FOUNDATIONS OF QUANTUM MECHANICS

VII. 1. LOCAL DETERMINISTIC HIDDEN VARIABLES 143
2
F (ϕ)
0
π/2
ϕ →
π
Figure VII. 3: The Bell inequality violated for every acute angle ϕ
We see that (VII. 14) is violated for every ϕ ∈ (0, 1 2 π). The maximum violation is F (60◦ ) = 5 2 ,
as can be seen in the figure.
Even larger violations are by the next configuration:possible in other configurations. The largest
violation is obtained in the configuration of figure VII. 4(with all vectors in a single plane),
leading to
E QM (⃗a, ⃗ b) = − cos 45 ◦ = − 1 2
√
2,
E QM (⃗a, ⃗ b ′ ) = − cos 135 ◦ = 1 2
√
2,
E QM (⃗a ′ , ⃗ b) = − cos 135 ◦ = 1 2
√
2,
E QM (⃗a ′ , ⃗ b ′ ) = − cos 135 ◦ = 1 2
√
2,
|E QM (⃗a, ⃗ b) − E QM (⃗a, ⃗ b ′ )| + |E QM (⃗a ′ , ⃗ b) + E QM (⃗a ′ , ⃗ b ′ )| = 2 √ 2. (VII. 15)
This is a violation of 41%. □
a
b
a ′ 45 ◦ b ′
Figure VII. 4: the configuration giving the largest violation of the Bell inequality (all vectors in the
same plane)