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Zurek's Born rule derivation

  1. Oct 17, 2011 #1


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    Zurek has proposed a http://lanl.arxiv.org/abs/1105.4810v1" (Phys.Rev.Lett.106:250402,2011) based on these assumptions:

    (i) States “live” in Hilbert spaces
    (ii) Evolutions (including measurements) are unitary.
    (o) Hilbert spaces of composite systems have tensor structure.
    (iii) Immediate repetition of a measurement yields the same outcome.
    (z) "Envariance also relies on locality of quantum dynamics (i.e., the fact that a unitary operation here cannot change a state there)"

    Is it correct and satisfactory? What is the BM take on this?
    Last edited by a moderator: Apr 26, 2017
  2. jcsd
  3. Oct 17, 2011 #2
    Hi atyy,
    Off the top of my head, I'm not sure. I'll have a look this evening if I can find some time.
    This, though, is something I've read about. In BM i.e. de Broglie-Bohm quantum mechanics, then Born's rule arises naturally because it's the highest entropy state. If you take a bunch of Bohmian particles distributed any way you like (including 'non-Born-rule' distributions) and allow them to evolve dynamically according to the Schroedinger probability current, then they are overwhelmingly likely over the course of time to become distributed as the square of the guiding wave function (like 'dust particles in a hurricane'). This is an entirely analagous process to classical particles reaching thermal equilibrium, which is why particle density = square of the wave function (Born's rule) is often called 'quantum equilibrium' in this context. BM is just the statistical mechanics of particles moving under a dynamical law different from Newton's equation, so this is hardly surprising - though not in fact very well known.

    This is discussed extensively and illustrated with numerical simulations in Towler, Russel and Valentini's paper http://uk.arxiv.org/abs/1103.1589".

    See also the Wikipedia article http://en.wikipedia.org/wiki/Quantum_non-equilibrium" [Broken] (though the article has clearly been written by someone who understands it imperfectly).

    Hope this helps. I'll get back to you on the Zurek stuff.
    Last edited by a moderator: May 5, 2017
  4. Oct 19, 2011 #3
    This deserved a reply.

    Zurek et al should demonstrate their derivation with a finite number of states, and show that it holds - can they do that?

    The Bohmian derivation of the Born Rule is interesting but not surprising, if a dynamical flow has an invariant probability density (which is |psi|^2) then as the flow evolves more and more states will fall into the invariant flow - so eventually the whole flow will follow the invariant distribution.

    Maybe the Bohmian argument has accidentally discovered the real reason for the Born Rule, but with the wrong underlying assumption of a deterministic evolution - ie the rule may also be a "thermodynamic limit" of non-deterministic evolution
  5. Oct 19, 2011 #4

    Ken G

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    Yes, I agree that's an interesting possibility. It's also something of a Catch-22 for Bohm.
  6. Oct 20, 2011 #5


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    I had the pleasure of attending a talk Zurek gave on this topic on the occasion of Zurek being awarded the Albert-Einstein professorship prize last thursday. The talk was consistent, quite good and his derivation of Born's rule made sense. None of the present "huge" guys gave any harsh criticism in the questions round after the talk. But on the other hand: Who would strongly criticize someone on the occasion of him getting a prize?
  7. Oct 20, 2011 #6
    And I'm sure you'll enlighten us with your demonstration that the underlying assumption is 'wrong'. What's that? You won't? Hmmmm.. As far as I know it's consistent with all known experimental facts.

    I also love your use of 'accidentally'.. Makes it sound like a monkey finally finishing his typescript of Hamlet. :smile:
    Last edited: Oct 20, 2011
  8. Oct 20, 2011 #7

    Ken G

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    I think you are missing unusualname's point here-- he did not claim to know that the Bohmian argument is wrong, he merely pointed out that if his alternative possibility were true, nothing in the Bohmian argument could distinguish it from their own perspective. That is a rather important point to make.
  9. Oct 20, 2011 #8


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    For most measurements, that assumption is not realized in nature. For example, a typical detection of a photon completely destroys the photon, so you cannot repeat the measurement.

    The above is true for the so-called no-demolition measurements, but most measurements are not such.
  10. Oct 20, 2011 #9
    Not so. The underlying assumptions of deBB allow one to contemplate the existence of non-Born-rule distributions (which are not even possible to contemplate in ordinary QM, where the Born rule is a postulate). And by their nature, non-Born-rule distributions are experimentally detectable.

