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Quantum Theory From Five Reasonable Axioms

Lucien Hardy

2001 · arXiv

PhysicsStatistics & probabilityLanguage models
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From the 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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Original HN comments, with labels from simple text rules. A label describes the whole comment and may miss context or irony. Read the discussion to make up your own mind. About the labels

10 selected excerpts · 10 total mentions

cevi
Neutral mention · rule
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, I would still expect the basic framework of quantum mechanics (i.e. unitary evolution, the Born rule for converting amplitudes to probabilities, density matrices describing mixed states) to be correct. If quantum computing was not possible, that would mean that everything we thought we knew about physics was hopelessly wrong. Physics would have to be much stranger - we'd have to give up on either…
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locality (i.e. causation being limited by the speed of light), reversibility (closely related to conservation of energy), or nondeterminism in order to make the math work out. There are several papers in the literature that derive quantum mechanics from these or related assumptions (often together with some extra technical detail, that differs between approaches): https://arxiv.org/abs/quant-ph/0101012 https://arxiv.org/abs/1011.6451 https://arxiv.org/abs/0911.0695
Original comment Physicists take a key step in correcting quantum computer errors
chriswarbo
Neutral mention · rule
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 pretty standard approach to relativity (I can't remember whether I covered it as an undergraduate, but I certainly encoutered it around that time). (Shannon) information theory is also closely related to probability, and quantum theory is a form of probability theory (e.g. https://arxiv.org/abs/quant-ph/0101012 ). Hence it's not entirely out-there to think that an informational/computational approach…
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like Wolfram's might bear fruit. In fact, I was hoping that someone smart(er than me) would investigate such connections a bit deeper; if only to rule them out! Note that "causal set theory" seems to be going that direction too (although I'm not overly familiar with it) https://en.wikipedia.org/wiki/Causal_sets I also find it promising that ideas from quantum computing (qbits, circuits, teleportation, protocols, etc.) have gone from a quirky engineering footnote (in the 90s), to become a core part of how many physicists now think about the world.
Original comment The Wolfram Physics Project: A One-Year Update
mikhailfranco
Neutral mention · rule
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 human categorization show QM structure [3]. Lucien Hardy's exquisite teasing out of the difference between classical and quantum probability: Quantum Theory From Five Reasonable Axioms [4] It turns out that the innovation, power and strangeness of QM is related to separating physical processes into: 1. Linear continuous unitary reversible evolution relying on complex amplitudes (wavefunction…
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propagation). 2. Non-linear discontinuous irreversible 'collapse' or 'measurement' or 'entanglement' or 'correlated branching', with probabilities based on the Born Rule (real values from the square of wave function amplitudes). Neural nets also decompose a problem into successive alternating layers of reversible continuous functions and discrete irreversible categorical decisions (e.g. softmax/sigmoid logistic classifiers). An even more obscure tangent is that most of the financial market is now constructed from 'options', which track continuous values with a continuous payoff, but offer a choice of execution to collapse the option and create a real financial result, e.g. see N.N.Taleb's dicussion of optionality [5]. [1] http://arxiv.org/pdf/1608.08225.pdf [2] http://www.youtube.com/watch?v=5MdSE-N0bxs [3] http://en.wikipedia.org/wiki/Diederik_Aerts#Quantum_structur... [4] http://arxiv.org/abs/quant-ph/0101012 [5] http://fooledbyrandomness.com/ConvexityScience.pdf
Original comment Deep Learning and Quantum Entanglement [pdf]
Xcelerate
Neutral mention · rule
> 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 evolution (the Schrödinger equation) for some ill-defined "system" while another postulate says that unitarity is broken at the system/environment boundary. While there's been plenty of attempts to work around this (e.g. https://arxiv.org/abs/quant-ph/0101012), I wouldn't say that anyone has formulated a consistent set of axioms that definitively resolves the issue. > We often describe quantum…
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systems using an entirely deterministic (Schroedinger) equation. But we don't know in what sense that equation describes the physical state of the experimenter + system, or in what sense it is "just" a probability model. Agreed. It's certainly a useful model but it leaves out all kinds of interesting phenomena that we observe in practice (namely, relativistic and radiative effects). Curiously though, if you turn to QFT for a better probabilistic model, Haag's theorem (https://en.wikipedia.org/wiki/Haag%27s_theorem) implies that a universal Hilbert space representation cannot describe both free and interacting fields (a problem that the non-relativistic Schrödinger equation doesn't have!)
Original comment Quantum Mechanics for Programmers
Totient
Neutral mention · rule
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 for what the Born probabilities are probabilities of. If you believe the MWI story, it seems like it's the probability you will "find yourself in the universe where this outcome happened" but even that sounds odd to me.
Original comment Quantum mechanics as a generalization of probability (2007)
magicalhippo
Neutral mention · rule
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 quantum theory. In that sense he argues that quantum theory is in a sense more reasonable than classical theory. There's also an interesting link between this and entanglement[2] which seems to rule out other probability theories, leaving only quantum theory able to exhibit entanglement. Not my field at all though, just find these foundational things interesting to ponder. [1]: https://arxiv.org/abs/quant-ph/0101012 [2]: https://arxiv.org/abs/0911.0695v1
Original comment Schrödinger equation emerges mathematically from classical mechanics (2012)
magicalhippo
Neutral mention · rule
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 here[2]. Entanglement has been shown to be intimately linked to this[3] result, which is interesting given the experimental evidence[4] for entanglement. Not my field, but I found this interesting at least. [1]: https://arxiv.org/abs/quant-ph/0101012 [2]: https://www.scottaaronson.com/democritus/lec9.html [3]: https://arxiv.org/abs/0911.0695 [4]: https://en.wikipedia.org/wiki/Quantum_entanglement#Notable_e...
Original comment Quantum theory based on real numbers can be experimentally falsified
mikhailfranco
Neutral mention · rule
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): http://www.iap.fr/activites/seminaires/GReCO/2013/presentati... When combined with Quantum Information Theory that shows QM as 'just' a probability theory, and simple axiomatic approaches like Lucien Hardy's "Quantum Theory From Five Reasonable Postulates": http://arxiv.org/abs/quant-ph/0101012 Some of the mystery of spooky QM, long perpetuated by the Copenhagen Interpretation, is finally falling away. Mik
Original comment Pilot waves: oil droplets model quantum behavior

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