Abstract
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.
Highlights
Not less important are the open question suggested by this research, e.g. what is the most convenient way of defining averages with respect to quantum imprecise probability, or are there even more general axioms that involve the density matrix non-lineary and reduce to the linear situation when (5, 6) holds
[3] W.N. Anderson and R.J. Duffin, Journal of Mathematical Analysis and Applications, 26, 576 (1969)
Summary
The operator ω(P, Q) − ω(P, Q) quantifies the uncertainty for joint probability, the physical meaning of this characteristics of non-commutativity is new. Means that the pair of projectors (P1, Q1) is surely more probable (on ρ) than (P, Q); see section 7 of the Supplementary Material for examples. My main message is that while joint precise probability for non-commuting observables does not exist, there are well-defined expressions for upper and lower imprecise probabilities Some conceptual issues involving probability in quantum mechanics, quant-ph/0001017 (2000). Entanglement, Upper Probabilities and Decoherence in Quantum Mechanics, in: M. Journal of Mathematical Analysis and Applications, 26, 576 (1969)
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