    One wouldn't expect to see them now because Schroedinger evolution causes the particles to become 'rapidly' Born-rule distributed over the course of time. One might therefore contemplate making predictions about the early universe - as discussed in the paper I cited in post #2 - which might be visible in the cosmic microwave background. It is possible, given the deBB assumptions, that the CMB had its origin in an epoch before 'quantum equilibrium' was fully established.

    A casual reader might certainly look at atyy's post and infer that the underlying assumptions of deBB have been shown to be 'wrong', rather than just 'experimentally unverifiable', in contrast to what you state. Nonetheless, neither view is correct, as I am sure he will admit.
  11. Oct 20, 2011 #10
    Actually that's probably a good basis for a "proof" that BM is wrong since Shakespeare used free-will to write Hamlet, which doesn't exist in bohmian world :wink:

    Unless of course you can show me Hamlet evolving in the Bohmian deterministic equations :smile:
  12. Oct 20, 2011 #11


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    That is not really what Zurek aims for. What Zurek is after is that every measurement will yield an outcome and leave a state that is "compatible" with the measurement performed. For simple experiments this postulate does not mean anything else than: measurements put the system under examination into an eigenstate of the operator corresponding to the measurement performed. It aloows him to derive orthogonality of outcome states.

    The problem of not really meeting this requirements in quantum optics experiments and why that is no real problem is explicitly discussed in W. Zurek, PRA 76, 052110 (2007) somewhere around page 3 I think.
  13. Oct 20, 2011 #12
    DeBB is not a theory of the world. It's a theory of non-relativistic quantum mechanics (with extensions to quantum-field theory where appropriate). When we have a complete theory of the world which includes everything, including gravity and everything that exists, then we look and see if that is deterministic before we make conclusions about free will.

    That said, if you want, Bohm himself showed how you can add a 'random noise' term - which you can imagine as arising from some kind of 'background stochastic quantum fluctuation process' - to the deBB deterministic equations, which render them non-deterministic. And yet, they give the same experimental predictions, and still lead to the above derivation of the Born rule (and clear explanations of everything else that is supposed to be mysterious about QM).

    Simplistic arguments about determinism are no proof of anything I'm afraid.
  14. Oct 20, 2011 #13
    We're getting a bit off-topic but I don't agree that BM gives clear explanations of much, simply because it doesn't explain much beyond a simplistic non-relativistic quantum mechanics. In fact this derivation of the Born Rule might be the only positive feature of BM. That's why I asked if Zurek's ideas could be demonstrated with a simple (finite) model on a computer perhaps - so I could better understand what is the crucial part of the argument that leads to the Born Rule probabilities (and whether these assumptions are much of an improvement on just assuming the BR as a postulate)
  15. Oct 20, 2011 #14
    ok, no replies, so let me say that I think Zurek et al are not doing physics, they are doing something akin to philosophy. The argument they have constructed is just over-thinking and obfuscation in the case of an unknown solution like most of philosophy is.

    Physics is a set of rules for how nature behaves, when you propose an idea in physics you should be able to demonstrate that nature could behave like this, in some concrete fashion.
  16. Oct 20, 2011 #15

    Ken G

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    OK, that is a potential application where the distinction could actually be made-- except that there are so many other unknowns involved in the inflationary epoch that I doubt a strong case could be made that one is really testing Bohmian aspects by looking at the CMB! That strikes me as one of the more pie-in-the-sky claims I've seen in awhile. But I'll grant you that it might be the one place where anything Bohmian actually amounts to something other than angels on the pin. It would be amazing if Bohmian mechanics could be verified by looking at the early universe, but there are a lot of other amazing things that might need to happen first, like we might figure out what the universe is made of and what dominates its dynamics.
    I think it remains quite true that at present, Bohmian mechanics is experimentally unverifiable-- all we have is a possible avenue that might one day allow that to not be true. It is of value to notice this possible avenue, but not to hype it. I don't know of any cosmologists who think the next place to take CMB observations is into the realm of testing Bohmian mechanics, there remain many more pressing and less speculative issues to iron out first.

