A continuation of the lecture transcription/ working out of idea for Boskone that I started in the previous post. There's a greater chance that I say something stupid about quantum measurement in this part, but you'll have to look below the fold to find out...

At the end of the previous post, I wrote:

We can verify this by doing the experiment with single particles, and what we see is exactly the prediction of quantum theory. If we send one electron at a time toward a set of slits, and detect the electron position on the far side, we see individual electrons arriving one at a time, in an apparently random pattern. If we repeat this many, many times, and add up the results, though, we recover the interference pattern. While the individual particles show up as discrete spots, there are places in the pattern where the probability of finding an electron is absolutely zero, and other spots where the probability is quite high. The electrons are detected as individual particles, but their arrangement shows signs that they were in a superposition state along the way-- the interference shows that each electron took two different paths to detection point.

Picking up from there, we come to:

V) This is where the idea of measurement first enters the picture. When we do the double-slit experiment with electrons, we detect the electrons as discrete particles, in a single position. Something must be happening that causes the wavefunction of the system to change from a weird, spread-out superposition state to a single outcome, determining the position of the electron to be a single point on the far side of the screen.

This is a complicated case of quantum measurement, but whatever the physical situation, the results are the same: when you make a measurement of the state of the system, you change the wavefunction from a weird quantum superposition of states to a single result. To take a simple example, if we set up a very small detector near one of the slits, and detect which slit the electron went through, we destroy the superposition state-- we go from a wavefunction that is a sum of the wavefunctions for the two individual slits to a wavefunction that is only one or the other (depending on whether the electron was detected passing through the slit we are monitoring).

This has an immediate and dramatic consequence for our experiment: the interference pattern goes away. As soon as we take the electrons out of a superposition state, we go from having a quantum interference pattern on the far side of the slits to just the simple two-lump pattern we expect for classical particles.

You can understand this process in terms of the uncertainty principle, if you like-- when you detect that the electron has gone through one slit, you make the uncertainty in its position very small, which in turn makes the uncertainty in the momentum very large. The wavelength of the electrons is determined by their momentum, so a big momentum spread gives you a big wavelength spread, and the interference of waves with very different wavelengths will wash out to give a classical-looking pattern. You can even come up with semi-classical explanations of why this happens, in terms of the physical mechanism used to measure the position-- Bohr was an absolute genius at this sort of thing.

That sort of explanation works pretty well for a beam of electrons, but it's not so good for the single electron. And, ultimately, everything comes back to the single-electron case. In order to understand that, you really have to think in terms of superposition states and the destruction thereof. When you don't measure the path, the electron goes both ways, and you get interference. When you do measure the path, it only takes one path, and everything looks classical.

VI) Of course, it gets even weirder (quantum mechanics almost always gets even weirder). It's not actually necessary for you to do the measurement of which path the electron took-- simply arranging things so you **could** measure it is enough to destroy the superposition, whether you record the path information or not.

You can also get the pattern back, by doing something to remove your ability to distinguish paths, after the fact. If you imagine doing the double-slit experiment with photons rather than electrons, you could (for example) cover one slit with a horizontal polarizer, and the other with a vertical polarizer. If you put a horizontal polarizer over your detector, you'll detect photons only from one slit, and see a classical pattern. If you put a vertical polarizer in front of your detector, you'll detect only photons from the other slit, and see a classical pattern.

But, if you put a polarizer in front of your detector that is oriented at 45 degrees from the vertical, and thus is equally likely to pass either horizontal or vertical polarizations, you'll see interference again. Once you can no longer distinguish between the two slits, you get back the original superposition state, and you see interference again. If you want to be really tricky, you can even decide which sort of detection you're going to do while the photons are in flight, and past the slits (imagine rotating your polarizer really quickly, based on some sort of random number generator), and you'll still get the appropriate type of pattern, when you add up the measurements later on.

These "quantum eraser" experiments are some of the most dramatic demonstrations that reality is much, much weirder than we think it ought to be.

