## 6.1 Hidden variables

The story of “hidden variables” dates back to 1935 and grew out of Einstein’s worries about the completeness of quantum theory.
Consider, for example, a single qubit.
Recalling our previous discussion on compatible operators (Section 4.6), we know that no quantum state of a qubit can be a simultaneous eigenstate of two *non-commuting* operators, such as ^{124}
Einstein felt very uncomfortable about all this: he argued that quantum theory is incomplete, and that observables **hidden variables**.

In this view, the indeterminacy found in quantum theory is merely due to our ignorance of these “hidden variables” that are present in nature but not accounted for in the theory.
Einstein came up with a number of pretty good arguments for the existence of “hidden variables”, perhaps the most compelling of which was described in his 1935 paper (known as “the EPR paper”), co-authored with his younger colleagues, Boris Podolsky and Nathan Rosen.
It stood for almost three decades as the most significant challenge to the completeness of quantum theory.
Then, in 1964, John Bell showed that the local^{125} hidden variable hypothesis can be tested and *refuted*.

Any theory can make predictions, but just because the predictions turn out to be correct, this does not make the theory true — there may be other, maybe equivalent, explanations.
The key to the scientific method is **falsifiability**: make one prediction incorrectly, and you have proven your theory is not true.

We already saw some no-go theorems in Section 5.9 that set limits on what we can do with quantum states.
In this chapter we’re going to see one no-go theorem relating to the *foundations* of quantum theory, in particular concerning these *local* “hidden variables”.
Again, there are many related no-go theorems, and again they fall beyond the scope of this book, but it’s worth mentioning them by name at least.
They all state that a certain type of (**realistic**, in some technical sense of the word) hidden-variable theory is inconsistent with reality:

**Bell’s theorem**(which we will see in Section 6.4) is for**local**hidden-variable theories.- The
**Kochen–Specker theorem**is for**non-contextual**hidden-variable theories. - The
**Pusey–Barret–Rudolph theorem**(often simply called the**PBR theorem**) is for**preparation independent**hidden-variable theories.

All together, these three theorems say that, if some hidden-variable theory does exist, then it has to be *non-local*, *contextual*, and *preparation dependent*.
But what do these words mean?

Preparation independence is the assumption that, if we independently prepare two quantum states, then their hidden variables are also independent.
Locality is the idea that things can only be directly affected by their surroundings, i.e. the exact opposite of “spooky action at a distance”.
Contextuality is a bit more subtle, and can actually be seen as a direct generalisation of non-locality (by **Fine’s theorem**), but it talks about how results of measurements depend on the commutator of the observable being measured, i.e. on its “context”.

A particularly useful way of formally defining non-locality and contextuality is by using the language of **sheaf theory**, which is an inherently topological and category-theoretic notion.
This approach was cemented by Abramsky and Bradenburger’s “The Sheaf-Theoretic Structure Of Non-Locality and Contextuality”, arXiv:1102.0264.