9.3 CHSH inequality

An upper bound on classical correlations.

We will describe the most popular version of Bell’s argument, introduced in 1969 by John Clauser, Michael Horne, Abner Shimony, and Richard Holt (CHSH). Let us assume that the results of any measurement on any individual system are predetermined. Any probabilities we may use to describe the system merely reflect our ignorance of these hidden variables.

Now, imagine the following scenario. Alice and Bob, two characters with a predilection for wacky experiments, are equipped with appropriate measuring devices and sent to two distant locations. Somewhere in between them there is a source that emits pairs of qubits that fly apart, one towards Alice and one towards Bob. Let us label the two qubits in each pair as \mathcal{A} and \mathcal{B} respectively, and let us assume that both Alice and Bob have well defined values of their observables. We ask Alice and Bob to measure one of the two pre-agreed observables. For each incoming qubit, Alice and Bob choose randomly, and independently from each other, which particular observable will be measured. Alice chooses between A_1 and A_2, and Bob between B_1 and B_2. Each observable has value +1 or -1, and so we are allowed to think about them as random variables A_k and B_k, for k=1,2, which take values \pm 1. Let us define a new random variable, the CHSH quantity S, as S = A_1(B_1 - B_2) + A_2(B_1 + B_2). It is easy to see that one of the terms B_1\pm B_2 must be equal to zero and the other to \pm 2, hence S=\pm2. The average value of S must lie somewhere in-between, i.e. -2 \leqslant\langle S\rangle \leqslant 2. That’s it! Such a simple and yet profound mathematical statement about correlations, which we refer simply to as the CHSH inequality. No quantum theory is involved because the CHSH inequality is not specific to quantum theory: it does not really matter what kind of physical process is behind the appearance of binary values of A_1, A_2, B_1, and B_2; it is a statement about correlations, and for all classical correlations we must have | \langle A_1 B_1\rangle - \langle A_1 B_2\rangle + \langle A_2 B_1\rangle + \langle A_2 B_2\rangle | \leqslant 2.

There are essentially two two assumptions here:

  1. Hidden variables: observables have definite values; and
  2. Locality: Alice’s choice of measurements (A_1 or A_2) does not affect the outcomes of Bob’s measurement, and vice versa.

We will not discuss the locality assumption here in detail but let me comment on it briefly. In the hidden variable world a, statement such as “if Bob were to measure B_1 then he would register +1” must be either true or false prior to Bob’s measurement. Without the locality hypothesis, such a statement is ambiguous, since the value of B_1 could depend on whether A_1 or A_2 will be chosen by Alice. We do not want this for it implies the instantaneous communication. It means that, say, Alice by making a choice between A_1 and A_2, affects Bob’s results. Bob can immediately “see” what Alice “does”.