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

Abstract

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 [1]. 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 transpositionfrom 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.

Bridging the gap between general probabilistic theories and the device-independent framework for nonlocality and contextuality

“…Among all possible theories, classical and quantum theories are the two prominent examples, due to their central role in physics. However, in order to understand what is specific about these two theories and to explore future generalizations, it is convenient to step back from their specific details and to place them in the wider context of general probabilistic theories [24,34,33,35,36,37,20,38,21,39,40]-see also the contributed volume [57] for an introduction to the different frameworks. Among the available frameworks, here we adopt the framework of operational-probabilistic theories (OPTs) [37,20,40,58], which extends the language of quantum circuits [59,60] to arbitrary physical theories, combining the categorical framework initiated by Abramsky and Coecke [61,62,32] with the toolbox of elementary probability theory.…”

“…Naturally, the principles used for reconstructing quantum theory presuppose that experimental data have already been organized in the basic structure of a physical theory, which has systems, states, transformations, and measurements at its backbone. For example, principles like Local Tomography [24,33,34] or Ideal Compression [20] explicitly refer to the "states of a given physical system", to the "measurements performed on a composite system", and to the "processes that transform a system into another". The formulation of these principles is based on the framework of general probabilistic theories [24,34,33,35,36,37,20,38,21,39,40], which describes on the same footing classical and quantum theory, as well as many hypothetical, postquantum theories.…”

Dilation of states and processes in operational-probabilistic theories

“…Note that the use of the "reference system" R is essential, unless the theory enjoys a property known as "local tomography". [27,8,24]. When local tomography is not satisfied, checking the validity of Eq.…”

“…The dilation approach is so common in quantum information that it earned itself a nickname-the "Church of the Larger Hilbert Space" 1 . However, the operational content of the dilation theorems is independent of the quantum framework: Even forgetting about Hilbert spaces and operator algebras, one can still express the notions of pure/mixed state, reversible/irreversible evolution, and sharp/unsharp measurement in a general framework of operational-probabilistic theories [27,8,24,5,6,12,25,13,7,28,29]. In this broader framework the dilation of states, evolutions and measurements can be promoted to the rank of axioms, from which (a number of features of) the theory is derived ( [12]) [13].…”

Quantum from Principles

“…Turning some of these reasons into axioms then appeared as a promising route towards a compelling axiomatization. Pioneering works along this route are those by Hardy [25] and D'Ariano [26,27]. More recently, the programme flourished, leading to an explosion of new axiomatizations [2,28,29,30,31,32,33,34].…”