semidefinite programs, the NPA hierarchy, that in general does not halt. The exact values are known for a few more Bell inequalities: For the Braunstein–Caves inequalities we have that For the WWŻB inequalities the Tsirelson bound is For the inequality the Tsirelson bound is not known exactly, but concrete realisations give a lower bound of , and the NPA hierarchy gives an upper bound of . It is conjectured that only infinite-dimensional quantum states can reach the Tsirelson bound. Derivation from physical principles Significant research has been dedicated to finding a physical principle that explains why quantum correlations go only up to the Tsirelson bound and nothing more. Three such principles have been found: no-advantage for non-local computation, information causality and macroscopic locality. That is to say, if one could achieve a CHSH correlation exceeding Tsirelson's bound, all such principles would be violated. Tsirelson's bound also follows if the Bell experiment admits a strongly positive quansal measure. Tsirelson's problem There are two different ways of defining the Tsirelson bound of a Bell expression. One by demanding that the measurements are in a tensor product structure, and another by demanding only that they commute. Tsirelson's problem is the question of whether these two definitions are equivalent. More formally, let be a Bell expression, where is the probability of obtaining outcomes with the settings . The tensor product Tsirelson bound is then the supremum of the value attained in this Bell expression by making measurements and on a quantum state : The commuting Tsirelson bound is the supremum of the value attained in this Bell expression by making measurements and such that on a quantum state : Since tensor product algebras in particular commute, . In finite dimensions commuting algebras are always isomorphic to (direct sums of) tensor product algebras, so only for infinite dimensions it is possible that . Tsirelson's problem is the question of whether for all Bell expressions . This question was first considered by Boris Tsirelson in 1993, where he asserted without proof that . Upon being asked for a proof by Antonio Acín in 2006, he realized that the one he had in mind didn't work, and issued the question as an open problem. Together with Miguel Navascués and Stefano Pironio, Antonio Acín had developed an hierarchy of semidefinite programs, the NPA hierarchy, that converged to the commuting Tsirelson bound from above, and wanted to know whether