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Collapse and Peres' Coarse Graining

  1. Jan 8, 2015 #1

    atyy

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    Collapse is the sudden change in wave function after a measurement. Unless otherwise stated, I will assume that neither the wave function nor collapse are necessarily physical, and are just calculational tools that are part of the experimental success of quantum mechancs. Collapse is in almost all major textbooks except Ballentine and Peres, and is used in a standard way in the Bell tests.

    Peres suggests that collapse can be replaced with coarse-graining. We know that collapse is verified by all experimental data to date, and that coarse-graining, if successful, must make exactly the same predictions as collapse. I think it is an interesting idea, but I have only ever seen it in Peres. Can coarse-graining without collapse successfully reproduce the predictions of quantum mechanics? In particular, can one do the coarse-graining without collapse explicitly and recover the correlations for the Bell tests? Is the coarse-graining in a Bell test a local procedure?
     
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  3. Jan 8, 2015 #2

    Demystifier

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    By looking at the Peres book, I can only find that he talks about coarse graining in the context of classical mechanics, to explain the classical irreversibility. Can you specify where exactly does he say that it can replace the quantum collapse?
     
  4. Jan 8, 2015 #3

    atyy

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    On p376 he uses the term "blurring", which is what I (and I think vanhees71) was referring to:

    "Consistency thus requires the measuring process to be irreversible. There are no superobservers in our physical world.

    Formally, the logical consistency of the "dequantization" of a measuring apparatus implies the equivalence of two different descriptions of the same process ... This reduced density matrix can be converted by means of Eq. (10.56) into a Wigner function, ##W_{A}(q, p)##. Some blurring (see page 316) converts the latter into a fuzzy Wigner function which is nowhere negative and may be interpreted as a Liouville density ##f_{A}(q,p)## if the ##\hbar^{2}## terms in the quantum Liouville equation (10.67) are negligible for the macroscopic apparatus."

    Some other references where Peres (and coauthors) mention similar ideas are:

    http://arxiv.org/abs/quant-ph/9712044
    Quantum and classical descriptions of a measuring apparatus
    Ori Hay, Asher Peres

    http://arxiv.org/abs/quant-ph/9906023
    Classical interventions in quantum systems. I. The measuring process
    Asher Peres
     
    Last edited: Jan 8, 2015
  5. Jan 8, 2015 #4

    vanhees71

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  6. Jan 8, 2015 #5

    atyy

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    Anyway, collapse works FAPP. Can coarse-graining really replace the collapse postulate? If it can, is the coarse-graining local?
     
  7. Jan 8, 2015 #6

    Demystifier

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    Thanks atty!

    The Peres's explanation of blurring is quite blurred, but it seems to be similar to the much better explored idea of decoherence. For the latter, it is already well known (and much discussed at this forum) in what sense it can and cannot explain the appearance of "collapse".
     
  8. Jan 8, 2015 #7
    How exactly is Collapse used in Bell tests? I think this isn't true.

    Again I don't believe this is true. How exactly is collapse "verified" by anything?
     
  9. Jan 8, 2015 #8

    Demystifier

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  10. Jan 8, 2015 #9

    atyy

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    Also, I guess I should say that Peres doesn't explcitly say that blurring can replace collapse. That is something that has come up more in discussions with vanhees71.

    I think one way in which Peres's blurring is a bit different from decoherence is that he goes from a Wigner function (with negative portions) to a Liouville density (positive all over). In a sense, once we have a Liouville density, things have collapsed since it is a classical probability distribution with definite outcomes. So I am willing to consider that it might be different from decoherence. But yes, blurring seems not as sharply defined as collapse (which is a sharp rule once one has made the classical/quantum cut, and has measurements with time stamps), and I doubt that if it works it can be a local procedure in the Bell tests.
     
  11. Jan 8, 2015 #10

    Demystifier

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    If nothing else, collapse is at least a useful bookkeeping device. Even if this is not something which exists in nature, it is "verified" through the mental practice of many theoretical physicists.
     
  12. Jan 8, 2015 #11

    Demystifier

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    Even if you get a positive Wigner function, this function still gives probabilities for different outcomes and no single outcome is picked out by it. So similarly to decoherence, you still have the single-outcome problem. In other words, you still need to postulate either collapse or some other additional assumption.
     
  13. Jan 8, 2015 #12

    atyy

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    Just a bit to add to my reply in post #9. I think you are right. One thing which is not obvious that coarse graining can do is that collapse links a classical outcome with a quantum outcome. In Peres's case, I think all he gets is a classical outcome for the apparatus. For the quantum system, one probably obtains a decohered density matrix, which after collapse will be a proper mixed state. This should not reproduce the Bell test prediction, since the proper mixed state only represents a non-selective measurement, whereas the Bell tests use selective measurements.
     
  14. Jan 8, 2015 #13

    atyy

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    Yes, I think you are right.
     
  15. Jan 8, 2015 #14

    vanhees71

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    I guess the point atyy makes is valid if you were exposed to the Copenhagen doctrine long enough, in the following sense. For definiteness let's discuss the Aspect experiment in ultrasimplified form, i.e., not considering wave packets for the photon states, which in principle is, what one must do for a fully correct description, also with respect to our discussion here, but I have to formulate this carefully, before I can write it down here. So let's do the handwaving arguments with considering polarization states only and just add the space-time aspects of the measurement procedure "by hand".

