[Paper Review] Information Theoretic Axioms for Quantum Theory

Marco Zaopo|arXiv (Cornell University)|May 10, 2012

Statistical Mechanics and Entropy4 references3 citations

TL;DR

This paper reconstructs quantum theory from four information-theoretic axioms—Distinguishability, Conservation, Reversibility, and Composition—showing that quantum mechanics is the only non-classical probabilistic theory satisfying them. It eliminates the need for the subspace axiom, classifies theories under subsets of axioms, and establishes a link between quantum logic and information-theoretic foundations, with real and quaternionic quantum theories emerging as consistent alternatives under modified tomography conditions.

ABSTRACT

In this paper we derive the complex Hilbert space formalism of quantum theory from four simple information theoretic axioms. It is shown that quantum theory is the only non classical probabilistic theory satisfying the following axioms: distinguishability, conservation, reversibility, composition. The new results of this reconstruction compared to other reconstructions by other authors are: (i) we get rid of axiom "subspace" in favor of axiom conservation eliminating mathematical requirements contained in previous axiomatics; (ii) we are able to classify all the probabilistic theories that are consistent requiring (a) only the first two axioms (b) only the first three axioms; this could be useful in experimental tests of quantum theory since it gives the possibility to understand whether or not other mathematical models could be consistent with such tests; (iii) we provide a connection between two different approaches to quantum foundations, quantum logic and the one based on information theoretic primitives showing that any theory satisfying the first two axioms given above either is classical or is a theory in which physical systems are described by a projective geometry.

Motivation & Objective

• To provide a new reconstruction of quantum theory based solely on information-theoretic principles, avoiding prior mathematical constraints like the subspace axiom.
• To classify all probabilistic theories consistent with subsets of the four axioms, enabling experimental testing of alternative models.
• To establish a formal connection between quantum logic (projective geometry of pure states) and information-theoretic foundations.
• To show that real and quaternionic quantum theories satisfy the axioms under 2-local tomography, suggesting they are viable alternatives if local tomography is relaxed.

Proposed method

• Derives the complex Hilbert space formalism from four axioms: Distinguishability, Conservation, Reversibility, and Composition.
• Uses the Conservation and Distinguishability axioms to classify all probabilistic theories where pure states form a projective space over a generic field.
• Applies the Reversibility axiom to restrict the state space to a hypersphere in d dimensions, leading to normed division algebras (reals, complexes, quaternions, octonions).
• Introduces 2-local tomography as a replacement for local tomography in real quantum theory, showing consistency with the axioms.
• Employs structural analysis of state spaces and transformations to characterize the set of theories satisfying subsets of the axioms.
• Uses results from Wooters and Hardy to argue that real quantum theory satisfies the axioms under 2-local tomography, suggesting it as a candidate alternative.

Experimental results

Research questions

• RQ1 Can quantum theory be reconstructed without the subspace or compression axioms, relying only on information-theoretic principles?
• RQ2 What class of probabilistic theories satisfy only the Distinguishability and Conservation axioms?
• RQ3 Which theories satisfy Distinguishability, Conservation, and Reversibility, and how do they relate to projective geometry and normed division algebras?
• RQ4 How does 2-local tomography compare to local tomography in characterizing composite systems in real quantum theory?
• RQ5 Is quantum theory over reals the only non-classical theory satisfying the four axioms if local tomography is replaced by 2-local tomography?

Key findings

• The only non-classical probabilistic theory satisfying all four axioms—Distinguishability, Conservation, Reversibility, and Composition—is quantum theory with complex amplitudes.
• Theories satisfying only Distinguishability and Conservation are either classical or have pure states forming a projective space over a generic field, generalizing quantum theory to real, complex, and quaternionic amplitudes.
• Adding Reversibility restricts the state space to a hypersphere in d dimensions, implying that amplitudes must belong to a normed real division algebra, including reals, complexes, quaternions, and octonions.
• Quantum theory over reals satisfies all axioms except local tomography, but is consistent with 2-local tomography, as shown in Wooters and Hardy.
• The paper conjectures that quantum theory over reals is the only theory satisfying Distinguishability, Conservation, Reversibility, and Composition with 2-local tomography.
• The work establishes a formal link between quantum logic (projective geometry) and information-theoretic foundations, showing that any theory satisfying the first two axioms is either classical or projective.