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9) As in the case of the measurement Π z,A ∈ Π z,B , the measurement outcomes are perfectly correlated. 1 Quantum Non-locality 29 by Eq. 10) and the outcomes are completely uncorrelated. To summarize this section, there are cases in which an entangled state exhibits a correlation between measurement outcomes whereas separable states do not. This implies that multiple joint probability distributions are necessary for testing quantum entanglement. 2 CHSH Inequality As shown in the previous section, entangled states have a specific property that separable states do not.

The difference between what we prepared and what we want to prepare. However we do no know ρˆ and so cannot calculate ∆(ρ, ˆ ρˆ∗ ). In many experiments, the accuracy of the preparation is evaluated by ˆ ρˆ∗ ) (see Fig. 1). This is not reasonthe quantity of ∆(ρˆ Nest , ρˆ∗ ) instead of ∆(ρ, able in general, but when N is sufficiently large, this can be justified as follows. Suppose that the loss function ∆ satisfies the triangle inequality. Then we have ˆ ρˆ Nest ) + ∆(ρˆ Nest , ρˆ∗ ). 39) When N is sufficiently large, the value of ∆(ρ, ˆ ρˆ Nest ) is negligible.

45) is obtained immediately from Eqs. 17). From Eq. 44), we can see that the range of the expected loss changes as follows: √ 8 , (2 2 < |CCHSH ( p)| → 4), N √ 12 CHSH → Δ¯ abs2 , (2 < |CCHSH ( p)| → 2 2), , C L | p) < N (X N 16 abs2 CHSH L ¯ → Δ N (X , (0 → |CCHSH ( p)| → 2). 48) corresponds to the super-quantum joint probability distributions. 3 Test of the CHSH Inequality 35 by quantum mechanics but cannot be described by any local hidden-variable model. 50) corresponds to those which can be described by local hidden-variable models.