    And as for determinism, I'd say it's pretty clear that the primary motivation for Bohmian mechanics has always been a deterministic description. Take that away, and it's hard to see any value in it at all. You say that its value is to provide an explanation of the Born rule, but at the cost of introducing a pilot wave. It is never hard to explain one thing by introducing one other thing instead, the trick is to explain many things with one thing.
    Last edited: Oct 20, 2011
  17. Oct 20, 2011 #16


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    I've not had alot of time lately, but that doesn't look satisfactory for me because I don't think it addresses the real points.

    I've seen other "derivations" and they tend to always postulate the keys that are exactly what should be explained.

    Ariel Caticha for example provided som arguments for born rule basically following from consistency requirements of how to manipulate information, but PROVIDED that it was given that information was represented by complex numbers.

    I'm currently trying to get some time to get back to my own work and one question I ask in this context is to understnad how information is optimally encoded. When you consider different ways to encode information in a sliding time history windows one immediately faces the problem how to COMBINE corresponding counter states of the original signal level, as well as counter states of a windowed say DFT transform. Then the problem becomes how to represent this union. Most probably the complex case will follow naturally from this and the born rule probably follows trivially as long as you keep track of your counting.

    Thus I tend to see th born rule as related to the decompression of an information code. The big question I ask, and that I want answered for anything "satisfactory" is WHY nature seems to CHOOSE THIS code.

    Here my tentative answer (remains to be proven) is that this code is the optimum information representation when considreing a compact code processing a windowed datastreem.

    I've got some ideas to try some numerical simulations of this and a I figured a worthy challange would be to test the algorithm of pseudorandom code. I a stable code can live in that datastream it means I've decoded the pesudo-random algorithm.

    So presumable matter systems in nature are the analogs of "optimal codes" and the lawd of physics should follow from the specific code. But the point is of course that the code isn't given, the code is exactly what erquires explanaion and I think an evolutionary context is unavoidable.

    In such a view the complex representation is poblaby a selected code component evolving early already at the state of combining statistics in time and frequence domain in the same code.

  18. Oct 21, 2011 #17

    Ken G

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    Perhaps we should not look at Zurek's goal as being to derive why the Born rule is true, but rather, to discover what the truth of the Born rule itself depends on. This is really all a derivation can ever do-- connect the theorem to the postulates required. So in that sense, it's goal is not to explain the Born rule, but to show what it is that must be explained if we wish to explain the Born rule. So the next question is, shall we simply regard that list as the axioms we will use when we use the Born rule, or is there some other set of axioms that seem more insightful or inherently true that we could use to derive those postulates as theorems?
  19. Oct 21, 2011 #18


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    Something like this is my take on the issue.

    There are some apparently deep connections between fourier transforms (which is unavoidably) a key transform in this context, and statistics that is clear already at very superficial level:

    For example that the unique distribution dictated by the central limit theorem, just happens to be the ONLY distribution that is it's own fourier transform. But you can turn the argument around, and instead of applying it to single out a distribution, it can be used to single out a coding transform if you consider a evolving coding system (which I of course secretly associate to a matter system). This QM tells us something about how one subsystem of nature encodes information about it's environemnt.

    Pursuing further thinking in this direction is what I'm doing.

  20. Oct 21, 2011 #19

    Ken G

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    That's an interesting insight, that stochastic processes lead to a distribution that is invariant under a change between two complementary bases. It sounds like the central limit theorem leads to a brand of partitioning of information between complementary bases, where only the raw "amount" of information is traded off against each other, but how that information is distributed reaches a common form. That might provide some insight into just what complementarity is in the first place.
  21. Oct 21, 2011 #20
    No we're not (you'll recall the OP had two questions, the second of which was "what is the BM take on this?" which I'm clearly attempting to answer). In fact, we're getting to the tedious final stage of any conversation involving BM on this forum - the bit where the clever supporter of orthodox QM realizes that his off-topic objections to BM are baseless, and he resorts to accusing me of being off-topic in order to save face and to divert attention from the fact that he's only stating what he's stating as a fact because that's what everybody else does, rather than because it's true, or relevant, or anything like that.. :smile:
    You're right, of course. Apart from deriving the Born rule, and providing clear explanations of quantum tunneling, scattering, quantum interference, the two-slit experiment, wave-particle duality, chemistry, measurement, spin, degeneracy pressure, everything supposedly 'mysterious about QM', better sanitation, medicine, education, irrigation, public health, roads and a freshwater system and baths and public order... what has de Broglie-Bohm theory ever done for us?
    Last edited: Oct 21, 2011
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