[Paper Review] A Royal Road to Quantum Theory (or Thereabouts), Extended Abstract

Alexander Wilce|arXiv (Cornell University)|Jan 3, 2017

Quantum Mechanics and Applications29 references34 citations

TL;DR

This paper presents a simplified, axiomatic derivation of finite-dimensional quantum theory using probabilistic models based on formally real Jordan algebras. By assuming homogeneity and self-duality of the state space, it derives a royal road to quantum mechanics that unifies real, complex, and quaternionic quantum theories while allowing limited extensions beyond standard quantum mechanics.

ABSTRACT

A representation of finite-dimensional probabilistic models in terms of formally real Jordan algebras is obtained, in a strikingly easy way, from simple assumptions. This provides a framework in which real, complex and quaternionic quantum mechanics can be treated on an equal footing, and allows some (but not too much) room for other alternatives. This is based on earlier work (arXiv:1206:2897), but the development here is further simplified, and also extended in several ways. I also discuss the possibilities for organizing probabilistic models, subject to the assumptions discussed here, into symmetric monoidal categories, showing that such a category will automatically have a dagger-compact structure. (Recent joint work with Howard Barnum and Matthew Graydon (arXiv:1507.06278) exhibits several categories of this kind.)

Motivation & Objective

• To provide a conceptually clear and mathematically simple derivation of finite-dimensional quantum mechanics from basic probabilistic assumptions.
• To unify real, complex, and quaternionic quantum mechanics within a single framework based on formally real Jordan algebras.
• To explore the conditions under which probabilistic models can be organized into symmetric monoidal categories with dagger-compact structure.
• To investigate the role of p-reversible processes and spectral structure in characterizing quantum-like theories.
• To determine whether the axioms of homogeneity, self-duality, and spectrality uniquely characterize Jordan models, thereby recovering quantum theory up to the Jordan algebraic level.

Proposed method

• Models are defined as pairs (M(A), Ω(A)), where M(A) is a test space and Ω(A) is a convex set of probability weights (states), with V(A) denoting the vector space spanned by states.
• Homogeneity and self-duality of the state space V(A) are posited as foundational physical assumptions, motivated by information-processing and symmetry principles.
• The Koecher-Vinberg theorem is applied to show that any finite-dimensional, self-dual, homogeneous ordered vector space with an order unit is isomorphic to a formally real Jordan algebra.
• Spectral structure is introduced via the existence of conjugate systems and p-reversible symmetric filters, leading to a functional calculus on effects.
• Composite systems are modeled via non-signaling composites AB with a monoidal product satisfying π(x,y) = xy ∈ V(AB)*, ensuring no signaling between subsystems.
• Dagger-compact structure is derived in a symmetric monoidal category of models under assumptions of sharpness, spectrality, and conjugate structure, using canonical isomorphisms V(A) ≃ V(A)*.

Experimental results

Research questions

• RQ1 Can finite-dimensional quantum mechanics be derived from simple, physically interpretable assumptions about probabilistic models?
• RQ2 To what extent do homogeneity and self-duality of the state space uniquely characterize formally real Jordan algebras as the underlying structure?
• RQ3 What conditions on processes (e.g., p-reversibility, symmetry) are sufficient to recover spectral decomposition and functional calculus?
• RQ4 Can a symmetric monoidal category of probabilistic models with dagger-compact structure be constructed without assuming local tomography?
• RQ5 Is there a converse to the result that sharp, spectral, conjugate models in a dagger-compact category must be Jordan models?

Key findings

• Under the assumptions of homogeneity and self-duality of the state space V(A), the Koecher-Vinberg theorem guarantees that V(A) is isomorphic to a formally real Jordan algebra.
• A probabilistic model is a Jordan model if and only if it satisfies: (a) existence of a conjugate and p-reversible symmetric filters for all non-singular states, or (b) sharpness, existence of conjugates, arbitrary p-reversible filters, and the correlation principle.
• Spectral decomposition is recovered: every effect a ∈ E(A) admits a unique representation a = ∑t_i e_i with t_i > t_j and e_i jointly orthogonal sharp effects, enabling a functional calculus.
• The Jordan product is uniquely determined as a·b = (a+b)^2 - a^2 - b^2, and bilinearity follows from homogeneity and the Koecher-Vinberg theorem.
• In a symmetric monoidal category of sharp, spectral, conjugate models satisfying additional structure (e.g., time-reversal compatibility), a canonical dagger-compact structure is induced.
• A dagger-compact category can be constructed that unifies real, complex, and quaternionic quantum mechanics, even without local tomography, by introducing a two-level classical bit to restore time-reversal symmetry in complex systems.