What Does a Hidden Variable Look Like?∗

The hidden-variable question is whether or not correlations that are observed in the outcomes of an experiment can be explained via introduction of additional (“hidden”) variables which are unobserved by the experimenter. The question arises most famously in quantum mechanics (QM), but can also be asked in the classical realm. The nature of the experiment will tell us how to model the observable variables–i.e., the possible measurements and outcomes. But, by definition, we cannot know what structure to put on unobservable variables. Nevertheless, we show that, under one condition, the hidden-variable question can always be put into a canonical form. The condition is that the spaces of possible measurements and the spaces of possible outcomes, viewed as measurable spaces, are separable (i.e., the σ-algebras are countably generated). An argument based on Maharam’s Theorem ([10, 1942]) then shows that the hiddenvariable space can always be taken to be the unit interval equipped with Lebesgue measure. As an application of our result, we give a hidden-variable characterization of the no-signaling property of QM.