# Chapter 9 Bell’s theorem

About

quantum correlations, which are stronger than any correlations allowed by classical physics, and about theCHSH inequalitywhich demonstrates this fact, and which is a variant ofBell’s theorem.

Every now and then, it is nice to put down your lecture notes and go and see how things actually work in the real world. What is wonderful (and surprising) about quantum theory is that it turns up in many places that we might not expect it to, such as in the polarisation of light, where we stumble across an intriguing paradox.

If we take two polarising filters, and place them on top of each other with their polarisations oriented at *more* light is let through than if the middle filter weren’t there at all.^{114}

This is intrinsically linked to Bell’s theorem, which proves the technical sounding statement that “any local real hidden variable theory must satisfy certain statistical properties”, which is *not* satisfied in reality, as many quantum mechanical experiments (such as the above) show!

## 9.1 Quantum correlations

Consider two entangled qubits in the singlet state
^{115}

## 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 (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 **CHSH quantity** **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

There are essentially two two assumptions here:

**Hidden variables**: observables have definite values; and**Locality**: Alice’s choice of measurements (A_1 orA_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 *prior to Bob’s measurement*.
Without the locality hypothesis, such a statement is ambiguous, since the value of

## 9.4 Quantum correlations revisited

In quantum theory the observables **CHSH matrix**

We can now evaluate

Thus

To be clear, this violation has been observed in a number of painstakingly careful experiments. The early efforts were truly heroic, and the experiments had many layers of complexity. Today, however, such experiments are routine.

The behaviour of entangled quantum systems cannot be explained by any local hidden variables.

## 9.5 Tsirelson’s inequality

An upper bound on

quantumcorrelations.

One may ask if **Tsirelson inequality**.

## 9.6 Remarks and Exercises

**!!!TODO!!!**

For the more visually inclined, there is a video on YouTube by minutephysics about this experiment, or you can play with a virtual version at Quantum Flytrap.↩︎

There are other, more elementary, ways of deriving this result but here I want you to hone your skills. Now that you’ve learned about projectors, traces, and Pauli operators, why not put them to good use.↩︎