Negative Probabilities, Fine's Theorem and Linear Positivity

Abstract: Many situations in quantum theory and other areas of physics lead to quasi-probabilities which seem to be physically useful but can be negative. The interpretation of such objects is not at all clear. In this paper, we show that quasi-probabilities naturally fall into two qualitatively different types, according to whether their non-negative marginals can or cannot be matched to a non-negative probability. The former type, which we call viable, are qualitatively similar to true probabilities, but the latter type, which we call non-viable, may not have a sensible interpretation. Determining the existence of a probability matching given marginals is a non-trivial question in general. In simple examples, Fine's theorem indicates that inequalities of the Bell and CHSH type provide criteria for its existence, and these examples are considered in detail. Our results have consequences for the linear positivity condition of Goldstein and Page in the context of the histories approach to quantum theory. Although it is a very weak condition for the assignment of probabilities it fails in some important cases where our results indicate that probabilities clearly exist. We speculate that our method, of matching probabilities to a given set of marginals, provides a general method of assigning probabilities to histories and we show that it passes the Di\'osi test for the statistical independence of subsystems.