How to Derive the Hilbert-Space Formulation of Quantum Mechanics From Purely Operational Axioms

TL;DR

This paper derives the Hilbert space formulation of quantum mechanics from operational axioms, emphasizing local observability and informational completeness, thus providing a foundational perspective rooted in experimental principles.

Contribution

It introduces a new operational axiomatization of quantum mechanics based on five simple postulates and the concept of informationally complete observables.

Findings

Derivation of real Hilbert space structure from operational axioms

Representation of elements as operators over complex Hilbert space in finite dimensions

Reconciliation of nonlocality holism with local observation reductionism

Abstract

In the present paper I show how it is possible to derive the Hilbert space formulation of Quantum Mechanics from a comprehensive definition of "physical experiment" and assuming "experimental accessibility and simplicity" as specified by five simple Postulates. This accomplishes the program presented in form of conjectures in the previous paper quant-ph/0506034. Pivotal roles are played by the "local observability principle", which reconciles the holism of nonlocality with the reductionism of local observation, and by the postulated existence of "informationally complete observables" and of a "symmetric faithful state". This last notion allows one to introduce an operational definition for the real version of the "adjoint"--i. e. the transposition--from which one can derive a real Hilbert-space structure via either the Mackey-Kakutani or the Gelfand-Naimark-Segal constructions. Here I analyze in detail only the Gelfand-Naimark-Segal construction, which leads to a real Hilbert space structure analogous to that of (classes of generally unbounded) selfadjoint operators in Quantum Mechanics. For finite dimensions, general dimensionality theorems that can be derived from a local observability principle, allow us to represent the elements of the real Hilbert space as operators over an underlying complex Hilbert space (see, however, a still open problem at the end of the paper). The route for the present operational axiomatization was suggested by novel ideas originated from Quantum Tomography.