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Recently there has been much interest in deriving the quantum formalism andthe set of quantum correlations from simple axioms. In this paper, we provide astep-by-step derivation of the quantum formalism that tackles both theseproblems and helps us to understand why this formalism is as it is. We beginwith a structureless system that only includes real-valued observables, statesand a (not specified) state update, and we gradually identifytheory-independent conditions that make the algebraic structure of quantummechanics be assimilated by it. In the first part of the paper, we deriveessentially all the "commutative part" of the quantum formalism, i.e., alldefinitions and theorems that do not involve algebraic operations betweenincompatible observables, such as projections, Specker's principle, and thespectral theorem; at the statistical level, the system is nondisturbing andsatisfies the exclusivity principle at this stage. In the second part of thepaper, we connect incompatible observables by taking transition probabilitiesbetween pure states into account. This connection is the final step needed toembed our system in a Hilbert space and to go from nondisturbing to quantumcorrelations.

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