Quantum Mechanics and Imprecise Probability

ARMEN E. ALLAHVERDYAN

It is known that non-commuting observables in quantum mechanics do not have joint probability. This statement refers to the precise (additive) probability model. I show that the joint distribution of any non-commuting pair of variables can be quantified via upper and lower probabilities, i.e. the joint probability is described by an interval instead of a number (imprecise probability). I propose transparent axioms from which the upper and lower probability operators follow. The imprecise probability depend on the non-commuting observables, is linear over the state (density matrix) and reverts to the usual expression for commuting observables.

Bassano Vacchini

Diederik Aerts

Stephan Hartmann

In his entry on "Quantum Logic and Probability Theory" in the Stanford Encyclopedia of Philosophy, Alexander Wilce (2012) writes that "it is uncontroversial (though remarkable) the formal apparatus quantum mechanics reduces neatly to a generalization of classical probability in which the role played by a Boolean algebra of events in the latter is taken over the 'quantum logic' of projection operators on a Hilbert space." For a long time, Patrick Suppes has opposed this view (see, for example, the paper collected in Suppes and Zanotti (1996). Instead of changing the logic and moving from a Boolean algebra to a non-Boolean algebra, one can also 'save the phenomena' by weakening the axioms of probability theory and work instead with upper and lower probabilities. However, it is fair to say that despite Suppes' efforts upper and lower probabilities are not particularly popular in physics as well as in the foundations of physics, at least so far. Instead, quantum logic is booming again, especially since quantum information and computation became hot topics. Interestingly, however, imprecise probabilities are becoming more and more popular in formal epistemology as recent work by authors such as James Joye (2010) and Roger White (2010) demonstrates.

J. J. Halliwell

We calculate the probability for a quantum-mechanical history consisting of imprecise samplings of position at two moments of time. In the limit of small time separation, this leads to an imprecise sampling of position together with a time-of-flight sampling of momentum. We also calculate the probability for the history consisting of direct momentum and position samplings a short time apart. In each case, we find that the resulting probability distribution on phase space is a smeared version of the Wigner function, and is positive. We show that these smearings belong to a class of smearings which make the Wigner function positive. In the case of the time-of-flight momentum sampling, it is more general than previously considered smearings, such as that of Husimi.

Carl Friedrich von Weizsäcker