Quantum Theory From Five Reasonable Axioms
2001/01/03 by Lucien Hardy, Lucién Hardy, Hardy, Lucien · 5 voices · 486 citations
Computer Science · Mathematics · Physics and Astronomy · #Advanced Algebra and Logic #Axiom #Geometry #Logic, Reasoning, and Knowledge #Mathematical economics #Mathematics #Physics #Pure mathematics #Quantum #Quantum Mechanics and Applications #Quantum mechanics #Theoretical physics #hep-th #quant-ph
paper · pdf · doi:10.48550/arxiv.quant-ph/0101012
published in arXiv (Cornell University) (Cornell University) · 34 pages. Version 4: Improved proofs of K=N^r and D=D^T. Discussion of state update rule after measurement added. Various clarifications in proofs Version 2: Axiom 2 modified and corresponding corrections made to proof in Sec. 8.1. Typos and minor errors fixed
openalex publication_date 2001/01/03 · arxiv created 2001/09/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.
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Discussions
- Quantum Theory from Five Reasonable Axioms (2001) [hn, 37 points, 6 comments]
- Quantum Theory from Five Reasonable Axioms [hn, 17 points, 1 comments]
- anyway looks pretty good to me @danabra.mov. Funny to think of a matrix mult as 'inspired by neural networks'. You might like this paper which does a minimal thing like you're going for [bsky, 4 points, 1 comments]
- Quantum Theory From Five Reasonable Axioms [hn, 1 points, 0 comments]
- Happy to! This is probably going to be more of a link dump, at least at the start. Scott Aaronson introduced me to that “QM is statistics in l2 norm” idea. See: www.scottaaronson.com/democritus/l... A [bsky, 0 points, 1 comments]
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