VII) Another interesting wrinkle on the whole measurement issue is seen from the EPR gedankenexperiment. Imagine taking a pair of particles, and putting them into a quantum state in which some property of each particle is correlated. For example, you take a pair of photons, and arrange so that one is vertically polarized, and the other horizontally polarized. Then you send them off in opposite directions, without measuring which is which.

According to quantum mechanics, that's a superposition state, and when you finally measure the state of one particle, you instantaneously determine the state of the other, no matter how far apart they are. Einstein **really** hated this part of quantum mechanics, but you can do some very clever experiments to show that this is really the way things work-- the first such experiment is the final entry in the Top Eleven, to be posted tomorrow.

The really interesting thing here is that the state of the photons in flight is indeterminate. It's not just that you don't know which photon is horizontal and which is vertical-- they don't know it either. The state of each photon is both horizontal and vertical at the same time, right up until the instant of measurement. That's a deeply odd and non-classical state of affairs.

VIII) So, what causes all this? Good question.

Sadly, I don't have a good answer. One of the maddening but fascinating things about quantum theory is that we're sort of stuck in that classic Sydney Harris cartoon, where Step Two is "Then a miracle occurs." Something happens during the process of measurement that changes the wavefunction, but it's not clear exactly what.

A lot of excellent work has been done on a process known as "decoherence," which uses interactions with the environment to destroy quantum superpositions. The basic idea is that as a particle moves around, it interacts with lots of other particles, and if we don't keep track of the states of all those other particles, we lose information about the state. That information loss amounts to the destruction of the superposition state, and we move into a scenario where the electron has followed a single definite path. (This is a little hard to explain in the usual wavefunction language, but I'm not about to introduce the density matrix formalism here...)

Decoherence gets you a lot of the way there, but it isn't a full explanation. It can get rid of the interference between terms of the wavefunction, and get you to a situation where you have a simple choice between two options, but it doesn't explain how you end up with one or the other. There are also some arrangements of systems where you can avoid decoherence entirely, even for some fairly complex systems, so there's work yet to be done.

There are various "interpretations" of quantum mechanics that deal with measurement in different ways. The grand-daddy of them all is the Copenhagen Interpretation, which has the killer "then a miracle occurs" statement. It holds that something in the process of observation causes a "collapse" of the wavefunction into only one of the possible states. What that something is is pretty nebulous-- some versions hold that sentient observers are required, which opens all sorts of annoying questions about who counts as an observer, and what counts as an observation, and if a tree falls in a forest does it make the sound of one hand clapping? It's great late-night dorm-room bull session stuff, but not too terribly satisfying.

Then there's the "Many Worlds Interpretation," which holds that there is no collapse of the wavefunction into a single state-- instead, the universe splits into two or more parallel universes, one for each of the possible outcomes. The wavefunctions continue on their merry way, evolving smoothly without a miraculous step, and it just happens that we only perceive one of the possible branches.

This isn't a whole lot better than the Copenhagen Interpretation in terms of actual answers. It really just sort of pushes the question back a level, as it's not clear why we see only one branch. Again, you have a host of questions to blow the minds of really stoned college students, about what happens to all those other universes, and why we only see one of them, and if what I see as "green" is really the same thing that other people see as "Red," can I get out of that traffic ticket?

Both of these interpretations have led to a wealth of vaguely annoying science fiction stories and pop-science books. They don't exhaust the possibilities by any stretch, though-- there are lots of other interpretations, each with its passionate supporters, and each with a host of unanswered questions.

IX) At the end of the day, most physicists end up with some form of the "Shut Up and Calculate" interpretation. What causes the weird behavior of quantum measurement is an open question at this point, but if you just try not to think about it, and roll with the theory as it exists, it's brilliantly successful. Using quantum mechanics, and the various outgrowths of it, you can predict some physical quantities to twelve or thirteen decimal places, and confirm those predictions by experiment. It's probably the most precisely tested theory in the history of science.

