* Bell’s inequality*, developed by John Bell in 1964, is a non-equal relation (between two expressions) that is based on the ERP paradox. It provides a way to determine whether quantum theory or the EPR paradox is correct. Bell suggested a unique way of measuring the spins of two spin- particles, which are generated from the decay of a spin-0 particle at rest: the two particles are to be passed through two Stern-Gerlach devices, each oriented along one of three non-orthogonal coplanar axes, which are specified by the unit vectors , and .

The average values of the product of the spins in units of (denoted by , , and ) are then calculated. As derived in an earlier article (see eq227), quantum mechanics expresses the average values as:

For the ERP paradox, let’s equate a spin-up measurement to +1 and a spin-down measurement to -1. The average value of the product of the measured spins, for example in the and directions, is:

where

i) is a hidden variable.

ii) is the probability density that is a function of , with and .

iii) is a function of the axis of measurement and . It is associated with the measurement made by the first Stern-Gerlach device, and has output values of .

iv) is a function of the axis of measurement and . It is associated with the measurement made by the second Stern-Gerlach device, and has output values of .

From experiments, we know that

Substitute eq230 in eq229

Similarly, and . Therefore,

Since , we can rearrange the above equation to

Taking the absolute value on both sides of the above equation and using the relation

###### Question

Why is ?

###### Answer

For all , we have . Therefore, or simply , where we have use the identity of .

Since

Since and , we can ignore the absolute value sign on RHS of the above equation:

Substitute and into the above equation,

Eq232 is the Bell’s inequality, which is based on the ERP paradox.

To show that quantum mechanics is incompatible with Bell’s inequality, we let , i.e. ** **bisects . From eq228,

Substitute eq233 and eq234 in eq232, we have , which is inconsistent with Bell’s inequality. Therefore, it is possible to experimentally measure the spins of the two particles at non-orthogonal angles to test the predictions of quantum theory versus the ERP paradox. In fact, the results of all experiments conducted at non-orthogonal angles were in agreement with quantum mechanics. This implies that all local hidden-variable hypotheses are invalid.