Reasonable conditions for joint probabilities of non-commuting observables

TL;DR

This paper explores the conditions under which joint probabilities for non-commuting quantum observables can be defined, proposing a specific mathematical form linked to weak measurement quasi-probabilities.

Contribution

It introduces criteria that uniquely determine joint probabilities in quantum mechanics, connecting them to experimentally observed weak measurement quasi-probabilities.

Findings

Joint probabilities can be defined by ordered products of projection operators.

The proposed joint probability aligns with experimentally observed weak measurement quasi-probabilities.

Conditions for defining joint probabilities are clarified within the quantum formalism.

Abstract

In the operator formalism of quantum mechanics, the density operator describes the complete statistics of a quantum state in terms of d^2 independent elements, where d is the number of possible outcomes for a precise measurement of an observable. In principle, it is therefore possible to express the density operator by a joint probability of two observables that cannot actually be measured jointly because they do not have any common eigenstates. However, such joint probabilities do not refer to an actual measurement outcome, so their definition cannot be based on a set of possible events. Here, I consider the criteria that could specify a unique mathematical form of joint probabilities in the quantum formalism. It is shown that a reasonable set of conditions results in the definition of joint probabilities by ordered products of the corresponding projection operators. It is pointed out that this joint probability corresponds to the quasi probabilities that have recently been observed experimentally in weak measurements.