Quantum Mechanical Bell Test Prediction - Demonstration of A Bell Inequality Violation

Demonstration of A Bell Inequality Violation

Now choose as the orientation angles of the transmission axes

a = 0, a′ = π/4, b = π/8, b′ = 3 π/8.

(6)

Then

EΦ(a, b) = cos 2(π/8) = 0.707,

(7a)

EΦ(a, b′) = cos 2(3π/8) = −0.707,

(7b)

EΦ(a′, b) = cos 2(−π/8) = 0.707,

(7c)

and

EΦ(a′, b′) = cos 2(π/8) = 0.707.

(7d)

Therefore the quantum mechanical prediction for the CHSH test statistic is

SΦ = EΦ(a, b) − EΦ(a, b′) + EΦ(a′, b) + EΦ(a′, b′) = 2.828,

(8)

exceeding the CHSH Bell test limit of 2 and thus completing the proof of a version of Bell's Theorem. In fact, all entangled quantum states yield predictions in violation of the inequality, as Gisin (1991) and Popescu and Rohrlich (1992) have independently demonstrated. Popescu and Rohrlich (1992) also show that the maximum amount of violation is achieved with a quantum state of maximum degree of entanglement, exemplified by |Φ> of Eq. (1).