And yet, there's this fundamental weirdness at the heart of it, where systems will happily sit in two (or more) states at the same time, and **stop** being in superposition states for reasons that we don't fully understand. It's an indication that for all our success in using the theory, we still don't fully understand it.

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My understanding of the "Many-Worlds" thing, which may be wrong, is that it's a bit more subtle than saying the "universe splits" into two alternatives. I think of it as the statement that there's really nothing particularly wrong with quantum mechanics as it is. Since the measuring device is quantum mechanical and it's state is coupled to the state of the thing its measuring, the measuring device also should exist in a superposition of states. So then the Many Worlds interpretation is really just the statement defying you to tell the difference between that and a world where the wavefunction collapses. Apparently, you can't.

Turning that into a real theory of measurement is beyond what I can do, and you probably need to have this occur in conjunction with some kind of decoherence. But written that way, it doesn't necessarily have the same repugnant feel to me.

Since I'm no expert, I will defer to your judgement on whether I have fairly characterized the Many Worlds interpretation.

I don't think that's an unfair summary of the "Many-Worlds" position, certainly no worse than my somewhat snarky recap. The "universe splits" part is a reference to the fact that, for whatever reason, we only perceive one branch of the wavefunction. Everything continues to evolve smoothly and all the states are quantum, but in order to match up with the reality we see, you have to do something that gets you only one answer, and that's where things get weird.

The mysteriousness of only getting one answer in the many-worlds picture can be exaggerated. Wigner claimed that pure time evolution of the wavefunction, combined with an assumed parallelism of experiences and physical phenomena, would imply that we should somehow feel as if we're in a superposition of states after a measurement occurs. But if you try to come up with some easily measurable operator that corresponds to that feeling of superposition, you find that there isn't one; so actually the parallelism implies the opposite of what he thought. He was confusing the god's-eye view of someone writing down the wavefunction with the POV of somebody inside it.

That said, I do personally think it's easier to treat the probabilistic interpretation of the wavefunction as a separate assumption, though once you assume that it's easy to show that the Born rule is the only one that works.

But my attitude is one that I inherited from John Baez a while back: when people talk about "collapse of the wavefunction" they're actually speaking of two logically separate things. (1) is the interpretation of the wavefunction as giving probabilities. (2) is the insistence that the wavefunction in some sense really collapsed, so that you *must* use the *post-collapse* wavefunction when making a calculation of some later probability.

But (2) is only necessary as a calculational shortcut if you fail to include the measuring apparatus, observer, rest of the world, etc. in the wavefunction (as the decoherence papers show). As a fundamental physical principle, it's purely optional and may even strictly be wrong. I tend to hold with (1) but not (2), except as a shortcut.

This stuff blows my mind, just like it did when I (briefly) brushed up against it in my college physics class, lo these many years ago..

Stupid question ("yes, there *is* such a thing as a stupid question"): Is the famous Shroedinger's Cat gedankenexperiment an example of this superposition, i.e. we don't know if the cat's dead due to decayed isotope (or however it goes--I'm fuzzy on the details, obviously) until we look?

Quick question: in this bit --

-- does "determine" have a formal meaning? The problem is that "instantaneous" has no meaning in special relativity. The usual example is two measurements on different axes, which can be seen in either time order in different reference frames. Can we say that either particle is not in a superposition before it is measured?

Relating this sort of mystery to the layman involves two problems. The first is that "interpretations" muck up the science by trying to explain things in a kind of god's eye view: what "really" happens, and why. There is really no "why" in physics, because you can chase all the "whys" back to one, final "why" that can only be answered, "Because that's the way it is." The other problem is that if you describe the strange and wonderful things that go on in this world to a lay audience, they will want to know why, and saying "shut up and calculate" isn't a very satisfying answer. Unfortunately, there is no satisfying answer that is even close to intuitive.

Matt McIrvin: The mysteriousness of only getting one answer in the many-worlds picture can be exaggerated. Wigner claimed that pure time evolution of the wavefunction, combined with an assumed parallelism of experiences and physical phenomena, would imply that we should somehow feel as if we're in a superposition of states after a measurement occurs. But if you try to come up with some easily measurable operator that corresponds to that feeling of superposition, you find that there isn't one; so actually the parallelism implies the opposite of what he thought. He was confusing the god's-eye view of someone writing down the wavefunction with the POV of somebody inside it.

