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Preferred basis in Relational Quantum Mechanics

  1. Jul 17, 2013 #1
    In RQM all systems are observers. Select the viewpoint with a system S and an observer O. The systema has 2 eigenfunctions |0> and |1> in a basis. Then the evolution from [itex]|init>_{O}[/itex][itex](|0>+|1>)_{S}[/itex]. Then the system evolutions to [itex]|O0>|0>_{S}[/itex] or to [itex]|O1>|1>_{S}[/itex] . But how does the measurement select the prefered basis? I think there is the same problem that in many worlds
     
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  3. Jul 17, 2013 #2
    In quantum mechanics, a measurement necessarily picks out a certain basis. A given measurement corresponds to a certain hermitian operator, and the observed outcomes of that measurement correspond to eigenstates of the corresponding operator. So the natural basis for that measurement/operator is the eigenbasis.

    I think what you are concerned with is that in certain problems we would like to classify statistical quantum states (ones described by a density matrix) as either a "pure state" or a "mixed state." This is a quantum statistical mechanics topic. However, whether something is a pure state or a mixed state depends on the basis, so you might object to the classification as arbitrary. But in those cases, one might argue that the physical scenario picks out the preferred basis; e.g. in a system with energy conservation, it's typical that the "preferred basis" is the energy eigenbasis. So this is a different sort of problem.

    In your analogy where an "observer's state" starts in some initially neutral state and evolves into states corresponding to the system's state, you are implicitly saying that the observer is choosing to measure along the basis that the system's state is defined with respect to. If the observer was making a different measurement, we would have change the basis with respect to which we write down the system's state, if we wanted to be consistent with the sort of notation Von Neumann was using when he created this model of the measurement process. But we can easily beef up Von Neumann's notation to be more general. Here's a little example:

    Let's say we have a spin 1/2 system, and we're only concerned with its spin state. Its state lives in a two-dimensional Hilbert space spanned by |up>n and |down>n, where n denotes an arbitrary axis.

    Now let's once and for all define our x, y, and z axes--no more redefining our axes from now on. Let's call our system's state |ψ>, and assume the system is initially in the state |up>x, an eigenstate of measurement along the x-axis. Here is a fact you can verify from any QM textbook:

    |ψ> = |up>x = 1/√2 (|up>z+|down>z)

    So if your observer is measuring along the z-axis, her basis states are |neutral>, |"I see up along z!">, |"I see down along z!">. Now a measurement of |ψ> would look like this more complicated process

    |neutral>|ψ> = |neutral>|up>x = |neutral>1/√2 (|up>z+|down>z) --> 1/√2 (|"I see up along z!">|up>z+ |"I see down along z!">|down>z)

    So the measurement "picks" the z-basis representation of |ψ>
     
    Last edited: Jul 17, 2013
  4. Jul 17, 2013 #3
    But I am speaking about Relational Quantum Mechanics Interpretation from Rovelli, nor about Von Neumann interpretation. In RQM any sistem is a measurement device. If I know what is the thing is measuring, then of course I know what is the preferred basis. This is a tautology
     
  5. Jul 17, 2013 #4
    ----
    Edit: Sorry that I kept editing my first post after your post came in. I added a pretty drawn-out example of how the process would look in more general notation to show that the basis can be arbitrary up until the measurement is performed. You may need to look back at what I added to my first post to understand what I'm saying.
    ----

    I'm not talking about any sort of interpretation. If you buy von Neumann's analysis of the measurement process (which aside from showing vividly where there "should" be collapse, is pretty much just a non-objectionable gedanken experiment), then I'm just explaining the notation for you. If you claim that all systems make measurements, then you're dabbling in vague interpretational ideas, and you'd have to be the one to tell me what measurements are being made and when.

    What I said above--the measurement process model of Von Neumann--has nothing to do with the "Von Neumann"/"Consciousness causes collapse" interpretation of QM. [In fact there is no collapse in the measurement process models we've written above.] The measurement process model of Von Neumann is applicable to all interpretations, except that different interpretations might explain why the observer does not see herself in a superposition in different ways. But the basic gedanken experiment can be performed in any interpretation--for example Sidney Coleman uses it to analyze both the Copenhagen and Many Worlds interpretations in his lecture "Quantum Mechanics In Your Face" which you can find on the Harvard website video archive.

