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arXiv:quant-ph/0101012quant-ph

Quantum Theory From Five Reasonable Axioms

Lucien Hardy

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.

arXiv comments
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
Journal reference
DOI
Subjects
quant-phQuantum Physics hep-thHigh Energy Physics, Theory

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Linked by 8 identifiable commenters in 12 comments across 12 threads, from Mar 2013 to Mar 2025. Found in 3 HN submissions from Aug 2010 to Oct 2019. Open submission discussions.

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Discussed in comments12 threads

  1. Math That Matters: The Case for Probability over Polynomials 1 comment

    • > Quantum Mechanics is essentially just probability with complex numbers For some more on that topic: https://arxiv.org/abs/quant-ph/0101012 https://www.scottaaronson.com/democritus/lec9.html

      magicalhippoView on HNParent context

  2. Schrödinger equation emerges mathematically from classical mechanics (2012) 1 comment

    • Reminds me of a paper by by Hardy[1] where he introduces five reasonable axioms (his words). Classical and quantum probability theory obeys the first four. However the fifth, which states that there exists continuous transformations between pure states, is only obeyed by the…

      magicalhippoView on HNParent context

  3. Quantum theory based on real numbers can be experimentally falsified 1 comment

    • Lucien Hardy wrote a paper[1] showing how one "naturally" ends up with a quantum theory by demanding a few reasonable axioms. The paper also goes into how this implies complex numbers (and rules out quaternions). Scott Aaronson has a more accessible (and humorous) article on it…

      magicalhippoView on HNParent context

  4. Physicists take a key step in correcting quantum computer errors 1 comment

    • Unlike perpetual motion or free energy, the math and physics behind quantum computing is rock solid - the possibility is baked into the framework of quantum mechanics at an absolutely fundamental level. Even if it turned out that all of quantum field theory was completely bogus…

      ceviView on HNParent context

  5. Category Theory Illustrated – Logic 1 comment

    • There is some: https://arxiv.org/abs/quant-ph/0101012 And according to the first sentence of the summary the current formulation is axiomatic as well...

      boris_mView on HNParent context

  6. The Wolfram Physics Project: A One-Year Update 1 comment

    • Keep in mind that 'information propagation' is pretty fundamental: we can use it to define light cones (if A can propagate information to B, then A/B are in each others past/future light cones). We can use light cones to define time (past/future) and space (neither). That's a…

      chriswarboView on HNParent context

  7. You thought quantum mechanics was weird: check out entangled time 1 comment
  8. Deep Learning and Quantum Entanglement [pdf] 1 comment

    • My random stream-of-consciousness reference reactions ... Tegmark & Lin's discussion of how deep learning maps onto the physical world: Why does deep and cheap learning work so well? [1][2] The work of Aerts et al. more than 10 years ago, including how vector space models of…

      mikhailfrancoView on HNParent context

  9. Quantum Mechanics for Programmers 1 comment

    • > and it is amusing how many professed non-many-worlders say such things. I'm not saying I do or don't believe any of this (if anything, I'm interpretation-agnostic at the moment). I'm just pointing out that there is a contradiction in having one postulate demand unitary state…

      XcelerateView on HNParent context

  10. Quantum mechanics as a generalization of probability (2007) 1 comment

    • I love this description of quantum. The paper Aaronson links to, "Quantum Theory from Five Reasonable Axioms" (http://arxiv.org/abs/quant-ph/0101012), is also a great read. The "probability in L^2" cleared up a lot of confusion for me, although I still have a very poor intuition…

      TotientView on HNParent context

  11. What Happened When We Tried to Publish a Real Paper Investigating Time Travel 1 comment

    • It's not surprising to me that no physics journals would accept it. Physics journals are, sadly, extremely wary of accepting anything that seems philosophical. For example, serious people writing serious papers on the foundations of quantum mechanics used to have a hell of a…

      bentonerView on HNParent context

  12. Pilot waves: oil droplets model quantum behavior 1 comment

    • Truly one of the most remarkable and important things I have seen in a very long time. Two page summary: http://www.tcm.phy.cam.ac.uk/~mdt26/tti_talks/deBB_10/bush_t... Longer presentation with lots of pictures (10MB)…

      mikhailfrancoView on HNParent context

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