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B Decoherence Clarification

  1. Aug 21, 2015 #1
    Despite the best efforts of some of PF's finest, I continue to struggle with the general concept of spontaneous quantum state reduction by means of environmentally induced decoherence.

    I think that much of my confusion lies in the confounding degree of ambiguity in the delineation between the "system" and the "environment". On a cosmological scale, this differentiation often seems to me to be somewhat arbitrary.

    Maybe part of my challenge is in simply not understanding the terms well enough. So, with regard to the three primary constituents of the decoherence process... System/Apparatus/Environment... could the physicists in the room please try to give me a conceptual definition using actual words (prohibiting any use of numbers, formulas or references to matrices). I recognize that this might feel like trying to teach me French without speaking French, but I'd greatly appreciate the effort. Consider it a charitable attempt at "No Fool Left Behind".

    A possible example of what I'm looking for might be something like (pending your correction of this concept)... "The apparatus is the thing by which a preferred basis of observation is isolated and/or determined".

    Any takers?
     
  2. jcsd
  3. Aug 21, 2015 #2

    jambaugh

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    Hmmm... I could use some more details as to your confusion but let me start with a general expo of my understanding. The system is of course that which you are representing with, presumably a density operator. As to the meaning of "the environment" that is any other system or systems which may interact with "The System". One would represent interaction by constructing a composite system representation (tensor product of Hilbert spaces and form density operator in the composite space). The interaction is represented by a joint Hamiltonian in which this composite system evolves. To revert to the original system description you would need to trace over the external system or environment component of the composite operator space. Decoherence manifest here in that entanglement occurs between system and environment and when we ignore the environment we see a less coherent system.

    Now it is hard to explain this without dragging you through the density operators and partial trace operations and such but let me try it this way.
    You meet a cute girl (or guy) who is a math geek and they left you their phone number in the form of two numbers on a piece of paper which add up to your number.
    Together the numbers represent "coherent" information but they then tear the paper in two giving you only one of the two numbers. You now have what amounts to a random number. This is not a perfect analogy because quantum mechanical correlation (entanglement) can be stronger than classical in a way we can't model with classical pieces of paper and classical information written on them. But the decoherence stage is analogous to this and not very much different from thermal randomization as described in the classical domain. The system entropy goes up. What the quantum mechanical description allows is that the entropy of the parts is greater than the whole because you can have the maximal information in the whole encoded in a way that is not compatible with subdividing the system into those particular parts (that's incompatible in the sense of momentum vs position type complementarity).

    So just as, say, a particle can evolve from a sharp state (or rather mode) of definite position to a sharp state where position is not well defined, a composite system can evolve from a sharp state describable as a composite of sharp states of its parts (system + environment e.g.) to another ***sharp*** state (of the whole) where in the best description of either part is not sharp. It has experienced decoherence.

    Now given you knew the original system state and exactly how it evolved you could in principle set things up to reverse this however it is usually the case that we do not know the sharp initial state of the secondary system since it is "everything else" and once interaction has occurred it the original system is entangled with an ever expanding sphere of the electromagnetic field in space. You physically cannot catch up with this in order to work with the whole composite system and reverse the decoherence.


    I'm fond of saying when I take this over the top "The entropy of the universe is 0! It's only when we look at separate parts that we get positive entropy!!!"

    I hope this clarifies more than it confuses.
     
  4. Aug 21, 2015 #3
    That was a very good effort! Thank you.

    But before I ask a follow up question, I'd like to make sure I understand this part. When you say "The system entropy goes up", are you referring to the COMPOSITE system or the original system?
     
  5. Aug 21, 2015 #4

    jambaugh

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    The original system, in the example I gave the composite retains zero entropy (sharpness is retained in the unitary dynamic evolution).
     
  6. Aug 21, 2015 #5
    .
     
  7. Aug 21, 2015 #6
    I think this is getting at my confusion regarding the mixed/pure state designations.

    Before interaction with the environment, we can say that the system was in a "pure" state with zero (informational) entropy. But after interaction (is considered), the original system becomes "mixed", with increased entropy. Is that right?

    But, isn't this really just a result of our ignorance of the quantum state, because if the quantum state is actually reduced as a result of the interaction, then the resultant state is still "pure" and has zero entropy.

    What am I missing here?
     
    Last edited: Aug 21, 2015
  8. Aug 27, 2015 #7

    jambaugh

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    Yes you have that right w.r.t. the first question. In your either/or question both are correct depending on answering the question "the state of what?" of the system vs of the system and environment together as a larger composite system.

    Both are occurring, the system "state" is reduced (but its really not a higher entropy state or reality but a higher entropy class of states) and the resulting state of the larger system is still "pure".

    Keep in mind that our wave functions and density operators (i. in the orthodox interp. and ii. especially when considering "mixed" "states") represent classes of possible systems and not specific systems. It is an important philosophical distinction.
     
