Quantum Mechanics II: Decoherence & States

Strictly speaking, most of the other rules about QM are already implicit in what I've already said.  But a few implications of this setup are worth pointing out.

First note that, in QM, the "state" includes information about every single object in the system.  So, when you add up the different histories, they only interfere if the final states are exactly the same in every respect.  If even one tiny particle is in a different place than it otherwise would be, then they don't interfere.  In that case, you just add up the probabilities normally.

This is why measurement is such a significant thing in QM.  If you try to catch out Nature by explicitly measuring which slit the particle went through, then YOU are now different as a result of you knowing which slit it went through.  As a result, the two histories don't interfere.  But it needn't be a person which does the "measurement".  Even if you refuse to look at it, the detector being different still prevents the interference from happening.  As far as we know experimentally, there is no special relationship between consciousness and QM (although some people have proposed interpretations of QM in which there is a connection between the two.).

Usually, once histories become sufficiently different from each other, for a long enough period of time, their random interactions with the environment will tend to be different, so that the chances of getting everything perfectly the same become tiny, and the histories won't interfere anymore.  This phenomenon is called decoherence.  People argue about what this tells us about the interpretation of QM, but the phenomenon itself can be studied in the laboratory, so my use of this word should not be regarded as an endorsement of any particular interpretation.

Secondly, if you have two or more distinct states, then it's possible to take a quantum superposition of the two states, formed by adding them up with complex coefficients.  For example, if X and Y are two distinct states, then

(\mathbf{X} + \mathbf{Y}) / \sqrt{2}

or

(\mathbf{X} - \mathbf{Y}) / \sqrt{2}

or

(2\mathbf{X} +i \mathbf{Y}) / \sqrt{5}

are all equally valid states!  (The reason for the square root in the denominator, is to make it so that, by the Born Rule, the total probability of the state is still 1.)  These states are just as much valid states as X or Y themselves would be.

The possibility of quantum superpositions is implicit in the quantum probability rules, since if you start with a particular state A, in general it will evolve to a superposition of different states as time passes.  And there's no particularly good reason you couldn't also have started out the experiment with a quantum superposition.

(Note that if we take any state like (\mathbf{X} + \mathbf{Y}) / \sqrt{2}, and we multiply it by a phase (a number on the unit circle of complex numbers, e.g. i, or -1, or (1+i)/\sqrt{2}) then we can't tell the difference between that and the original state in any way!  That's because, when we work out the patterns of interference, we only care about the relative phases between different histories, not the absolute phase of the whole system.  So it's good to remember that there is a slight redundancy in our description here: two states that differ by a phase are really the same state.)

Now if we have a system with N possible states, then we can imagine a higher dimensional geometry consisting of all possible superpositions of these N possible states (including, for mathematical convenience, those for which the probability doesn't add to 1).  This is called the Hilbert Space of that system.  It is a kind of vector space with N complex dimensions, which means in terms of real numbers it's a 2N-dimensional space.  But don't worry about these details for the moment.

(It's kind of hard to visualize a Hilbert space when N is greater than about 2, but it's still very useful mathematically!)

The simplest nontrivial Hilbert Space is the one with N = 2 states.  (I'll give a physical example in a moment.)  This would normally involve a 4-dimensional space, but to keep things as simple as possible, I give you permission to ignore the bit about complex numbers and just think about a 2-dimensional plane.  (This is the space of all states of the form

a\mathbf{X} + b\mathbf{Y}

where a and b are now real numbers.)  Then we can think of X as a unit vector pointing along the x-axis, and Y as a unit vector pointing along the (wait for it...) y-axis.

Perhaps a picture will help:

The Hilbert space for a system with 2 states.

As you can see, the Hilbert space has an origin, which is the point in the middle which represents "zero".  Each state is a represented by a vector coming out of the origin, pointing in some direction.  (But remember that -X is really the same state as +X, since they differ by a -1 phase.  I didn't draw -X on the picture, but if I had it would be 180º around from X.)  The Born Rule tells us that length = total probability squared.  That means that in order for a vector to be a state-in-good-standing, it needs to be length 1.  (In other words, by the Pythagorean Theorem, the sum of the squares of its (x,y) coordinates needs to add up to 1).  So don't ask me what the physical meaning of the "zero" vector is, since it doesn't have one.