    Measurements picking out a basis is tautological. There is no problem. There is only a problem when you deal with states described by density matrices. However, you are free to introduce speculative ideas such as "all systems perform measurements" into your interpretation, and these strange ideas may lead you to philosophical/nonscientific problems, but there is no real answer to those nonscientific problems since there is no evidence for or against any interpretation.
     
    Last edited: Jul 17, 2013
  6. Jul 18, 2013 #5
    But it is an aseveration from Relational Quantum Mechanics interpretation. I am not suscripting this interpretation, I simply wanna know what is the solution to this question in that interpretation, no a discalification of that interpretation.
     
  7. Jul 18, 2013 #6
    Starsruler, I would appreciate if you would spell-check your posts to avoid things like "suscripting" and "discalification".

    Anyway, as far as I know, the "solutions" to problems in Relational Quantum Mechanics never rely on some mystery measurement machine making an undetermined measurement. Typically the "solution" to a problem like
    |neutral>1/√2 (|up>z+|down>z) --> 1/√2 (|"Bob sees up along z!">|up>z+ |"Bob sees down along z!">|down>z)
    would be that Bob perceives himself in one of the two eigenstates [akin to collapse] but outside observers would agree with the superposition. RQM says both descriptions are valid since descriptions are in general observer dependent.

    This "solution" would not involve some lurking measurement device performing some byzantine series of measurements. RQM wouldn't dictate whether Bob measures along z or not--that is his choice. He sets the z eigenbasis. So there is no need to set up a rule for how non-autonomous observers pick which measurements they're performing/preferred basis. [In this last sentence, by "autonomous" I do NOT mean having free will or consciousness, I merely mean that there is a definite setting for what measurement is being made. It could be that Bob has made his own mind up that he measures z, it could be that Bob's Ph.D. advisor told him to, or it could be that Bob is actually just a piece of electronics with its measurement setting knob turned to z. This is a crucial point for RQM.]

    I am no expert on RQM, however, so please let me know if my characterization seems wrong to anyone.
     
    Last edited: Jul 18, 2013
  8. Jul 18, 2013 #7
    Sorry for my misspellings , subscrip and disqualification are the correct spellings, of course.

    Anyway, then is RQM "agnostic" about basis selection and collapse time? maybe does it resort just to decoherence for this questions, like nother moderns interpretations?
     
  9. Jul 18, 2013 #8
    Well in the measurement process problem we were talking about above, the basis gets picked, in any interpretation, by what Bob chooses to measure. So there is no ambiguity there--why would you call it "agnostic?"

    Collapse time and basis selection with regards to decoherence go back to the problem I mentioned in my earlier posts--those topics have to do with quantum statistical mechanics, i.e. states described by density matrices. So this is quite different from the problem we've been speaking of. Problems of this sort appear regardless of your interpretation and have nothing to do with how Bob's basis gets picked in RQM. Collapse time and pure-state/mixed-state basis selection are decoherence topics, and decoherence is an experimentally observed phenomenon, so obviously it is not a matter of which interpretation one ascribes to. [However, decoherence is a topic that is quite relevant for interpretational ideas.] So there is a kind of statistical basis selection--a decoherence topic, and then there is the measurement basis selection--an elementary QM topic, but these are two separate concepts.

    Make sure to keep these concepts separate because people often use decoherence to analyze the measurement process by modeling the observer as a quantum statistical ensemble. So you'll run into problems applying one idea to the other if you can't keep the two ideas distinct.
     
    Last edited: Jul 18, 2013
  10. Jul 20, 2013 #9
    Anyway, I would like to return to the original question of the thread. How is solved the preferred basis problem and how we can know the collapse time in RQM??
     
  11. Jul 20, 2013 #10

    tom.stoer

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    This is not correct.

    A generalized state = a density operator ρ can be expressed in terms of a basis |n> as

    ##\rho = \sum_{mn}\rho_{mn}\,|m\rangle\langle n|##

    with the additional requirements that ρ is positive semidefinite, hermitean, and

    ##\text{tr}\,\rho = 1##

    But whether this generalized state is a pure (mixed) state depends only on the fact whether ρ is a projection operator (or not). So a generalized state is a pure state if and only if

    ##\rho^2 = \rho##

    and therefore

    ##\rho = |\psi\rangle\langle\psi|##

    for some state psi.

    But this property does not depend on the basis!
     
    Last edited: Jul 20, 2013
  12. Jul 21, 2013 #11
    You can solve another dudes in another threats. I would like to return to the original question of the thread. How is solved the preferred basis problem and how we can know the collapse time in RQM??
     
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