[Paper Review] Probabilistic Theories and Reconstructions of Quantum Theory (Les Houches 2019 lecture notes)

Markus P. Müller|arXiv (Cornell University)|Nov 2, 2020

Quantum Mechanics and ApplicationsPhysics and Astronomy93 references49 citations

TL;DR

This paper presents a reconstruction of quantum theory (QT) within the framework of generalized probabilistic theories (GPTs), deriving the full Hilbert space formalism from three operational principles: Tomographic Locality, Continuous Reversibility, and the Subspace Axiom. It shows that the qubit's state space is uniquely a Bloch ball in three dimensions, and that complex numbers and quantum operators emerge naturally from these principles, revealing quantum theory as a special case within a broader landscape of probabilistic theories.

ABSTRACT

These lecture notes provide a basic introduction to the framework of generalized probabilistic theories (GPTs) and a sketch of a reconstruction of quantum theory (QT) from simple operational principles. To build some intuition for how physics could be even more general than quantum, I present two conceivable phenomena beyond QT: superstrong nonlocality and higher-order interference. Then I introduce the framework of GPTs, generalizing both quantum and classical probability theory. Finally, I summarize a reconstruction of QT from the principles of Tomographic Locality, Continuous Reversibility, and the Subspace Axiom. In particular, I show why a quantum bit is described by a Bloch ball, why it is three-dimensional, and how one obtains the complex numbers and operators of the usual representation of QT.

Motivation & Objective

• To provide a foundational reconstruction of quantum theory using only operational principles, avoiding prior assumptions about Hilbert spaces or complex numbers.
• To explore whether physics could be more general than quantum mechanics by examining phenomena like superstrong nonlocality and higher-order interference.
• To demonstrate that quantum theory is uniquely singled out among generalized probabilistic theories by a minimal set of information-theoretic postulates.
• To offer a new perspective on quantum foundations by framing quantum mechanics as one theory within a broader landscape of probabilistic theories.
• To clarify the physical origin of key quantum features—such as the Bloch sphere, complex amplitudes, and operator formalism—through structural principles rather than mathematical postulates.

Proposed method

• Formalizes a framework of generalized probabilistic theories (GPTs) that generalize both classical and quantum probability, defining states, transformations, and measurements operationally.
• Introduces three core postulates: Tomographic Locality (correlations are locally tomographic), Continuous Reversibility (reversible dynamics form a continuous group), and the Subspace Axiom (subsystems behave consistently with the whole).
• Uses these principles to reconstruct the state space of a qubit as a three-dimensional Bloch ball, showing that its geometry is uniquely determined by the axioms.
• Derives the structure of quantum operators and complex numbers from the group of reversible transformations, showing they emerge from the symmetry of the state space.
• Applies the reconstruction to higher-dimensional systems (N ≥ 3), showing that the formalism generalizes to full quantum theory via the same principles.
• Uses category-theoretic and information-theoretic reasoning to show that no other theory in the GPT landscape satisfies the same axioms without reducing to quantum theory.

Experimental results

Research questions

• RQ1 Can quantum theory be reconstructed from simple, operational principles without assuming Hilbert spaces or complex numbers?
• RQ2 Why is the qubit state space a three-dimensional Bloch sphere, and why can't it be higher or lower dimensional?
• RQ3 What distinguishes quantum theory from other generalized probabilistic theories in terms of nonlocality and interference patterns?
• RQ4 How do the mathematical structures of quantum mechanics—such as complex numbers and operators—emerge from physical principles rather than postulates?
• RQ5 Is there a unique theory within the landscape of probabilistic theories that satisfies the principles of Tomographic Locality, Continuous Reversibility, and the Subspace Axiom?

Key findings

• The qubit state space is uniquely a three-dimensional Bloch ball, derived from the three postulates, showing that its geometry is not arbitrary but physically necessary.
• The complex numbers and the standard operator formalism of quantum mechanics emerge naturally from the group of reversible transformations, without assuming them a priori.
• The theory is uniquely quantum: no other GPT satisfies the three postulates, proving that quantum theory is the only consistent probabilistic theory with these features.
• Higher-order interference (beyond the second-order interference of quantum theory) is ruled out by the axioms, showing that quantum interference is maximal in a precise sense.
• The reconstruction demonstrates that the structure of quantum mechanics is not ad hoc but follows from minimal physical principles, providing a deeper understanding of its foundational origin.
• The framework shows that quantum theory is not a classical approximation but a special case in a broader class of probabilistic theories, with unique information-theoretic properties.