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Quantum Theory from Five Reasonable Axioms (2001)

37 points by ClintEhrlich 11y ago

ClintEhrlich 11y ago
If the paper is a little dry but you are interested in the concept of axiomatically deriving quantum theory, try this wonderful written lesson from Scott Aaronson: http://www.scottaaronson.com/democritus/lec9.html http://www.scottaaronson.com/democritus/lec9.html The lecture is part of his course, "Quantum Computing Since Democritus," which features some of the clearest prose you will find anywhere on topics like complexity theory and P vs. NP.
j2kun 11y ago
> Axiom 5 (which requires that there exist 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. I actually recently read this paper and wondered the following question: what experiment supports the need for axiom 5, distinguishing quantum mechanics from classical probability theory? This came up because I always hear about "2-norm preserving unitary operators" as the only reasonable theory for quantum computing, which is of course different from classical probability. Is it just Occam's razor that to achieve the same results in experiments one would need to impose some large number of new states to a particle? Most of my understanding comes from a light reading of Nielsen-Chuang and Aaronson's stuff. (A minor tangent: a theorem from computer science informs me that complex numbers aren't needed if you're willing to get a "good enough approximation" and polynomial blowup, but this paper argues complex numbers are necessary, even for finite/countable state spaces; I want to read the paper a bit closer to figure out where this discrepancy is). This culminated in the following physics stackexchange question, which was probably not worded in the best way for the physics community[1]. I still don't really understand any of the answers. Maybe someone on HN can elucidate it for me :) [1]: http://physics.stackexchange.com/questions/205742/what-experiment-supports-the-axiom-that-quantum-operations-are-reversible http://physics.stackexchange.com/questions/205742/what-exper...
- ClintEhrlich 11y ago
  Time-reversal is a confusing topic, both because it involves innately complex issues, and because scientists are not always consistent about the terminology they use. This article does a good job untying the Gordian knot: Jacobs, Tim, and Christian Maes. "Reversibility and irreversibility within the quantum formalism." arXiv preprint quant-ph/0508041 (2005). http://arxiv.org/pdf/quant-ph/0508041v1.pdf http://arxiv.org/pdf/quant-ph/0508041v1.pdf
scottlocklin 11y ago
I remember when this came out I found it less than satisfying. Quantum mechanics is a theory of physics, and those axioms are not physical. Rovelli's earlier ideas are mathematically equivalent, far more physical and come closer to answering some real questions. http://fr.arxiv.org/abs/quant-ph/9609002 http://fr.arxiv.org/abs/quant-ph/9609002
- DennisP 11y ago
  "the theory describes only the information that systems have about each other"...it's almost tempting to see it as the universe's solution to a distributed systems problem.
- j2kun 11y ago
  For mathematicians like me it's great, though. I personally wonder if anyone can really understand physical laws until they're expressed in pure mathematical language.

37 points by ClintEhrlich 11y ago

If the paper is a little dry but you are interested in the concept of axiomatically deriving quantum theory, try this wonderful written lesson from Scott Aaronson: http://www.scottaaronson.com/democritus/lec9.html http://www.scottaaronson.com/democritus/lec9.html The lecture is part of his course, "Quantum Computing Since Democritus," which features some of the clearest prose you will find anywhere on topics like complexity theory and P vs. NP.

j2kun 11y ago

> Axiom 5 (which requires that there exist 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. I actually recently read this paper and wondered the following question: what experiment supports the need for axiom 5, distinguishing quantum mechanics from classical probability theory? This came up because I always hear about "2-norm preserving unitary operators" as the only reasonable theory for quantum computing, which is of course different from classical probability. Is it just Occam's razor that to achieve the same results in experiments one would need to impose some large number of new states to a particle? Most of my understanding comes from a light reading of Nielsen-Chuang and Aaronson's stuff. (A minor tangent: a theorem from computer science informs me that complex numbers aren't needed if you're willing to get a "good enough approximation" and polynomial blowup, but this paper argues complex numbers are necessary, even for finite/countable state spaces; I want to read the paper a bit closer to figure out where this discrepancy is). This culminated in the following physics stackexchange question, which was probably not worded in the best way for the physics community[1]. I still don't really understand any of the answers. Maybe someone on HN can elucidate it for me :) [1]: http://physics.stackexchange.com/questions/205742/what-experiment-supports-the-axiom-that-quantum-operations-are-reversible http://physics.stackexchange.com/questions/205742/what-exper...

ClintEhrlich 11y ago
Time-reversal is a confusing topic, both because it involves innately complex issues, and because scientists are not always consistent about the terminology they use. This article does a good job untying the Gordian knot: Jacobs, Tim, and Christian Maes. "Reversibility and irreversibility within the quantum formalism." arXiv preprint quant-ph/0508041 (2005). http://arxiv.org/pdf/quant-ph/0508041v1.pdf http://arxiv.org/pdf/quant-ph/0508041v1.pdf

scottlocklin 11y ago

I remember when this came out I found it less than satisfying. Quantum mechanics is a theory of physics, and those axioms are not physical. Rovelli's earlier ideas are mathematically equivalent, far more physical and come closer to answering some real questions. http://fr.arxiv.org/abs/quant-ph/9609002 http://fr.arxiv.org/abs/quant-ph/9609002

DennisP 11y ago
"the theory describes only the information that systems have about each other"...it's almost tempting to see it as the universe's solution to a distributed systems problem.
j2kun 11y ago
For mathematicians like me it's great, though. I personally wonder if anyone can really understand physical laws until they're expressed in pure mathematical language.