    You start with a two-photon Fock state with entangled polarization states (usually prepared, using parametric downconversion and appropriate phase shifters for one of the photons), represented by the following ket:

    $$|\Psi \rangle=\frac{1}{\sqrt{2}} (|HV \rangle-|VH \rangle).$$

    The single photons are in mixed states (using the standard reduction formalism, by taking the trace over B's photon to get A's photon's state and vice versa) is for both
    $$\hat{\rho}_A= \hat{\rho}_B=\frac{1}{2} (|H \rangle \langle H|+|V \rangle \langle V|,$$
    i.e., the polarizations of both single photons is maximally uncertain (maximum von Neumann entropy).

    Now Alice and Bob have detect at far-distant places at the polarization of one of the photons respectively. Due to the geometrical setup you can by a precise enough measurement of the time of the photon detection make sure that A and B always look at two photons belonging to the entangled pair. Suppose now that Alice is much closer to the photon-pair source than Bob, so that she detects her photon way before Bob.

    I'd like to also discuss the most simple case, where Alice uses a polarization filter letting horizontally polarized photons through and Bob one letting vertically polarized photons through. I think there's no debate about the outcome of the measurement when A and B compare their measurement protocols (modulo detector inefficiencies which can be made arbitrarily small nowadays, so that we can neglect it for our idealized description): If A detects a then necessarily H-polarized photon, Bob also detects his then necessarily V-polarized photon, and if A doesn't detect her photon, also B doesn't detect his.

    Now let's discuss the experiment from the point of view of a Copenhagen-collapse interpreter (which I heavily disagree with) and from the point of view of a minimal interpreter (I heavily agree with)

    Copenhagen-collapse interpreter's point of view

    Suppose, A detects her photon (which happens with 50% probability). Since it's then for sure H-polarized according to the Copenhagen collapse mechanism, after this measurement the entire state collapses instantaneously to the state described by
    $$|\Psi' \rangle=|HV \rangle \; \Rightarrow \; \hat{\rho}_B'=|V \rangle \langle V|.$$
    I've already renormalized the ket to be of norm 1 again. So taken the subensemble, where A detects her photon, then B for sure also detects his photon, because it's vertically polarized.

    Criticism against this view

    This point of view violates relativistic causality and contradicts the very foundations of QED, which is (I think also undoubtedly) the correct model to describe this experiments, because if there were a collapse like this, the detection of A's photon must instantaneously change the state of B's photon from ##\hat{\rho}_B## to ##\hat{\rho}_B'##, i.e., from "maximally uncertain" to "determined".

    Now, by construction, this contradicts QED by construction: The interaction of A's photon with her polarization foil and photon detector is local from the point of view of QED, because QED is constructed as a local relativistic QFT, and there can be no FTL signal propagation (note that such signals are described by the retarded propagator not the Feynman propagator as in classical electrodynamics!).

    So, as atyy said, if you want to invoke the collapse argument, you must not misinterpret it as a real physical process, and the 100% correlation between the outcome of A's and B's polarization measurement cannot be satisfactorily explained by the collapse. It's an ad-hoc assumption to apparently simplify the prediction of the outcome of the measurement.

    Minimal interpreter's point of view

    There's no need for the collapse to explain the result of the experiment in terms of QED. You just evaluate the transition probabilities according to the corresponding S-matrix elements. In this case, you simply have to take "wave-packet states", leading to a detection probabilities as a function of the times and locations of A's and B's detection event ("click of the photon detectors"). The result is of course the same: 100% correlation between the single-photon polarizations, but nowhere did I invoke a collapse argument.

    As I said, I should work out this in mathematical form using standard QED (quantum optics to be more precise, because one has to use the standard effective theory to describe the optical instruments involved, i.e. in this case, polarizers).
     
  16. Jan 8, 2015 #15

    atyy

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    Could I see an explicit calculation or a reference?
     
  17. Jan 8, 2015 #16
    By the same line of argument, anyone can argue that "collapse" is a useful bookkeeping device in classical probability, and "verified" through the mental practice of probability.
     
  18. Jan 8, 2015 #17

    atyy

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    To add to my request in post #15 for an explicit calculation, I'd also like to ask what the calculation looks like in the Schroedinger picture. I assume we are using free fields (either Maxwell or Dirac), so the theory should rigourously exist. The real experiments are done using Maxwell fields, but it may be easier to use Dirac fields, which also show entanglement and violate Bell inequalities.
     
    Last edited: Jan 8, 2015
  19. Jan 8, 2015 #18

    zonde

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    As much as I like ensemble interpretation I have to note that your argument is flawed. First, Bell test is not about explaining 100% correlated measurements.
    And then when we consider correct Bell test it turns out that it is not possible to get violation of Bell inequality in ideal experiment without FTL signal. (you can easily verify that using Nick Herbert's simplified proof, see for example this thread for discussion about it thread)
     
  20. Jan 9, 2015 #19

    vanhees71

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    You get a Bell test by choosing certain relative angles of A's and B's polarizers. Nothing changes in the argument by just setting the polarizers at different relative angles than ##\pi/2## as I've chosen to simplify the discussion. There's no FTL signal necessary to explain the violation of Bell's (or related) inequalities, because there's nothing traveling faster than light. This is so by construction of QED.
     
  21. Jan 9, 2015 #20

    vanhees71

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    Ok, I'll see that I get this done over the weekend, but I'll not use the Schroedinger picture, because that's very inconvenient in relativistic QFT, but of course, there are only free fields as usual in quantum optics. Then you only need a "wave-packet description" for the photons. The polarizer is described as ideal in terms of a projection operator located at Alice's and Bob's place. Everything works of course in the two-photon Fock space.
     
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