9.2 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.157 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 (local) hidden variable hypothesis can be tested and refuted.

  1. Here it’s important that we’re really talking about so-called local hidden variable theories. We discuss this assumption briefly after talking about the CHSH inequality in Section 9.3.↩︎