You've obviously thought about this a lot more than I have, but this sounds a little circular to me, in the "easily measurable operator" area. I'm not sure I understand exactly what you're getting at.

But (2) is only necessary as a calculational shortcut if you fail to include the measuring apparatus, observer, rest of the world, etc. in the wavefunction (as the decoherence papers show). As a fundamental physical principle, it's purely optional and may even strictly be wrong. I tend to hold with (1) but not (2), except as a shortcut.

I think I might agree with that, though "shortcut" is an awfully dismissive word for a step that's probably necessary to allow you to do some calculations at all...

Trent: Stupid question ("yes, there *is* such a thing as a stupid question"): Is the famous Shroedinger's Cat gedankenexperiment an example of this superposition, i.e. we don't know if the cat's dead due to decayed isotope (or however it goes--I'm fuzzy on the details, obviously) until we look?

Yes, absolutely. The weirdness of superposition states is what the cat gedankenexperiment is all about.

(My actual lecture on this topic contained references to both Schroedinger's cat and quantum computation. I left those out here in an effort to make room for EPR.)

Jeff Klein (Re: EPR): does "determine" have a formal meaning? The problem is that "instantaneous" has no meaning in special relativity. The usual example is two measurements on different axes, which can be seen in either time order in different reference frames. Can we say that either particle is not in a superposition before it is measured?

The problem with "instantaneous" is exactly where the EPR paper takes issue with QM, because it seems to contradict relativity. I'll say a little more about this in tomorrow's Top Eleven post.

Well, this is where I wave my hands a lot and say that I haven't actually done the derivations. But I'm pretty sure that if you want to measure, say, an off-diagonal operator in the basis of states like |I saw the pointer move left> and |I saw the pointer move right>, you have to do something on the order of a simultaneous interference experiment on every atom in your body, and probably a large chunk of the rest of the world.

I heard a physicist say once that the idea of That Darn Cat ending up in a superposition of alive and dead states was absurd, because you could just measure an off-diagonal operator and resurrect the cat with a probability of 50 percent. But to actually do that would probably be somewhat harder than just disassembling the dead cat into atoms and reassembling the atoms into a live cat, so maybe it's not so paradoxical.

though "shortcut" is an awfully dismissive word for a step that's probably necessary to allow you to do some calculations at all...

Well, there are shortcuts you can take without talking about collapse. For instance, suppose that I've got a two-state system with states |1> and |2>, and suppose that I measure which state it's in so that the state of the rest of the world becomes correlated with the system, and call those two possibilities |w1> and |w2>. Assume that the measurement was perfect; then they're orthogonal states. Then the state of the whole world after the measurement, assuming no collapse of the wave function, is

c1|w1>|1> + c2|w2>|2>

Now you compute the expectation value of any operator that just acts on the little two-state system and not the rest of the world, all you need to know about the rest of the world is the orthogonality of those two states, and you'll find that you get exactly the same answer that you'd get if the two-state system were described by a density matrix with |c1|^2 and |c2|^2 along the diagonal.

Maybe the difference is just a language thing, but this seemed like a revelation to me when I finally absorbed it.

The thing is, as was emphasized to me by Michael Nielsen, the use of the reduced density is, in a sense, equivalent to the collapse postulate.

As you say, decoherence makes it very, very hard to measure macroscopic superpositions, but I'm not sure that helps us when looking for an ontology. To me, it just makes life difficult for the experimentalists.

What Aaron said.

Tracing over the states of the environment to get the reduced denisty matrix isn't exactly the same as wavefunction collapse-- it just gets you down to a classical mixture of probabilities-- but it accounts for most of the process. All that's left is determining which of the two possibilities you actually see in any given measurement, and I don't know of any theory that has a mechanism for that.