## 9.3 CHSH inequality

An upper bound on

classicalcorrelations.

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^{158} values **CHSH quantity**

By a case-by-case analysis of the four possible outcomes for the pair **CHSH inequality**.

There is absolutely *no quantum theory involved* in the CHSH inequality

There are essentially *two* (very important) assumptions here:

**Hidden variables.**Observables have definite values.**Locality.**Alice’s choice of measurements (choosing betweenA_1 andA_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 us just give one brief comment.
In the hidden variable world a, statement such as “if Bob were to measure *prior to Bob’s measurement*.
Without the locality hypothesis, such a statement is ambiguous, since the value of *instantaneous communication* — it means that, say, Alice by making a choice between

Now let’s see how quantum theory *fundamentally disagrees* with the CHSH inequality.

The phrase “well defined” corresponds to our “hidden variable” assumption, i.e. that the observables

*always*have*definite*values.↩︎