A physical example of an N = 2 state system would be the polarization of a photon coming straight at you from your computer screen.  Light can be either horizontally polarized (the X state, corresponding to an electric field that points in the x direction) or it can be vertically polarized (the Y state, corresponding to an electric field that points in the y direction).  Now since physics is rotationally symmetric, it's obvious that if light can be horizontal or vertical, it can also be diagonal.  So you might have naïvely thought the photon would have infinitely many possible states.  And in a sense this is true, but each of these diagonal states is really just a quantum superposition of the X and Y states.

Yet on a plane, the choice of axes is arbitrary.  You can rotate the coordinate system by 45º, and it would be just as good as the original coordinate axis.  In the same way, we are currently thinking of X and Y as the two possible states of the system (with every other state being a superposition of X and Y)—but this is an arbitrary choice!  We could just as well say that every state is a superposition of (\mathbf{X} + \mathbf{Y}) / \sqrt{2} and (\mathbf{Y} - \mathbf{X}) / \sqrt{2}!  So actually every state is a quantum superposition, of certain other states.

Although the choice of coordinate axis is arbitrary, it is important that the states you pick are all "orthogonal" to each other (i.e. at right angles in the Hilbert space).  That is what tells you that it represents a set of  mutually exclusive possibilities.  Any such set of N orthogonal states is called a basis of the Hilbert space.  (The plural of "basis" is "bases", pronounced BASE-EES.  Just like the plural of "index" is "indices".)  A basis gives the possible set of outcomes for some particular way to measure the system.

For example, suppose we start with a diagonal photon in the (\mathbf{X} + \mathbf{Y}) / \sqrt{2} state, and we measure it to see whether it is horizontally or vertically polarized.  (Maybe by passing it through some kind of material in which these two polarizations follow different trajectories.)  What happens?

Well, people disagree about interpretation (what is ultimately going on), but everyone agrees on the practical set of rules you'd use in the laboratory.  We just look at the state (\mathbf{X} + \mathbf{Y}) / \sqrt{2}.  It has an amplitude of 1/\sqrt{2} to be X, and also 1/\sqrt{2} to be Y.  By the Born Rule, we've got to square these numbers, so we get a 1/2 chance for it to be horizontal, and a 1/2 chance for it to be vertical.

Let's suppose it turns out to be vertical (the Y state).  Then from now on, the particle behaves just as if it had been in the Y state all along.  (This is called "projection" or sometimes "collapse of the wavefunction"; but see my remarks on decoherence earlier in this post.)  For example, if we measure it a second time to see if it is in the Y state.  If we check to see whether it is in the X state, it is definitely not.

But now we can ask a separate question: is it in the (\mathbf{X} + \mathbf{Y}) / \sqrt{2} state, or the (\mathbf{Y} - \mathbf{X}) / \sqrt{2} state?  This corresponds to sending it through a different kind of filter, which discriminates between the two 45° diagonal polarization choices.  We would then find a 1/2 chance of it being the former, and a 1/2 chance of it being the latter.

Supposing it turns out to be (\mathbf{Y} - \mathbf{X}) / \sqrt{2}, this is a bit paradoxical.  Since if we had just started off asking whether the (\mathbf{X} + \mathbf{Y}) / \sqrt{2} photon was in the (\mathbf{Y} - \mathbf{X}) / \sqrt{2} state, Nature's answer would have been "Nope.  Definitely not.  Those states are orthogonal and therefore if it's the one, it's not the other!"

But somehow, merely by answering a series of questions about the photon's polarization, we managed to trick Nature into converting the photon from its original polarization to one 90° away, which is inconsistent with the first.  By measuring the photon we have affected it!