  9. Dec 9, 2015 #8
    I've been rolling this around in my head, and I keep hitting the same cognitive stumbling block.

    I think that the "pure" vs "mixed" designation continues to lie at the heart of my confusion. Again... IF environmental decoherence triggers state reduction (I'm purposefully avoiding the "collapse" term to avoid unnecessary debate), then it seems to me that the reduced state should be considered pure and have zero (informational) entropy at the moment of reduction. I'm not sure what "reality" refers to in your statement here. However, IF decoherence can be considered the triggering mechanism by which state reduction occurs, then I would think it would be accurate to say that one of the potential quantum states of the reduction would be "realized".

    Yet, this portion of your post seems to imply that the wave function is not "really" reduced by decoherence, but that decoherence simply limits which system states CAN occur upon reduction.

    Can anyone please clarify this distinction further?
     
    Last edited: Dec 9, 2015
  10. Dec 9, 2015 #9
    Maybe what you're missing is this? Suppose a system is comprised of two parts as the tensor product of the corresponding Hilbert spaces, and is in a pure state with the corresponding density matrix; in general, the density matrix of one of the two single parts (the trace over the corresponding Hilbert space of the single part) represents a mixed state, except for a specific non-entangled case (which doesn't apply here).

    If you want I can prove this to you, but there will be math. Or you can take it as a dogma if you don't want the math. Honestly I don't think it's possible to explain this detail without the math, but if you just believe it I don't see the problem (or I don't see by which intuition it would have to be otherwise, it's not like the converse would be intuitive either).

    So once you know this, apply it to a system + environment case: the environment interacts with the system and becomes entangled with it, the total is a pure state but the system is now a mixed state.
     
    Last edited: Dec 9, 2015
  11. Dec 9, 2015 #10
    The system is in a "mixed" state because there are multiple "potential" states of the system?
     
  12. Dec 9, 2015 #11
    No. "Mixed state" and "pure state" are technical terms that refer to properties of the density matrix associated to a quantum state. A pure state's density matrix is just a projector. Both the pure and mixed states represent a probability amplitude for an observable outcome, if that's what you mean. In fact, if you write both the pure and the mixed states with a state vector and a density matrix respectively, using a particular eigenbasis, the probability outcomes of a measurement of an observable with that eigenbasis can be exactly the same. Only with a non-commuting observable do you see the difference, which lies in an interference term given by the relative phase between the pure state vector coefficients (which are complex numbers); whereas a mixed state density matrix is written using real numbers, which, in the above case of there being the same probability distribution, correspond to the former complex numbers squared. That's why the density matrix has less information than a state vector.

    So you should not confuse the quantum-informational entropy with the quantity of measurement outcomes: after all, if you had only one outcome for an observable, you could get more with another observable not commuting with the first (which is typical with polarization, for example). Quantum-informational zero entropy refers to the uniqueness of the state vector, represented by complex coefficients in some basis; whereas a mixed state's density matrix, which can be seen as a mix of pure states, has real coefficients (the square of the pure states' complex coefficients, losing the phase information) for all observables, which means it represents an incoherent mix.

    Edit: if by "potential states" you mean the pure states the mix is comprised of, then yes.
     
    Last edited: Dec 9, 2015
  13. Dec 9, 2015 #12
    I think these passages from 'Quantum Enigma' by Bruce Rosenblum and Fred Kuttner may assist (pg 209, 2nd edition):

    (without referencing the experimental set-up discussed)
    and

    EDIT: realised the emphasis on some words, as per the book, were not present.
     
    Last edited: Dec 9, 2015
  14. Dec 9, 2015 #13
    I'm afraid that it's precisely this ambiguity that confuses me. I thought that is what I meant, but now I'm not sure.
    Let me try this a different way, and maybe it will help you help me (and I do very much appreciate the effort).

    My confusion initially began during a previous PF thread (can't put my finger on it immediately) that was discussing the ontological "reality" of the wave function. I was trying to understand how this might relate to the process of environmental decoherence on a cosmological scale.

    In that thread, there seemed to be a school of thought among some of the participants that, on a cosmological scale, the delineation between the "system" and the "environment" might be somewhat arbitrary. The question of exactly when, and why, actual state reduction occurred in an ontologically "real" wave function (secondary to decoherence) lead to the discussion of mixed vs pure quantum states of the environment/system complex. It seemed that some were suggesting that the "mixed" state (mixture of potential pure states) after interaction was a reflection of ignorance of outcome, and that the "pure" (realized?) state could not yet be determined. Others appeared to argue that the differentiation was mathematically irrelevant, and any suggestion to the contrary was utterly philosophical.

