I Is the Born Rule Derivable from Other Postulates in Quantum Mechanics?

[tomdodd4598]

TL;DR Summary

Can the Born rule of quantum mechanics be derived from more fundamental principles?

Hey there!

I’ve been thinking about the Born rule recently and whether it can be derived from the other postulates from QM.

I’ve done a bunch of google searching, across PF, stackexchange and the arxiv, but most of it has felt a little opaque, particularly on whether anything has been ‘accepted’ as a derivation yet.

I decided to just try myself: using the assumptions that probability needs to be conserved under unitary transformations, that the rule holds for all quantum systems, and that the probabilities are functions of the respective amplitudes, I think I’ve showed that the Born rule must hold.

Do those assumptions go beyond the other postulates of QM, or are they perhaps in any sense less ‘fundamental’ than the Born rule itself?

Thanks in advance!

 

[vanhees71]

That's very interesting. As far as I know so far nobody has been able to derive Born's rule from the other postulates without introducing any other assumption(s). So it would make and interesting research paper!

The issue is very nicely discussed in Weinberg's textbook on quantum mechanics:

S. Weinberg, Lectures on Quantum Mechanics, Cambridge University Press

 

[tomdodd4598]

Well, I'll write up about it properly and talk to a tutor or something, always a decent chance I've make a mistake or assumed something implicitly somewhere. I just so happen to have that book, so I'll have a read :)

 

[Demystifier]

I’ve been thinking about the Born rule recently and whether it can be derived from the other postulates from QM.

The Born rule in the arbitrary basis can be derived from the Born rule in the position basis. See the paper linked in my signature below.

 

[atyy]

As @vanhees71 mentions above, there is no known derivation of the Born rule "purely" from the other postulates of QM.

However, if one supplements the postulates or uses other postulates, the Born rule can be derived.

Gleason's theorem
Hardy, Quantum Theory from Five Reasonable Axioms
G. Chiribella, G. M. D'Ariano, P. Perinotti, Informational derivation of Quantum Theory
Lluís Masanes, Thomas D. Galley, Markus P. Müller, The measurement postulates of quantum mechanics are operationally redundant

 

[tomdodd4598]

These are exactly the sorts of papers and ideas which I’ve found rather ‘opaque’ :P
Maybe I just don’t know enough maths to understand what exactly they’re doing, but perhaps I’ll go through them again. Just seems to me to be an awful lot more complicated than I imagine it could be...

 

[bobob]

Summary: Can the Born rule of quantum mechanics be derived from more fundamental principles?

Hey there!

I’ve been thinking about the Born rule recently and whether it can be derived from the other postulates from QM.
Thanks in advance!

Possibly, but I'm not sure if there is any point. Quantum mechanics is essentially an an extension of classical probability theory, so I'm not sure the Born rule is really out of place as a fundamental axiom. In any axiomatic stystem, you have a choice of which statements take the role of axioms and which statements are left to be proved from those axioms. Since quantum mechanics is a probabilistic theory, the Born rule would seem to be a good choice to consider fundamental to the theory. I really don't see any reason for a lot of angst and hand wringing over this.

 

[N88]

Summary: Can the Born rule of quantum mechanics be derived from more fundamental principles?

Hey there!

I’ve been thinking about the Born rule recently and whether it can be derived from the other postulates from QM.

I’ve done a bunch of google searching, across PF, stackexchange and the arxiv, but most of it has felt a little opaque, particularly on whether anything has been ‘accepted’ as a derivation yet.

I decided to just try myself: using the assumptions that probability needs to be conserved under unitary transformations, that the rule holds for all quantum systems, and that the probabilities are functions of the respective amplitudes, I think I’ve showed that the Born rule must hold.

Do those assumptions go beyond the other postulates of QM, or are they perhaps in any sense less ‘fundamental’ than the Born rule itself?

Thanks in advance!

Tom,

I suspect that you might like my friend Fritz Fröhner (1988):

“Missing link between probability theory and quantum mechanics: the Riesz-Fejér theorem."

Z. Naturforsch. 53a, 637-654. http://zfn.mpdl.mpg.de/data/Reihe_A/53/ZNA-1998-53a-0637.pdf

All the best; N88

 

[Mentz114]

I suspect that you might like my friend Fritz Fröhner (1988):

“Missing link between probability theory and quantum mechanics: the Riesz-Fejér theorem."