So we see that, somehow, we can get the photon to be definitely - or | polarized, or definitely / or \ polarized.  But we can't get both of these things to be definite simultaneously.  This is an uncertainty relationship.  It's analogous to the "Heisenberg uncertainty principle" where you can't measure position and momentum at the same time; so that measuring one makes the other uncertain.  (Although it's not exactly the same, since position and momentum are continuous variables, while each polarization choice is a yes-no question.)

In the case we are considering, we're been lucky that the Hilbert space is directly related to two dimensions of the physical space.  That means that the rotation of axes in the Hilbert space is the same thing as a rotation of physical space.  In general, however, we are not so lucky and the Hilbert space is more abstract.  But it is still true that there are a bunch of different possible bases of the Hilbert space, that are related by rotations in the Hilbert space.  (Since the Hilbert space is complex, we are really only interested in those rotations that don't mess with the notion of "multiplying-by-i".  These are called unitary transformations.)

As long as I'm talking about complex numbers, I should mention that there's also such a thing as circularly polarized photons, which involve complex superpositions like (\mathbf{X} \pm i \mathbf{Y}) / \sqrt{2}.  But most of the bizarreness of superpositions can be illustrated without thinking about complex numbers.

About Aron Wall

I am a postdoctoral researcher studying quantum gravity and black hole thermodynamics at the Institute for Advanced Study in Princeton. Before that, I read Great Books at St. John's College (Santa Fe), got my physics Ph.D. from U Maryland, and did my first postdoc at UC Santa Barbara.
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28 Responses to Quantum Mechanics II: Decoherence & States

  1. Bob Kurland says:

    Nice examples, Aron. With respect to your first argument about measurement
    "But it needn't be a person which does the "measurement". Even if you refuse to look at it, the detector being different still prevents the interference from happening. As far as we know experimentally, there is no special relationship between consciousness and QM (although some people have proposed interpretations of QM in which there is a connection between the two.)."
    are you going to discuss the "Wigner's Friend" gedanken experiment?

  2. Aron Wall says:

    Bob,
    I didn't have any particular plans to do so, but feel free to describe it yourself in the comments here! It might clarify some things for some people.

  3. Bob Kurland says:

    Thanks for the invitation, Aron. I'll not describe the "Wigner's Friend" paradox in detail, other than to say it deals with the paradox involved in describing the states of an observer (human) of an observer (human) of an experimental result. A summary is given by Stapp: www-physics.lbl.gov/~stapp/WignersFriendStapp.doc.
    Although I do defer to your superior knowledge of physics, I'm not sure I entirely agree with you that decoherence gives the answer to the measurement problem or how consciousness enters in to it. There's a good account of this by Bernard d'Espagnat (he of the Aspect experiments and "On Physics and Philosophy").
    Here's the web site:
    http://arxiv.org/pdf/quant-ph/0402121v2.pdf
    I'd very much appreciate critical comments on either of these.
    Thanks.

  4. kashyap vasavada says:

    I would like to see your answer to the following question. I have seen different answers from different people. Quantum mechanical processes went on during the early period of universe when there were no human observers or machines to observe. Did collapse take place at that time? One answer is the delayed choice. But frankly I am not happy with that. Of course some people may say that consciousness was there all the time! But that is a slippery slope!

  5. Aron Wall says:

    kashyap,
    Your question seems to presume that consciousness causes wavefunction collapse (you also mention the possibility of machines, although I don't see why a machine designed by human beings would be any different from a naturally occurring physical object prior to human existence).

    But I don't believe that consciousness causes the collapse of the wavefunction. (I also don't disbelieve it, but I don't see why I should believe it either. Both consciousness and QM are deep mysteries which might or might not be connected.)

    You also seem to be assuming that the collapse should be thought of as some kind of event occuring at some particular moment in history. But maybe it's not like this. Maybe we should think about it in a more timeless way. Maybe the wavefunction doesn't "collapse" until the end of time, retroactively picking out which of the possible histories of the universe really happened!

  6. kashyap vasavada says:

    Hi Aron,
    Thanks for the reply. Yes. This is all a big puzzle! Do I understand from your last paragraph that you may be inclined towards delayed choice?

