[Paper Review] Quantum Theory From Five Reasonable Axioms

Lucién Hardy|ArXiv.org|Jan 3, 2001

Logic, Reasoning, and KnowledgeComputer Science5 references457 citations

TL;DR

This paper derives quantum theory from five intuitive axioms, showing that the key difference between quantum and classical probability lies in the requirement for continuous reversible transformations between pure states (Axiom 5). When this continuity postulate is removed, classical probability theory emerges, revealing why quantum theory necessitates complex Hilbert spaces and the trace formula for probabilities.

ABSTRACT

The usual formulation of quantum theory is based on rather obscure axioms (employing complex Hilbert spaces, Hermitean operators, and the trace rule for calculating probabilities). In this paper it is shown that quantum theory can be derived from five very reasonable axioms. The first four of these are obviously consistent with both quantum theory and classical probability theory. Axiom 5 (which requires that there exists continuous reversible transformations between pure states) rules out classical probability theory. If Axiom 5 (or even just the word "continuous" from Axiom 5) is dropped then we obtain classical probability theory instead. This work provides some insight into the reasons quantum theory is the way it is. For example, it explains the need for complex numbers and where the trace formula comes from. We also gain insight into the relationship between quantum theory and classical probability theory.

Motivation & Objective

• To identify a minimal set of physically reasonable axioms that uniquely recover the structure of quantum theory.
• To clarify why quantum theory requires complex numbers and the trace formula, by deriving them from foundational principles.
• To show that classical probability theory arises naturally when the continuity postulate (Axiom 5) is removed.
• To provide a deeper conceptual understanding of quantum theory by grounding it in axioms that could have been posited before empirical data.
• To offer a framework that may guide extensions beyond quantum theory, such as quantum gravity, by clarifying the theory's foundational structure.

Proposed method

• Define the dimension $N$ as the maximum number of distinguishable states in a single-shot measurement, and $K$ as the number of real parameters needed to specify a state.
• Use Axiom 1 (frequentist convergence) and Axiom 2 (simplicity) to constrain $K = K(N)$, with $K$ minimized for each $N$, leading to $K = N^2$ for quantum theory.
• Apply Axiom 3 (subspaces) to show that systems constrained to $M$-dimensional subspaces behave like $M$-dimensional systems.
• Use Axiom 4 (composite systems) to enforce $N = N_A N_B$ and $K = K_A K_B$, ensuring consistency under composition.
• Use Axiom 5 (continuous reversible transformations between pure states) to rule out classical theory ($K=N$) and force $K=N^2$, implying a complex Hilbert space structure.
• Derive the state as a real vector $\mathbf{p}$, with measurement probabilities given by linear functionals $\mathbf{r} \cdot \mathbf{p}$, and show that the most general evolution is a superoperator acting on density operators.

Experimental results

Research questions

• RQ1 What minimal set of physically reasonable axioms can uniquely derive the structure of quantum theory?
• RQ2 Why does quantum theory require complex numbers and the trace formula for probabilities, and can these be derived from first principles?
• RQ3 How does the requirement for continuous reversible transformations between pure states distinguish quantum theory from classical probability theory?
• RQ4 What happens to the theory if the continuity postulate (Axiom 5) is removed—does it recover classical probability theory?
• RQ5 Can this axiomatic framework provide insight into the interpretation of quantum mechanics and guide extensions beyond quantum theory?

Key findings

• Quantum theory is uniquely derived from five axioms, with the key distinction from classical probability theory being the requirement for continuous reversible transformations between pure states (Axiom 5).
• When Axiom 5 is dropped, the simplicity axiom leads to $K = N$, which corresponds to classical probability theory, showing that continuity is the essential difference.
• The state space is shown to be isomorphic to a real vector space of dimension $K = N^2$, which implies the need for complex Hilbert spaces and the density operator formalism.
• The most general evolution of a quantum state is shown to be a superoperator, consistent with both unitary evolution and state collapse.
• The probability of a measurement outcome is given by a linear functional $\mathbf{r} \cdot \mathbf{p}$, and the state of a composite system is represented by a positive operator on the tensor product of Hilbert spaces.
• The framework naturally incorporates collapse interpretations, as the most general evolution includes maps from pure to mixed states.