Z. Naturforsch. 53a, 637-654. http://zfn.mpdl.mpg.de/data/Reihe_A/53/ZNA-1998-53a-0637.pdf

All the best; N88

The Fröhner paper is a blast. It "... demystifies quantum mechanics to quite some extent" as promised.

I look forward to spending some time reading the paper fully. Thanks for the link.

 

[HomogenousCow]

Summary: Can the Born rule of quantum mechanics be derived from more fundamental principles?

I decided to just try myself: using the assumptions that probability needs to be conserved under unitary transformations, that the rule holds for all quantum systems, and that the probabilities are functions of the respective amplitudes, I think I’ve showed that the Born rule must hold.

Haven't you assumed the born rule in order to derive the born rule?

 

[johnkclark]

Gleason's Theorem, proven in 1957, says that the Born rule is the only one that is unitary, the only one where all the probabilities add up to exactly 1. So if you want probabilities its got to be proportional to the square of the magnitude of the particle's wave-function and not the cube or something else. However why we must use probabilities at all and not certainties is unknown.

John K Clark

 

[vanhees71]

Just to add: It has also been tested that it's $|\psi|^2$ and not some other power to provide the probabilities (or probability distributions):

With photons:

https://physicsworld.com/a/quantum-theory-survives-its-latest-ordeal/

With molecules:

https://physicsworld.com/a/quantum-superposition-still-adds-up-in-three-slit-experiment/

 

[atyy]

Gleason's Theorem, proven in 1957, says that the Born rule is the only one that is unitary, the only one where all the probabilities add up to exactly 1. So if you want probabilities its got to be proportional to the square of the magnitude of the particle's wave-function and not the cube or something else. However why we must use probabilities at all and not certainties is unknown.

As I learned from @bhobba, this is not correct. Probability does not preclude certainty. The crucial assumption in Gleason's Theorem is not the assumption of probability, but the assumption of contextuality.

 

[tomdodd4598]

Bit late back to this, but thanks for the replies everyone. I'm going to conclude that what I was doing was a rough/lenient form of Gleason's result.

 

[Michael Price]

Bit late back to this, but thanks for the replies everyone. I'm going to conclude that what I was doing was a rough/lenient form of Gleason's result.

BTW you might find Zurek's derivation useful, which I've tried to present here: https://www.quora.com/How-does-the-...ding-to-the-Born-rule/answer/Michael-Price-29

 

[DarMM]

Bit late back to this, but thanks for the replies everyone. I'm going to conclude that what I was doing was a rough/lenient form of Gleason's result.

Gleason's theorem can be summarised by the statement that if you assume that dichotomic measurements on physical systems are modeled by projectors from a C*-algebra (loosely a Projector is an operator with $P^2 = P$ and a C*-algbera is the algebra of operators you'd find in QM) then non-contextual probabilities for those measurements have to obey the Born rule, i.e. come from a density matrix.

It's a difficult theorem to prove. Though many say it still leaves the mystery as to why the algebra of measurements is that of a C*-algebra.

 

[vanhees71]

That's easy to answer: Because it works so well!

That's the answer to all these questions, like "why is nature behaving the way she does?" This is not a sensible answer to ask in the realm of the natural sciences. It's the other way around: Natural science starts with objective reproducible observations of phenomena. Tha very fact that such reproducible objective phenomena exist is already an observation. Then on discovers some regularities, i.e., causal structure. This means that if I set up something in a specific way, I get after some time something different, which is predictable. QT tells us simply that there's less predictable than within classical physics: In QT, given a certain situation not all observables take determined values but you know the probabilities to find a given possible value of each observable when you measure it. As it has turned out QT is very successful in predicting these probabilities, and so far we haven't found any way for such a causal description which leads back to a deterministic theory which as successfully describes the phenomena as QT.

So the answer to the question, why nature behaves according to the laws described in QT is simple: QT has been discovered by making more and more detailed experiments and the attempt to find a mathematical theory to describe their outcomes. As it turned out, this attempt was very successful, i.e., QT survived all the empirical tests done to check it's accuracy. So it's not that "nature behaves according to the laws of QT" but rather "QT is discovered from observations how nature behaves and tailored such as to describe nature's behavior accurately".

 

[Michael Price]

Just to add: It has also been tested that it's $|\psi|^2$ and not some other power to provide the probabilities (or probability distributions):

It always amuses me that Max Born originally got the power wrong. He originally gave the rule as $|\psi|$ in his 1926 paper - $|\psi|^2$ only appears as a proof correction, in a footnote! ☺