Quantum Theory From Five Reasonable Axioms
Lucien Hardy
TL;DR
<3-5 sentence high-level summary> Hardy derives quantum theory from five intuitive axioms, showing that classical probability arises only if the continuity axiom is dropped. He develops a vectorized, bilinear framework using fiducial states and measurements and proves that the number of degrees of freedom satisfies K=N^2, yielding the density-operator, POVM, and CP-map structure of quantum theory; the tensor-product rule for composites follows from the axioms. The approach clarifies why complex amplitudes and the trace rule emerge and highlights a sharp boundary between quantum and classical probability. It also discusses infinite-dimensional generalizations and interpretational implications for quantum information and foundational physics.
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.