Bell's Theorem

What does Bell Theorem mean?

Bell’s theorem states that

There is no physical theory for local hidden variables which can reproduce the quantum mechanics predictions.

The theorem is named after John Stewart Bell. Hidden variables are the microscopic properties of particles that are difficult to be observed with the existing microscope and according to the Heisenberg Uncertainty Principle, these variables do not exist outside the context of observation.

Bell’s Theorem Formula

According to the theorem if X, Y and Z are the photon measurement variables then,

P (X = Y) + P (Y = Z) + P (Z = X) ≥ 1

Where,

  • P(X=Y) is the probability of X = Y which is mathematically relevant

What is Bell’s Inequality?

Bell’s inequality can be best explained with the help of quantum mechanics. In quantum mechanics, it is believed that when electrons are sent across the magnetic field, half of the electrons get deflected towards the right and the other half to the left. The right half of the electrons are again sent across another magnetic field which is perpendicular to the first, then they split into half in such a way that few go up and other few go down. This is known as the randomness of the electron which is studied using Bell’s theorem.

What is Local Realism?

Local realism is a concept that is formalised to state and prove Bell’s theorem with Alice and Bob (the outcome of random sampling). There are two observed values by Alice and Bob with detector setting and they are a is A(a,λ) and b is B(b,λ) respectively.

Above is the expression of local realism.

Example of Bell’s Theorem

  • Photon polarisation: Polarisation of light is the example that explains Bell’s theorem and Bell’s inequality as it involves the study of particles.

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