The I3322 inequality is the simplest bipartite two-outcome Bell inequality beyond the ClauserHorne-Shimony-Holt (CHSH) inequality, consisting of three two-outcome measurements per party. In case of the CHSH inequality the maximal quantum violation can already be attained with local two-dimensional quantum systems, however, there is no such evidence for the I3322 inequality. In this paper a family of measurement operators and states is given which enables us to attain the largest possible quantum value in an infinite dimensional Hilbert space. Further, it is conjectured that our construction is optimal in the sense that measuring finite dimensional quantum systems is not enough to achieve the true quantum maximum. We also describe an efficient iterative algorithm for computing quantum maximum of an arbitrary two-outcome Bell inequality in any given Hilbert space dimension. This algorithm played a key role to obtain our results for the I3322 inequality, and we also applied it to improve on our previous results concerning the maximum quantum violation of several bipartite two-outcome Bell inequalities with up to five settings per party.

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We have determined the maximum quantum violation of 241 tight bipartite Bell inequalities with up to five two-outcome measurement settings per party by constructing the appropriate measurement operators in up to six-dimensional complex and eight-dimensional real component Hilbert spaces using numerical optimization. Out of these inequalities 129 has been introduced here. In 43 cases higher dimensional component spaces gave larger violation than qubits, and in 3 occasions the maximum was achieved with six-dimensional spaces. We have also calculated upper bounds on these Bell inequalities using a method proposed recently. For all but 20 inequalities the best solution found matched the upper bound. Surprisingly, the simplest inequality of the set examined, with only three measurement settings per party, was not among them, despite the high dimensionality of the Hilbert space considered. We also computed detection threshold efficiencies for the maximally entangled qubit pair. These could be lowered in several instances if degenerate measurements were also allowed.PACS numbers: 03.65.Ud, 03.67.-a I. INTRODUCTIONSuppose two classical systems which are separated from each other. Let us make some local measurements on them. Then the results of these measurements may be correlated, which may be explained by shared randomness experienced in the past. For a given number of possible measurement choices (inputs) and results (outputs) the set of correlations forms a polytope whose finite number of vertices correspond to all the deterministic assignments of outputs to inputs. The facets of the polytope, which form the boundary of the classical region, correspond to tight Bell inequalities [1,2]. Thus Bell inequalities has no a priori relation to quantum physics. However, quantum physics violates them, which has been verified experimentally in numerous occasions up to some technical loopholes (see e.g., [3]). Indeed, this makes Bell inequalities very interesting.On the other hand, one may ask what is the achievable set of correlations if one allows the two parties to share quantum resources as well over shared randomness. This is a convex set, such as for the classical case, however it cannot be described by a finite number of extreme points [4]. Nevertheless, one can construct so called quantumBell inequalities which bound the correlations achievable by quantum physics [4-6] (see also item 5 in Sec. III.C of Ref. [7]). A simple way to form quantum-Bell inequalities is to use the coefficients of known Bell inequalities, and determine the maximum value one can get by performing measurements on quantum systems of arbitrary dimensions.The simplest result in this respect is the Tsirelson bound [8] stating that quantum correlations cannot vio- * Electronic address: kfpal@atomki.hu † Electronic address: tvertesi@dtp.atomki.hu late the CHSH inequality [2] by more than ( √ 2 − 1)/2. Note, however that the requirement of non-signaling alone allows a higher bound of 1 [9,10] (the maximum allowed value in a local classical theory is 0). I...

We propose a new optimization method based on a demagnetization procedure well known in magnetism. We show how this procedure can be applied as a general tool to search for optimal solutions in any system where the configuration space is endowed with a suitable "distance." We test the new algorithm on frustrated magnetic models and the traveling salesman problem. We find that the new method successfully competes with similar basic algorithms such as simulated annealing.

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