Regular article

Hierarchical axioms for quantum mechanics^⋆

¹ Department of Physics, Indian Institute of Technology, Madras, Chennai 600036, India
² Jaypee Institute of Information Technology, A10, Sector 62, Noida 201309, India
³ Poornaprajna Institute of Scientific Research, Bangalore, Karnataka, India

^a e-mail: anirban.pathak@gmail.com

Received: 31 August 2018
Revised: 9 August 2019
Published online: 24 September 2019

Abstract

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 unconnected 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. By hierarchy is meant an axiomatic scheme where the succeeding axioms can be regarded as superstructure built on top of the structure provided by the preceding axioms. In a sense, the latter are necessary, but not sufficient, for the succeeding axioms. In our scheme, the axioms briefly are: (Q1) incompatibility and uncertainty; (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), relevant toy GPTs are introduced at each layer when useful to illustrate the action of the nonclassical features associated with the particular layer.