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Bell's Inequality

If two particles is generated spontaneously from some energy, the both stay entangled and in superposition. If we measure one's spin and find (+), it will instantaneously determine spin of other one as (-) no matter how far apart they are, if spin is measured in same direction perpendicular of its motion path.

When we generate two entangled particles they start travelling opposite direction of each other. We have a setup to measure both in three perpendicular direction of its travelling path. The measurement direction is separated equidistantly to each other (120 degree). We will say this directions as A, B and C. So if A is on 0 degree then B will be on 120 degree and C on 240 degree on the perpendicular plane.

We have two observer Alice and Bob very far apart of each other. There is no way they can communicate with each other. They can measure each particle spin in any direction from the three directions.

As stated earlier if they measure both on same direction they get opposite spin. But, what if they measure both in different direction. They get probabilistic result in this case.

The idea of instantaneous communication and probabilistic result bother Einstein. As instantaneous communication breaks the rule of special relativity that nothing can go faster than light and Einstein also thinks that "god does not play dice" which are not matching with probabilistic result. So he along with Rosen and Podolsky came up with "Hidden variables theory".

They suggest that there would be deterministic result and no instantaneous communication. The both idea can be satisfied with the hidden variables theory. The name hidden variables says that we can not know before hand the values of the variables until we measure it.

The theory says that each pair of particles contain hidden variables to determine which spin it shows when measured in particular direction. (e.g one particle have configuration to show (+) spin in A direction and other entangled particle have configuration to show (-) spin in B direction.) This way measurement on one particle does not influence other particle. So, no faster then light communication is happening and no more probabilistic results.

Both theory tells that what would be situation before measurement.

John S. Bell came up with an experiment and an inequality to determine that which is correct, quantum mechanics or hidden variables theory. If the inequality stays true with suggested experimental results than it proves that hidden variables theory is correct. 

Suppose if hidden variables theory is correct than both particles being measured in three equidistance direction can have below hidden variables configurations.

 Figure 1

There are 8 types of pairs for three angles. Suppose there are total number of measurement done is Nt (one measurement means measurement on pair by both observer). Suppose, pair a to d occurs for N1, N2, N3 and N4 times respectively out of Nt. If we look closely pair e, f, g, and h are identical but just opposite to d, c, b, and a respectively. So they also should have same occurrence as a to d.

Pair a and h give 100% opposite result with whatever direction chosen by both observer, there are no probability in results of the measurement when this pair occurs. We are interested in pair b to g. If we look closely again we can see that each configuration has two arrows in same direction and one in different direction. So b to g are mathematically equivalent pairs. We can choose any one to check inequality based on experimental results. 

Figure 2

Lets choose (A+, B+, C-)

The configuration looks like below.

Figure 3

Let's write bells inequality for above configuration.


P(A+, B+) means probability of measurement of (+) spin at direction A by Alice and measurement of (+) spin at direction B by Bob. Same goes true for P(A+, C-) and P(C-, B+).

From figure 1, we can check inequality as below.


We can clearly see that the inequality satisfies if we assume that hidden variables theory is correct. On the other hand quantum mechanics shows probability based on below equation.

When we really perform experiment we find that measurement do not follow hidden variables theory and we are getting below information.


Lets see how we can explain this with quantum mechanics.

This shows that Bell's inequality does not satisfy experimental results and thus hidden variables theory could not correct.
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