The 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 (whence the name “CHSH”).
Let us start by making this assumption 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.
Imagine the following scenario.
Alice and Bob, our two characters with a predilection for wacky experiments, are equipped with appropriate measuring devices and sent to two distant locations.
Assume that Alice and Bob each have a choice of two observables that they can measure, each with well defined values +1 and -1.
Let’s say that Alice can choose between observables A_1 and A_2, and Bob between B_1 and B_2.
Now, somewhere in between them there is a source that emits pairs of qubits that fly apart, one towards Alice and one towards Bob.
For each incoming qubit, Alice and Bob choose randomly, and independently from each other, which particular observable will be measured.
This means we can think of the observables as random variables A_k,B_k (for k=1,2) that take values \pm1.
Using these, we can define a new random variable: the CHSH quantity
S = A_1(B_1 - B_2) + A_2(B_1 + B_2).
By a case-by-case analysis of the four possible outcomes for the pair (B_1,B_2), we see that one of the terms B_1\pm B_2 must be equal to zero and the other to \pm 2 (basically depending on if B_1=B_2 or not), and so (looking at the four possible outcomes for the pair (A_1,A_2)) we see that S=\pm2.
But the average value of S must lie in between these two possible outcomes, i.e.
-2 \leqslant\langle S\rangle \leqslant 2.
Such a simple and yet profound mathematical statement about correlations, which we refer simply to as the CHSH inequality.
There is absolutely no quantum theory involved in the CHSH inequality
-2 \leqslant\langle S\rangle \leqslant 2
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 merely 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
(which is just another way of phrasing the CHSH inequality).
There are essentially two (very important) assumptions here:
- Hidden variables. Observables have definite values.
- Locality. Alice’s choice of measurements (choosing between A_1 and A_2) does not affect the outcomes of Bob’s measurement, and vice versa.
We will not discuss the locality assumption right now in detail (see Section 6.7), but let us just give one brief comment.
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 (and not “undecidable” or some other such thing!) 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 since it implies 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”.
Now let’s see how quantum theory fundamentally disagrees with the CHSH inequality.