Bell's Inequality

CHSH Inequality

In a classical experiment, we have the following setup where Alice can choose to measure either Q or R and Bob chooses either S or T. The measurements are performed imultaneously and far off from each other.

$$E(QS) + E(RS) + E(RT) - E(QT) = E(QS + RS + RT-QT)\\ $$

$$\implies E(QS) + E(RS) + E(RT) - E(QT) = \sum_{q,r,s,t}p(q, r, s, t)(qs + rs + rt - qt)\\ $$

$$\implies E(QS) + E(RS) + E(RT) - E(QT) \leq \sum_{q,r,s,t}p\times 2 = 2$$

and thereby we obtain the inequality $E(QS) + E(RS) + E(RT) - E(QT) \leq 2$.

This is one of the set of Bell inequalities, the first of which was found by John Bell. This one in particular is named CHSH inequality.

Quantum Anomaly

In the quantum case, let us consider the measurements to be based on the following observables over the EPR pair $|\psi\rangle = \frac{|01\rangle - |10\rangle}{\sqrt 2}$.

$$Q = Z_1, R= X_1,\\ S = \frac{-Z_2-X_2}{\sqrt 2},\\ T = \frac{Z_2 - X_2}{\sqrt 2}$$

Then, we have the following result.

$$E(QS) + E(RS) + E(RT) - E(QT) = \frac{1}{\sqrt 2} + \frac{1}{\sqrt 2} + \frac{1}{\sqrt 2} - \frac{1}{\sqrt 2} = 2\sqrt 2 > 2$$

Thus, in other words, CHSH inequality doesn't hold.

Interpretation

The fact that CHSH doesn't hold in the quantum scenario implies that two of the major assumptions about nature is wrong in case of the classical experiment.

The assumptions are:

Realism: Q, R, S, T are physical quantities which have defininte values irrespective of observation.
Locality: Alice's measurement doesn't influence that of Bob's.

Thus, the result of CHSH being false when accounted for the quantum mechanical properties of nature (we can perform the experiment in a lab with particles) suggests that nature cannot be locally real and neither can any true mathematical representation of it be locally real.