On classical finite probability theory as a quantum probability calculus

Abstract

This paper shows how the classical finite probability theory (with equiprobable outcomes) can be reinterpreted and recast as the quantum probability calculus of a pedagogical or "toy" model of quantum mechanics over sets (QM/sets). There have been four previous attempts to develop a quantum-like model with the base field of $\mathbb{C}$ replaced by ${\mathbb{Z}}_2$, but they are all forced into a merely "modal" interpretation by requiring the brackets to take values in ${\mathbb{Z}}_2$ (1 = possible, 0 = impossible). But the usual QM brackets <ψ|ϕ> give the "overlap" between states ψ and ϕ, so for subsets S,T of U, the natural definition is $<S|T>=|S\cap T|$. This allows QM/sets to be developed with a full probability calculus that turns out to be the perfectly classical Laplace-Boole finite probability theory. The point is not to clarify finite probability theory but to elucidate quantum mechanics itself by seeing some of its quantum features (e.g., two-slit experiment) in a classical setting.

arXiv admin note: substantial text overlap with arXiv:1310.8221, arXiv:1210.7659

Cited by in corpus (2)

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