Quantum Probability: Sets & Complex Numbers

[closet mathemetician]

Classically, probability can be described using axiomatic set theory, where probabilities are related to the cardinality (size) of the sets involved.

For a quantum probability, the probability of a particular state is the squared magnitude of the wave function (eigenfunction) for that state's eigenvalue.

Relating this back to sets, what sort of a set has a complex number representing its cardinality? Or, more specifically, how can a complex number represent the "size" of a set?

 

[Bob_for_short]

...how can a complex number represent the "size" of a set?

And a couple of real numbers, can they?

 

[mathman]

Classically, probability can be described using axiomatic set theory, where probabilities are related to the cardinality (size) of the sets involved.

Probability axiomatics uses set theory (sigma algebras and all that), but cardinality has nothing to do with it. The axioms for probability resemble those for measure theory, plus the addition that the probability (measure) of the entire space is one.