Non-additive measures for quantum probability?

Abstract

It is well-established that quantum probability does not follow classical Kolmogorov probability calculus. Various approaches have been developed to loosen the axioms, of which the use of signed measures is the most successful (e.g. the Wigner quasiprobability distribution). As part of our larger effort Assumptions of Physics, we have been considering the various roles of measures, which are used in physics not only for probability, but also to quantify the count of possible states and configurations. These measures play a crucial role in classical mechanics, as they effectively define its geometric structure. If one tries to construct a parallel in quantum mechanics, the measure to quantify the count of states turns out to be non-additive. The proper extension of probability calculus may require the use of non-additive measures, which is something that, to our knowledge, has not yet been explored. The purpose of this paper is to present the general idea and the open problems to an audience that is knowledgeable of the subject of non-additive set functions, though not necessarily in quantum physics, in the hope that it will spark helpful discussions. We go through the motivation in simple terms, which stems from the link between the entropy of a uniform distribution and the logarithm of the measure associated to its support. If one extends this notion to quantum mechanics, the associated measure is non-additive. We will explore some properties of this "quantum measure", its reasonableness in terms of the physics, but its peculiarity on the math side. We will explore the need for a set of properties that can properly characterize the measure and a generalization of the Radon-Nikodym derivative to define a properly extended probability calculus that reduces to the standard additive one on sets of physically distinguishable cases (i.e. orthogonal measurement outcomes).