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 \sigma_x and \sigma_z. Physically, this means that if the qubit has a definite value of \sigma_x then its value of \sigma_z must be indeterminate, and vice versa. If we take quantum theory to be a complete description of the world, then we must accept that it is impossible for both \sigma_x and \sigma_z to have definite values for the same qubit at the same time.124 Einstein felt very uncomfortable about all this: he argued that quantum theory is incomplete, and that observables \sigma_x and \sigma_z may both have simultaneous definite values, although we only have knowledge of one of them at a time. This is the hypothesis of 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 local125 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:

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.

  1. Here it’s important that we’re really talking about so-called local hidden variable theories. We discuss the technical details in 6.7.↩︎

  2. This key word “local” is very important for those who care about the subtle technical details, but we won’t explain it here.↩︎