User:Boris Tsirelson/Sandbox1

Classical physics obeys the counterfactual definiteness and therefore negates entanglement. Classical apparata A, B cannot help Alice and Bob to always win (that is, agree on the intersection). What about quantum apparata? The answer is quite unexpected.

First, quantum apparata cannot ensure that Alice and Bob win always. Moreover, the winning probability does not exceed

${\frac {2+{\sqrt {2}}}{4}}=0.853\dots$

no matter which quantum apparata are used.

Second, there exist quantum apparata that ensure a winning probability higher than 3/4 = 0.75. This is a manifestation of entanglement, since under the three classical assumptions (counterfactual definiteness, local causality and no-conspiracy) the winning probability cannot exceed 3/4 (the classical bound). But moreover, ideal quantum apparata can reach the winning probability $(2+{\sqrt {2}})/4$ (the quantum bound), and non-ideal quantum apparata can get arbitrarily close to this bound.

Third, a modification of the game, called "magic square game", makes it possible to win always. To this end we replace 2x2 matrices with 3x3 matrices, still of numbers 0 and 1 only, with the following conditions:

• the parity of each row is even,
• the parity of each column is odd.

The classical bound is equal to 8/9; the quantum bound is equal to 1.