? We show that there are only three bipartite Bell inequalities with quantum violation associated with the simplest graph of relationships of exclusivity with a quantum-classical gap. These are the most elementary logical Bell inequalities. We show that the quantum violation of some well-known Bell inequalities is related to them. We test the three Bell inequalities with pairs of polarization-entangled photons and report violations in good agreement with the quantum predictions. Unlike other experiments testing noncontextuality inequalities with pentagonal exclusivity, the ones reported here are free of the compatibility loophole.Comment: REVTeX4, 6 pages, 4 figure

Graph-Theoretic Approach to Quantum Correlations

“…For a given Bell or NC inequality (i.e., for a given S), ϑ(G, w) only provides an upper bound to its quantum maximum [41]. Then, a natural question is whether, given G, there is a NC inequality that reaches ϑ(G).…”

“…For example, if G is a pentagon, there is no Bell inequality reaching ϑ(G) [41]. This is due to the extra constraints imposed by the Bell scenario which enforce a specific labeling of the events (see [41] for details). This shows the advantage of discussing quantum correlations in the framework of NC inequalities not referring to any specific experimental scenario.…”

Maximum quantum violations of a class of Bell inequalities

“…We would like to point out a very nice discussion by Gisin, which summarizes what was known and what was not as of 2007 [8]. We would also like to mention approaches to Bell inequalities based on nonlocal games [9], graph theory [10], and logical consistency conditions [11].…”

Exclusivity principle forbids sets of correlations larger than the quantum set

“…The principle of local orthogonality [14] may be seen as the E principle restricted to Bell scenarios. However, when this restriction is removed is when the E principle shows itself more powerful, since while for a given graph G, there is always a NC inequality for which QT reaches ϑ (G) [33], this is not true if "NC inequality" is replaced by "Bell inequality" [21].…”

“…A second approach consists of identifying principles that explain the set of quantum nonlocal correlations [6][7][8][9][10][11][12][13][14]. The third approach consists of identifying principles that explain the set of quantum contextual correlations [15][16][17][18][19] without restrictions imposed by a specific experimental scenario [20][21][22][23].…”