
Quantum Theory from Five Reasonable Axioms (2001) - ClintEhrlich
http://arxiv.org/abs/quant-ph/0101012
======
j2kun
> 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-
exper...](http://physics.stackexchange.com/questions/205742/what-experiment-
supports-the-axiom-that-quantum-operations-are-reversible)

~~~
ClintEhrlich
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)

------
ClintEhrlich
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.

------
scottlocklin
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
"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.

