Step-by-step derivation of the algebraic structure of quantum mechanics (or from nondisturbing to quantum correlations by connecting incompatible observables)

Abstract

Recently there has been much interest in deriving the quantum formalism and the set of quantum correlations from simple axioms. In this paper, we provide a step-by-step derivation of the quantum formalism that tackles both these problems and helps us to understand why this formalism is as it is. We begin with a structureless system that only includes realvalued observables, states and a (not specified) state update, and we gradually identify theoryindependent conditions that make the algebraic structure of quantum mechanics be assimilated by it. In the first part of the paper (sections 3-8), we derive essentially all the "commutative part" of the quantum formalism, i.e., all definitions and theorems that do not involve algebraic operations between incompatible observables, such as projections, Specker's principle, and the spectral theorem; at the statistical level, the system is nondisturbing and satisfies the exclusivity principle at this stage. In the second part of the paper (sections 9-10), we connect incompatible observables by taking transition probabilities between pure states into account. This connection is the final step needed to embed our system in a Hilbert space and to go from nondisturbing to quantum correlations. Contents 4 The category of observables and the completely mixed state 18 5 Projections and their traces 25 6 Compatibility (or binary products in the category of observables) 30 7 Specker's principle (or finite products in the category of observables) 40 8 Algebraic aspects of the "commutative part" of quantum mechanics 45 8.