  7. Aron Wall says:

    kashyap,

    There are certain QM experiments called "delayed choice experiments", which some people take to have certain interpretational consequences. But I'm not familiar with the use of the term "delayed choice" to refer to a specific interpretation of QM. Do you have a particular source in mind? Thanks!

  8. kashyap vasavada says:

    Aron,
    If my understanding is correct, some people interpret delayed choice experiments as suggesting that until you observe a quantum system, it remains in a kind of suspended status (superposed) and when you observe the system, the past (history if you like) materializes.Materializes may not be the correct word for whatever is happening! Personally I do not understand or like it. But experiments have been clear cut! I am not sure which particular interpretation of QM is consistent with them. People who believe in consciousness collapsing wave function say that universal consciousness (whatever it means) was there all the time, long before humans came up. So for them there is no problem!!

  9. Aron Wall says:

    kashyap,
    I think I was trying to suggest that consciousness per se does not have anything to do with the "collapse", but that since two histories only interfere if they end up in the same configuration, at the end of the universe (if the universe does end) it would be completely clear which histories interfere and which do not. And so at that point there would be some clearly disjoint possibilities, of which it could be consistently said that only one of them happens. "Collapse" may be the wrong way of thinking about it.

    Since I'm a theist, I guess I do believe in a certain sort of "universal consciousness". (Although "God" seems like a far more precise term.) But there's a serious difficulty in saying that the UC's perception automatically cause wavefunction collapse---then you have to explain why collapse doesn't happen in the middle of a two slit experiment, since the UC presumably perceives everything (otherwise it isn't so "universal" after all)! So I don't see that this improves the situation.

  10. kashyap vasavada says:

    Hi Aron,
    Thanks for discussing your views on QM. We can use more discussions like these. I do not want to push this idea of consciousness collapsing the wave function too much, because I do not understand it completely anyway. However let me mention one thing before quitting the discussion. The idea of universal consciousness affecting quantum phenomena may not be that ridiculous. As I mentioned in my previous discussions, according to the Hindu Vedic concept, Brahman (UC) is in every particle. It was there for all the time and will be there for ever! That might explain collapse. Obviously this is not rigorous and has certain amount of hand waving!! So all of this remains a big puzzle!

  11. Mactoul says:

    That God or UC or Brahman may explain some facet of QM is classic God of the Gaps. It is not physics and it does not actually explain anything. And theologically, it is bad--essentially a category confusion between the Primary Cause and the Secondary Causes.

  12. Aron Wall says:

    Mactoul,

    I don't think all God of the Gaps arguments are necessarily fallacious. (Although everything is caused to exist by God, there could be some events (like miracles, or perhaps a beginning of time) which are caused only by primary causation, without any "secondary causation".) However, I agree that this one doesn't work very well as an explanation. A good theory of "collapse"---assuming such a thing does occur---needs to explain why it happens in some cases but not others.

  13. Mactoul says:

    Aron.
    Miracles are not studied in physics and physics is not competent to pronounce upon them. To speculate upon the dynamics of the Virgin Birth or Parting of Red Sea is just to impute doubt on the the miraculousness on these events.

    But wavefunction collapse is not an event outside the domain of physics. More than that, it is not an "event" in the sense that it exists ONLY in the formalism of physics. It is something postulated in physics.
    Thus, any explanation MUST lie in the domain of physics.

  14. Aron Wall says:

    Mactoul,
    Well, physics can study miracles in a negative sense, by determining that there is no plausible mechanism by which it could have happened nonmiraculously.

    In principle, one could also use physics to look for the downstream causal consequences of the event after the miracle has ceased. But I don't know of any good examples of this. For the most common sort of miracle (healing) you'd want a physician, not a physicist, to look at the patient afterwards.

    There are indeed some very silly people who try to find "physics explanations" for the miracles in the Bible, as though this were some sort of rapproachment between science and religion. This is obviously dumb because (as you say) finding an ordinary explanation would imply it wasn't miraculous in the first place.