    Now, I am fully aware that I probably misunderstood the discussion, and "proper" vs "improper" mixes were additional sources of confusion for me, so it's entirely possible that I've also confused the "pure/mixed" and the "proper/improper" terms. Also, I'm confident that the answer to my question is highly interpretation dependent. However, IF we are considering a "universal" wave function on a cosmological scale... and IF we are trying to consider this wave function as being ontologically "real" (whatever that means)... I still have trouble understanding how decoherence "triggers", by direct causation, quantum state collapse (or state reduction, if you prefer). It seems, to my befuddled brain anyway, that the mathematical formalism of decoherence simply places logical limitations on what quantum states can be observed... defining the possible "subsystems" of the universal wave function describing the cosmological system/environment complex.

    Is this assessment even remotely close to reasonable? If not, can you identify where (in the likely long chain of errors) my conception is in error?
     
    Last edited: Dec 9, 2015
  15. Dec 9, 2015 #14

    Both of these postings seem to suggest a similar concept, though without the necessity of the wave function being ontologically "real".
     
  16. Dec 10, 2015 #15
    Okay. Indeed, the mixed state is the analogue of the classical probability density in phase space. Let me break this down more clearly.

    Classical mechanics: an ideal state is a point in phase space, the corresponding density is a Dirac delta in phase space. A statistical density (of microstates) instead assign a probability distribution to possible ideal states the system is in: it's a "smeared cloud" in phase space if you want to visualize it. A classical physical state is always non-ideal due to measurement errors.

    Quantum mechanics: a pure state is a ray (or orbit, if you take Gauge transformations into account) in Hilbert space, which for simplicity we call a state vector. The density is now a "matrix" (actually can also have a term with an integral over continuous observables) which is just a projector over that pure state vector. A statistical mixture - a mixed state - assigns probabilities (hence real numbers) for a quantum system to be found in a number of pure states. A quantum pure state CAN be realized physically.

    As you can see there are fundamental differences: in quantum mechanics, a pure state is already of statistical nature it representing a probability amplitude. So you can consider a mixed state to be just another state: the pure state was a superposition of observable eigenstates, the mixed state is a superposition of pure states. In the formalism, there's much less "ontological" distinction between pure and mixed states than there is between ideal and ensemble states in classical mechanics. However, you can devise an interpretation in which such an ontological distinction is recovered (I guess, for a psi-ontologists that's evident).

    Now, if your question is, how does a psi-ontologist justify the fact that a composite pure state has its parts taken separately that are mixed states, I guess the trivial answer is "you're not addressing the whole wave-function but only a part, hence whatever you do to a part behaves as a statistical mix to you but the whole is still a pure state".

    Be warned that this is NOT the mathematical formalism of decoherence per se. It's the mathematical formalism of all of quantum mechanics. It's just a mathematical lemma: given a vector in a tensor product of two Hilbert spaces, in general the density operator on just one of the two spaces is not a projector (and is never a projector if the total state is entangled). This means that in all interpretations, in all of quantum mechanics, an entangled pure state is always a mixed state for just one part of it. So the "direct causation" you're talking about is simply the entanglement that happens between the environment and the system (which is inevitable): the "logical limitations" you talk about are inevitable maths, decoherence theory didn't postulate them or anything.

    The point of decoherence is that keeping track of pure states becomes impossible since they're scattered in bits and pieces all over the place, while you want to look at just one system. Suppose your system is hit by a number of photons: each of them becomes entangled with it and then scatters away. You'd have to run after each photon to recover and study the pure state, looking at the system you'll see a mixed state.
     
  17. Dec 11, 2015 #16

    Demystifier

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    I think you are right. The split into the "system" and the "environment" is quite arbitrary. In practical calculations this is usually not the problem because physicists have a good intuition about what is a "natural" split in given circumstances. So if you think of QM only as a practical mental tool for human physicists (including cosmologists), then there is no any serious problem. But if you think of QM as a fundamental law obeyed by nature itself, irrespective of humans, then there is a deep problem.

    How to resolve the problem? Well, to resolve it, the minimal quantum formalism is not enough. You must use some interpretation of QM, and any choice of interpretation is somewhat controversial.

    Personally, I like the Bohmian interpretation. Among other things, this interpretation gives a preferred status to the position observable, which circumvents the problem of "arbitrary split into system and environment".

    For a related discussion see also
    https://www.physicsforums.com/threads/many-worlds-proved-inconsistent.767809/
     
  18. Dec 11, 2015 #17
    Can you provide an example in which only an intuitive but not arbitrary choice provides a correct prediction, while another choice is wrong? Of course if calculations turned out correct in all arbitrary choices there wouldn't be a problem.
     
  19. Dec 11, 2015 #18

    Demystifier

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    There are some examples in Sec. 4 of
    http://arxiv.org/abs/1210.8447
     
  20. Dec 11, 2015 #19
    Thank you. This is a fascinating paper. I think this will address my confusion directly. It's very similar to a position I've read by Lee Smolin from the cosmological perspective. I'm hoping that I can just be a spectator now and watch while you professionals discuss this.
     
  21. Dec 11, 2015 #20

    zonde

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    Do I understand it correctly? The author of this paper gives an argument why universal wave function equipped with unitary evolution can not predict classical world (via decoherence induced branching).
     
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