## BELL’S INEQUALITIES

John Stewart Bell was originally from Ireland. He started his professional career at Harwell in England, and then at CERN in Geneva. By his own admission, John Bell was a follower of Einstein. He believed in Einstein’s view about realism although he was probably somewhat misled in believing that Einstein supported hidden variables in quantum mechanics. In the early 1960s, Bell wrote two extremely influential papers related to the theory of hidden variables and the EPR paradox. The paper entitled “On the Einstein Podolsky Rosen Paradox” contained the celebrated Bell’s theorem which states: Any local hidden variable theory is inherently incompatible with quantum mechanics.

The EPR argument—quantum mechanics is not a complete theory but must be supplemented by additional variables to restore causality and locality—was shown by Bell to be mathematically incompatible with the statistical predictions of quantum mechanics. Bell’s theorem (a set of inequalities) dispelled the idea that there are undiscovered hidden variables in quantum theory that determine particle states and laid to rest the claims that quantum mechanics can be replaced by a local hidden variable theory.

To understand how, we will utilize a property of subatomic particles, called spin. Spin is analogous in some respects to the spin angular momentum of a macroscopic body such as our earth. For our purposes it is sufficient to say that the spin of a particle is represented by a vector, or an arrow, attached to the particle. A projection of this vector onto any axis in three-dimensional space is the component of the spin along that axis. Spin of a particle can either be up or down along an axis, designated by plus and minus.

According to EPR, if two particles in a particular quantum mechanical configuration called a singlet state are separated, the same component of spin measured on both particles will always be plus for one and minus for the other, or in other words, a strict negative correlation will be observed between spin components of the particles, no matter how far they are spatially separated. Here we assume there are no perturbing influences, of course. The particles are entangled but there is no way of predicting which particle will have the plus component and which the minus component.

We now consider a typical “Bell experiment”, where a pair of particles in a singlet state are each measured by one of two distant observers, Alice and Bob. Each particle is then tested randomly for just one spin component along the A, B or C directions. If Alice and Bob measures the same component in a pair accidentally, those results will be discarded, since they provide no new information. The remaining pairs must then be made up of three populations:

• AB: one particle tested along axis A and one tested along axis B, or
• AC: one tested along axis A and one along axis C, or
• BC: one along axis B and one along axis C.

We label the $A^+ B^+$ pair as the pair that on testing yields the results $A^+$ for one particle and $B^+$ for the other. The number of such pairs observed are represented by $n[A^+ B^+]$. We use the same labelling scheme for the other pairs.

For any large sample of singlet particle pairs, John Bell proved that the local realistic theories impose a limit on the extent of correlation that can be expected when different spin components are measured. The limit is expressed in the form of an inequality, called the Bell inequality, which states that the number of $A^+ B^+$ pairs cannot exceed the sum of the number of $A^+ C^+$ pairs plus the number of $B^+ C^+$ pairs

$\displaystyle n[A^+ B^+]\leq n[A^+ C^+]+n[B^+ C^+]$

Many similar inequalities can be constructed by transposing various symbols or by reversing their signs. All such formulations are interchangeable because the directions along which the spin components are defined were chosen arbitrarily.

Another form of the form of Bell’s inequality (S), given below, is directly applicable to the experiments:

$\displaystyle -2\leq S\leq 2$

where the inequality S compares the sum of all possible results of Alice and Bob’s experiment. The inequality above is called the Clauser-Horne-Shimony-Holt (CHSH) inequality, after the names of its four discoverers. CHSH were building on earlier ideas of John Bell.

Quantum theory predicts probabilities for outcomes, not the definite outcomes. Alice and Bob find that the outcomes match more often than by pure chance. The correlations could, in principle, either be an artifact of quantum mechanics or arise due to hidden variables. But Bell showed that these two descriptions predicts different degrees of correlation. The mathematics of hidden variables predicts that the measurements would match 33% of the time while quantum mechanics, with no hidden variables, predicts a match no more than 25% of the time.

Thus, Bell’s inequalities are experimentally testable. The question of whether a complete formulation of quantum mechanics is only possible using hidden variables and at the same time satisfies Einstein’s locality condition is thus not just an open-ended debate, it can actually be settled by rigorous arguments and experimentation. This precisely is why Bell’s theorem is so remarkable. One can now draw conclusions about assumptions made at the level of individual measurements by appealing to experiments.

Bell’s theorem has deeply influenced our perception and understanding of physics, and arguably ranks among the most profound scientific discoveries ever made. With the advent of quantum information science, a considerable interest has been devoted to Bell’s theorem and a wide range of concepts and technical tools have been developed for describing and studying the nonlocality that is the hallmark of quantum theory.