On quantum vs. classical probability

TL;DR

This paper explores the shared foundations of quantum and classical probability theories, highlighting their core similarities and the key distinction of joint decidability versus smoothness, and argues that no other generalizations are relevant to physics.

Contribution

It clarifies the common core of quantum and classical probability and identifies the fundamental difference as joint decidability versus smoothness, with implications for understanding quantum theory's uniqueness.

Findings

Shared core properties in six areas of quantum and classical probability

Distinction between joint decidability and smoothness as fundamental difference

No other generalizations of classical probability are relevant to physics

Abstract

Quantum theory shares with classical probability theory many important properties. I show that this common core regards at least the following six areas, and I provide details on each of these: the logic of propositions, symmetry, probabilities, composition of systems, state preparation and reductionism. The essential distinction between classical and quantum theory, on the other hand, is shown to be joint decidability versus smoothness; for the latter in particular I supply ample explanation and motivation. Finally, I argue that beyond quantum theory there are no other generalisations of classical probability theory that are relevant to physics.