    Although, let's be careful here. Suppose for the sake of argument that there were a physical mechanism for parting the Red Sea, which is in principle possible but extremely rare. Then the fact that it happened right when Moses raised his staff and prayed, would still make it pretty obviously a divine intervention!

    Just because something was a miracle doesn't strictly imply there was no "secondary causation" involved. God could still accomplish the effect by means of other causal agents, as long as he does so in a way which does not follow the usual pattern of secondary causation found in Nature. For example, the book of Exodus states that "the Lord drove the sea back with a strong east wind" (14:21). The fact that the wind was involved does not make the event nonmiraculous.

    I agree with you that wavefunction collapse, if it happens, should in some way be accounted for by physical laws. But I don't think I agree with your underlying principle that something which is described by physics must therefore be explained by physics. Once we identify the most fundamental laws of physics that there are, surely we can only explain why they are that way by appealing to the one who created the physical universe, right?

  15. Mactoul says:

    "something which is described by physics must therefore be explained by physics."

    I would rephrase it as:
    Something which is posited in physics must therefore be explained by physics.

    The wave function collapse is something that is posited in physics.

    Regarding the "most fundamental laws of physics that there are", there are two objections to such a thing:
    1) Even if a set of laws is found that encapsulates and explains all the phenomena, there is no telling if suddenly some new phenomena is observed tomorrow.
    2) there is a Goedelian objection that is made by Fr Stanley Jaki in his article "Late Awakening to Goedel's theorem in physics."--it is available online.

  16. Aron Wall says:

    Macotul,
    I don't see why the statement is any more true when I substitute "posited" for "described". A field of study can perfectly well posit that something is the case, without providing the final explanation for why it should be that way. For example, Darwinian evolution posits that at least one life form exists, but does not (by itself) explain why the first life form came into existence. Similarly, Coulomb's law assumes the existence of charges, it does not explain their existence.

    Regarding your two objections to fundamental laws of physics:

    (1) is irrelevant because it is an epistemological claim, not an ontological one. True, we cannot ever be totally sure that any given current best understanding of physics is the final one. But that does not imply that those laws cannot exist in reality in a mind-independent way. Nor does it imply that they cannot be comprehended by the human mind and posited to be complete. Therefore my claim stands.

    (2) I read the article and I think he fundamentally misunderstands Godel's theorem and it's relationship to the laws of physics. There is a reason why none of the famous physicists he was talking to took him seriously. You may also have noticed the absence of math in a paper which is purporting to talk about a mathematical theorem. This is a big yellow flag!

    Godel's theorem applies to formal theorem proving systems which include the laws of arithmetic. But the laws of physics are not a formal system in the requisite sense! All of the specific predictions of the laws (as currently formulated) could in principle be calculated by a sufficiently large computer in a finite amount of time. Such systems do not exhibit incompleteness properties. Only complicated propositions involving the evaluation of infinitely many arithemetic facts can be subject to incompleteness.

  17. Mactoul says:

    Aron,
    "Posited" vs "describes"

    Physics describes lightening but posits electromagnetic field to explain it.
    Darwin does not posit life forms --we know of life forms before and independent of Darwin but natural selection is posited in Darwin to explain features of life forms.

    Regarding Goedel, while I am totally incompetent to speak on the matter, I note that Jaki isn't speaking of physics or mathematics but of a metaphysical implication of a mathematical theorem. Thus, there is no maths. That the great physicists did not pay heed to him is neither here or there. Great physicists are not necessarily great metaphysicians. Witness Hawking's pronouncements on theology and philosophy.

  18. Mactoul says:

    "those laws cannot exist in reality in a mind-independent way"

    The laws of physics encapsulate our attempts to understand physical phenomena. I do not know in what sense they may be said to "exist in reality" either in a mind-dependent or mind-independent way.

    What "exists in reality" in the most unqualified sense are things, the most ordinary objects and all knowledge proceeds from them. As Heisenberg is supposed to have said--the sky is blue and birds fly in it.

