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Hierarchical axioms for quantum mechanics

The origin of nonclassicality in quantum mechanics (QM) has been investigated recently by a number of authors with a view to identifying axioms that would single out quantum mechanics as a special theory within a broader framework such as convex operational theories. In these studies, the axioms tend to be logically independent in the sense that no specific ordering of the axioms is implied. Here, we identify a hierarchy of five nonclassical features that separate QM from a classical theory: (Q1) Incompatibility and indeterminism; (Q2) Contextuality; (Q3) Entanglement; (Q4) Nonlocality and (Q5) Indistinguishability of identical particles. Such a hierarchy isn't obvious when viewed from within the quantum mechanical framework, but, from the perspective of generalized probability theories (GPTs), the later axioms can be regarded as further structure introduced on top of earlier axioms. Relevant toy GPTs are introduced at each layer when useful to illustrate the action of the nonclassical features associated with the particular layer.