Partial list of bipartite Bell inequalities with four binary settings

We give a partial list of 26 tight Bell inequalities for the case where Alice and Bob choose among four two-outcome measurements. All tight Bell inequalities with less settings are reviewed as well. For each inequality we compute numerically the maximal quantum violation, the resistance to noise and the minimal detection efficiency required for closing the detection loophole. Surprisingly, most of these inequalities are outperformed by the CHSH inequality. Finding all the Bell inequalities for a given number of measurement settings and outcomes is a difficult problem [1]. Even for the case of binary settings (two-outcome settings), few is known. All tight Bell inequalities, i.e. facets of the local polytope [1, 2], have been listed for the following cases: 2222, M222 (M ≥ 3), 3322 and 4322, where ijmn refers to the situation where Alice chooses among i settings with m outcomes and Bob among j settings with n outcomes. For the cases 2222 and M222 (M ≥ 3), there is only one Bell inequality [3, 4]: the famous Clauser-Horne-Shimony-Holt (CHSH) [5] inequality CHSH ≡ −1 0 −1 1 1 0 1 −1 ≤ 0 . (1) Here the notation represents the coefficients that are put in front of the probabilities, according to P (rB = 0|y) P (rA = 0|x) P (rA = rB = 0|xy) , (2) where x (y) denotes the measurement setting of Alice (Bob) and rA (rB) its result. In the 3322 case, only one new inequality [4] appears I3322 ≡ −1 0 0 −2 1 1 1 −1 1 1 −1 0 1 −1 0 ≤ 0 (3) Note that by first inverting the output of Alice’s first setting as well as Bob’s second and third settings, and then relabelling the settings, one gets a symmetric version of I3322 Ĩ3322 ≡ −1 −1 0 −1 0 1 1 −1 1 −1 1 0 1 1 −1 ≤ 0 . (4) For the case 4322, there are three new Bell inequalities [4]: I 1 4322 ≡ −1 0 0 −2 1 1 1 −1 1 −1 1 −1 1 1 −1 0 1 −1 −1 ≤ 0 I 4322 ≡ −2 −1 0 −1 1 1 1 0 0 1 −1 0 1 −1 0 0 1 0 −1 ≤ 0 I 4322 ≡ −1 −1 0 −2 2 1 1 −1 −1 1 1 0 0 1 −1 0 1 −1 −1 ≤ 0