[Paper Review] Four and a Half Axioms for Finite Dimensional Quantum Mechanics

Alexander Wilce|ArXiv.org|Dec 30, 2009

Quantum Mechanics and Applications25 references35 citations

TL;DR

This paper proposes four and a half axioms—grounded in symmetry, classicality under single measurements, and bipartite non-signaling correlations—that uniquely characterize finite-dimensional quantum mechanics up to the structure of formally real Jordan algebras. The axioms yield a real ordered Hilbert space representation of states and measurements, and a final minimization principle forces homogeneity and self-duality of the state cone, leading to the standard quantum formalism in finite dimensions.

ABSTRACT

I discuss a set of strong, but probabilistically intelligible, axioms from which one can {\em almost} derive the appratus of finite dimensional quantum theory. Stated informally, these require that systems appear completely classical as restricted to a single measurement, that different measurements, and likewise different pure states, be equivalent (up to the action of a compact group of symmetries), and that every state be the marginal of a bipartite non-signaling state perfectly correlating two measurements. This much yields a mathematical representation of measurements and states that is already very suggestive of quantum mechanics. In particular, in any theory satisfying these axioms, measurements can be represented by orthonormal subsets of, and states, by vectors in, an ordered real Hilbert space -- in the quantum case, the space of Hermitian operators, with its usual tracial inner product. One final postulate (a simple minimization principle, still in need of a clear interpretation) forces the positive cone of this space to be homogeneous and self-dual and hence, to be the the state space of a formally real Jordan algebra. From here, the route to the standard framework of finite-dimensional quantum mechanics is quite short.

Motivation & Objective

• To derive the mathematical framework of finite-dimensional quantum mechanics from a small set of operationally meaningful axioms.
• To identify conditions under which the state space of a physical system becomes homogeneous and self-dual, key properties for quantum theory.
• To explore the role of composite systems and non-signaling correlations in reconstructing quantum mechanics.
• To clarify the operational and information-theoretic meaning of the canonical inner product in symmetric state spaces.
• To investigate connections between this axiomatization and existing reconstructions, such as those by Hardy, Rau, and D'Ariano.

Proposed method

• Axiom 1 requires that each measurement appears completely classical, meaning the system behaves as a classical probability space when restricted to a single observable.
• Axiom 2 enforces symmetry: all basic measurements and pure states are equivalent under a compact group of physical symmetries.
• Axiom 3 states that every state arises as a marginal of a bipartite non-signaling state that perfectly correlates a pair of measurements.
• Axiom 4 ensures that the space of observables forms a finite-dimensional ordered real Hilbert space, with measurements represented as orthonormal subsets and states as vectors.
• Axiom 5 (the 'half' axiom) is a minimization principle that forces the positive cone of the Hilbert space to be homogeneous and self-dual.
• The Koecher-Vinberg theorem is applied to show that homogeneous, self-dual cones correspond to formally real Jordan algebras, leading to the standard quantum formalism in finite dimensions.

Experimental results

Research questions

• RQ1 Can finite-dimensional quantum mechanics be reconstructed from symmetry, classicality per measurement, and non-signaling correlations in composite systems?
• RQ2 What is the operational or information-theoretic meaning of the canonical invariant inner product on the state space?
• RQ3 How do the proposed axioms relate to those of Hardy, Rau, and D'Ariano, particularly regarding tensor products and local tomography?
• RQ4 Under what conditions do systems satisfying these axioms admit non-signaling tensor products that preserve the axioms?
• RQ5 Can the symmetry group structure (e.g., compact Lie group) provide additional leverage to eliminate or reinterpret the minimization postulate?

Key findings

• The axioms yield a real ordered Hilbert space representation of measurements as orthonormal subsets and states as vectors, with symmetries acting unitarily.
• The final minimization postulate forces the positive cone of the state space to be homogeneous and self-dual, a key step toward Jordan algebra structure.
• By the Koecher-Vinberg theorem, homogeneous and self-dual cones in finite dimensions correspond to formally real Jordan algebras.
• The only such systems that support a reasonable tensor product with a qubit are those whose Jordan algebras are the Jordan parts of C*-algebras, leading to complex quantum mechanics.
• The scalar field is fixed to be the complex numbers via the local tomography condition, completing the derivation of standard finite-dimensional quantum mechanics.
• The approach provides a natural path to quantum theory that emphasizes symmetry, composition, and operational clarity, with the exceptional octonionic case and spin factors as potential physical alternatives.