  19. Aron Wall says:

    QED is a good description of the behavior of electrons and photons, even in times prior to the exsitence of the human mind. Therefore, whatever your system of metaphysics, there ought to be a way of saying that "QED describes electrons and photons" in a way which is talking about electrons and photons, not just our own thoughts. Metaphysicians argue about the exact meaning of "laws" (are they merely descriptive, or do they have actual causal influence or what) but whatever the laws are, they involve some true propositions about Nature, which were true before there were any human beings.

    Then I can call this set of propositions, whatever it turns out to be, "the Laws of Nature", and ask what if anything explains it. If it does have an explanation, it must exists outside of physics.

    As for the article by St. Jaki you cite, he never uses the word "metaphysics" anywhere in it. Nor does he make any particularly metaphysical arguments. His claim is that physicists are missing the basic, obvious mathematical meaning of Godel's theorem, not some esoteric extra principle he deduced from it.

  20. Mactoul says:

    "QED is a good description of the behavior of electrons and photons, even in times prior to the exsitence of the human mind. "

    QED does describe electrons but QED itself is a human construct. IS there a QED independent of human minds or culture? There might well be a better theory than QED in future. So, what happens to the picture of human-independent QED then?

  21. Aron Wall says:

    Mactoul,

    In that case, QED will be an extremely good approximation to the true theory of nature in those circumstances, and this is also a mind-independent fact about the relationship between Nature and certain mathematical structures.

    (In fact we already know QED is not the best theory of Nature, because it is just one component of the Standard Model, which has other forces in it besides electromagnetism.)

  22. Mactoul says:

    Aron,
    You say that the observer causes wave function collapse for this reason:
    "If you try to catch out Nature by explicitly measuring which slit the particle went through, then YOU are now different as a result of you knowing which slit it went through. As a result, the two histories don't interfere"

    But this reason would hold ONLY if you have a QM theory of consciousness and knowing. The knowing is not a physical change. I may either know A or know B. But this is no correlation between my knowing and the physical state relevant to QM discussion.

  23. Aron Wall says:

    Mactoul,
    Regardless of the nature of consciousness and knowing, whenever we experimentally measure which slit it went through, this always involves some physical object becoming different based on which slit the particle goes through.

    Perhaps the point would be made clearest by saying that if we build a machine that measures which slit the particle goes through, then IT will now be different as a result of the measurement, and the interference goes away REGARDLESS of whether any human being or other conscious being looks at the readout of said machine. Thus, in some sense, consciousness is not particularly relevant to the situation.

    If a human being does make a consicous observation, in my view this will necessarily involve some change to the physical state of their brain. However, before this happens there will be a change to the physical state of their sense organs, and before that happens there will be some kind of change to the environment (because that is what sense organs detect). Thus my statement was a bit of an exaggeration since (at least in practically possible situations) it is never the case that ONLY a conscious observer changes as a result of a measurement.

  24. David says:

    Has anyone ever tried to formulate an objective collapse theory based on decoherence? I've heard of spontaneous localization, where (partial) collapse is just something that happens to wavefunctions, and of Penrose's gravity collapse theory, where collapse occurs when the things in superposition have a total mass equal to the Plank mass, but never of one that uses decoherence as it's starting point. Why is that? Does decoherence only look like it can be related to wavefunction collapse from the layman's perspective?

  25. Andrew says:

    Aaron is there a particular site you used to create that image? Thanks.

  26. Aron Wall says:

    Andrew,
    I made it using Open Office's drawing tool. (Also, my name has only 1 "A" in it.)

  27. Andrew says:

    Thanks Aron. Sorry I think that was auto-spell check.

  28. Mactoul says:

    Aron,
    I am still unclear which QM interpretation you actually favor and why.
    To me, realism about ordinary objects, which is the absolute bedrock of science including quantum mechanics itself, demands that the wavefunction collapse objectively. That there should not remain any superposition. So decoherence does not do. You still have one branch in which Jesus was resurrected and another in which he was not. Unless God set-up a special quantum state.
    But again, there is one branch in which I am saved and another in which I am not saved. So \, does God sets up special quantum state for me too?
    Indeed, as I understand from your exposition of Many Worlds, it is Schrodinger's equation plus decoherence.

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