To complete the proof of Bell's theorem it is necessary to demonstrate that quantum mechanics makes a prediction that violates a "Bell inequality".

The following is based on section 2 of an encyclopaedia article written by Abner Shimony, one of the authors of the original Clauser, Horne, Shimony and Holt article (1969) after which the CHSH Bell test is named (Shimony, 2004).

Shimony's derivation of the quantum mechanical prediction

Let the system consist of a pair of photons A and B propagating respectively in the z and −z directions. The two kets |x>_j and |y>_j constitute a polarisation basis for photon j (j =A, B), the former representing (in Dirac's notation) a state in which the photon A is linearly polarized in the x-direction and the latter a state in which it is linearly polarized in the y-direction. For the two-photon system the four product kets |x>_A |x>_B, |x>_A |y>_B, |y>_A |x>_B, and |y>_A |y>_B constitute a polarisation basis. Each two-photon polarisation state can be expressed as a linear combination of these four basis states with complex coefficients. Of particular interest are the entangled quantum states, which in no way can be expressed as |φ>_A|ξ>_B, with |φ> and |ξ> single-photon states, an example being

(1)    |Φ> = (1/√2) [ |x>_A |x>_B + |y>_A |y>_B ],

which has the useful property of being invariant under rotation of the x and y axes in the plane perpendicular to z. Neither photon A nor photon B is in a definite polarisation state when the pair is in the state |Φ>, but their potentialities (in the terminology of Heisenberg 1958) are correlated: if by measurement or some other process the potentiality of photon A to be polarised along the x-direction or along the y-direction is actualised, then the same will be true of photon B, and conversely.

Suppose now that photons A and B impinge respectively on the faces of birefringent crystal polarisation analysers, with the entrance face of each analyser perpendicular to z. Each analyser has the property of separating light incident upon its face into two outgoing non-parallel rays, the ordinary ray and the extraordinary ray. The transmission axis of the analyser is a direction with the property that a photon polarised along it will emerge in the ordinary ray (with certainty if the crystals are assumed to be ideal), while a photon polarized in a direction perpendicular to z and to the transmission axis will emerge in the extraordinary ray. (See diagram.)

The crystals are also idealised by assuming that no incident photon is absorbed, but each emerges in either the '+' or the '−' channel. Quantum mechanics provides an algorithm for computing the probabilities that photons A and B will emerge from these idealised analysers in specified rays, as functions of the orientations a and b of the analysers, a being the angle between the transmission axis of the 'A' analyser and an arbitrary fixed direction in the x–y plane, and b having the analogous meaning for B:

(2a)    prob_Φ (j, k |a, b) = | <Φ|θ_j>_A |ξ_k>_B |².

Here j is a quantum number associated with the ray into which photon A emerges, taking values +1 or −1 depending on which channel it emerges from, while k is the analogous quantum number for photon B; and | <Φ|θ_j>_A |ξ_k>_B | is the ket representing the quantum state of photons A and B with the respective quantum numbers j and k. Calculation of the probabilities of interest from Eq. (2a) can be simplified by using the invariance noted after Eq. (1) and rewriting |Φ> as

(3)    |Φ> = (1/√2) [ |θ₁>_A |θ₁>_B + |θ₋₁>_A |θ₋₁>_B ].

Eq. (3) results from Eq. (1) by substituting the transmission axis of the A analyser for x and the direction perpendicular to both z and this transmission axis for y.

Since |θ₋₁>_A is orthogonal to |θ₁>_A, only the first term of Eq. (3) contributes to the inner product in Eq. (2a) if j = k = 1; and since the inner product of |θ₁>_A with itself is unity because of normalisation, Eq. (2a) reduces for j = k = 1 to

(2b)    prob_Φ (1, 1 |a, b) = 1/2 |  _B<θ₁ | φ₁>_B  |².

Finally, the expression on the right hand side of Eq. (2b) is evaluated by using the law of Malus, which is preserved in the quantum mechanical treatment of polarisation states: that the probability for a photon polarised in a direction n to pass through an ideal polarization analyser with axis of transmission n′ equals the squared cosine of the angle between n and n′. Hence

(4a)    prob_Φ (1, 1 |a, b) = 1/2 cos²σ

where σ is b − a. Likewise,

(4b)    prob_Φ (−1, −1 |a, b) = 1/2 cos²σ, and

(4c)    prob_Φ (1, −1|a, b) = prob_Φ (−1, 1|a, b) = 1/2 sin²σ.

The expectation value of the product of the results j and k of the polarisation analyses of photons A and B by their respective analysers is

(5)    E_Φ (a, b) = prob_Φ (1, 1 |a, b) + prob_Φ (−1, −1 |a, b) − prob_Φ (1, −1 |a, b) − prob_Φ (−1, 1 |a, b)

= cos²σ − sin²σ

= cos 2σ.

The required quantum mechanical predictions are thus 1/2 cos²σ for coincidence probabilities and cos 2σ for quantum correlations, where σ is the angle between the detectors.

Demonstration of infringement of a Bell inequality

Now choose as the orientation angles of the transmission axes

(6)    a = 0, a′ = π/4, b = π/8, b′ = 3 π/8 .

Then

(7a)    E_Φ(a, b) = cos2(π/8) = 0.707,

(7b)    E_Φ(a, b′) = cos2(3π/8) = −0.707,

(7c)    E_Φ(a′, b) = cos2(−π/8) = 0.707, and

(7d)    E_Φ(a′, b′) = cos2(π/8) = 0.707.

Therefore the quantum mechanical prediction for the CHSH test statistic is

(8)    S_Φ = E_Φ(a, b) − E_Φ(a, b′) + E_Φ(a′, b) + E_Φ(a′, b′) = 2.828,

exceeding the CHSH Bell test limit of 2 and thus completing the proof of a version of Bell's Theorem. It is important to note, however, that 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).

References

• Clauser, J. F., M. A. Horne, A. Shimony and R. A. Holt [1969], Proposed experiment to test local hidden-variable theories, Physical Review Letters 23, 880-884
• Gisin, N. [1991], Bell's inequality holds for all non-product states, Physics Letters A 154, 201-202 [Note: the title of this paper is erroneous and should be replaced by Bell's inequality is violated by all non-product states]
• Popescu, S. and D. Rohrlich [1992], Generic quantum nonlocality, Physics Letters A 166, 293-297
• Shimony, A., Bell's Theorem [2004], Stanford Encyclopedia of Philosophy online