What follows is a fairly complete transcript of a discussion about quantum physics on May 19, 1994, the last day of a workshop in Santa Fe, N.M. It begins with some technical issues, posed by John Denker of Bell Labs, concerning projection operators, mathematical expressions involved in representing quantities that can be observed in quantum measurements. It soon evolves into a more general discussion of the interpretation of quantum mechanics and the quantum measurement problem.
John Denker, displaying a diagram of a double-slit experiment: “Let’s be a little bit careful about what’s going on here. We start with some source of state here and there’s some amplitude to get from the source to B. There’s some amplitude to get from the source to A. Some amplitude to get through the two slits, and there’s some amplitude to propagate all the way over to the receiver. And you add those things up and multiply them in the right way and it’s all hunky dory. And we’ve been doing that since we were babies.
“The unit operator is basically in the appropriate space that we care about right now is this outer product, but this outer product is represented by that, and the projection operator about which I have immoderate feelings is written like this, and it’s this outer product plus zero. And the question is, is this an OK thing to use, or is this a shorthand for something else? And in particular, when we put a brick in front of that slit, what is the appropriate quantum mechanical representation for that brick? If we treat this as a measuring apparatus, what’s the appropriate quantum mechanical [unintelligible] about measuring apparatus. Well, rather than put a brick in front of this slit, I’m going to put here what really you can put there. You can put an antenna there attached to a resistor which is attached to a cold load which is attached to the rest of the universe, and it actually conserves energy. So let’s do the quantum mechanics of this thing. It’s the same propagator and source of this thing and [unintelligible] A slit but what about the B slit? There’s some chance that the B slit is going to get tossed into the dissipation, which is represented as D, and there’s equally something here that says a fluctuation is going to come out of this cold load due to the slit and off to the receiver. The unit operator in this enlarged space ADBF looks like that, and by putting this block in front of the B slit we do not get a projection operator. What happens is this little blue block here gets permuted one and we get this thing here. D is the dissipation and F is the fluctuation and they’re related by the fluctuation dissipation theorem. They’re both part of the heat bath. One is the mode going into the heat bath, the other is the mode coming out of the heat bath. And the magnitude of the fluctuation is going to be equal to the magnitude of the dissipation times some function of temperature. The function does not go to zero even at zero temperature, which means that I can see the fluctuation even at zero temperature, even in the ground state, I can build a quantum nondemolition voltmeter that’s sensitive enough to see the fluctuations coming off of your brick. Notice that the upper left corner two by two piece of this thing looks exactly like the projection operator that you thought you had when you had a brick that you didn’t really understand. But you see what this really is, is shorthand for a permutation matrix that permutes one of your modes into the heat bath and permutes the corresponding thing out of the heat bath. This is why I sit in the back of the room and growl every time I see this formalism of the entire universe as a bunch of projection operators times projection operators times projection operators times projection operators. They are maybe the shorthand for this, but maybe not....
“If you expand unity as a complete sum of projection operators, OK, that’s mathematics, you can do that, no problem. It’s not OK in general, even as a shorthand, for we know it went through the A slit and not through the B slit. But the bottom line is yeah, what it really is, is a shorthand not for the projection operator, a shorthand for a rotation operator that rotates something in the heat bath and rotates something out. And that’s OK, if you can’t tell the ground state from zero times the ground state. I’ve tried to make this point in various ways, and a lot of people say I’m wrong about this, but I would dearly love to have somebody explain why it’s not exactly right....”
Neil Gershenfeld: “What is the conclusion you draw about building quantum computers?”
Denker: “I think that you will probably wind up using a lot of quantum nondemolition measurement techniques in your quantum computer, and you need to worry about what’s terminating the other port of your beam splitter. So I think that certainly when you’re thinking about the foundations and probably when you’re trying to implement the foundations....”
William Unruh: “What you’re essentially saying here is that quantum mechanics has unitary transformations, not projection operators.”
Denker: “Yes, and when you use projection operators as a shorthand you might get away with it and you might not.”
Unruh: “But it’s not a shorthand, because the projection operators are a statement about the interpretation of quantum mechanics, the fact that when one goes out one finds definite things happening. What you’re raising here is the whole quantum measurement problem. In my language, that a determination is not the same as a measurement.”
Denker: “My view of the quantum measurement problem is very different from some other people’s view of it.... A lot of times you can get what you want in the appropriate limit by making, by tracing over a low temperature or zero temperature distribution of the heat bath. What you do is you amplify the signal you care about to the point where it’s a hell of a lot bigger than your temperature, or a hell of a lot bigger than your ground state fluctuations, and then you average over that thing and you get a measuring device that you understand in detail without ever appealing to a projection operator. You appeal to a rotation operator with the thing that your rotated in the [unintelligible].”
Unruh: “You still have to appeal finally to a projection operator because ultimately you have to, ultimately you have to, ultimately you have to [Zurek shaking his head] ultimately (unintelligible.)”
Denker: “He’s telling you what I have to do in order to measure a voltage, and I’m telling him that in my experience in the laboratory, I don’t have to do that, ultimately or otherwise. And that this is, at least in the situations that I understand a sufficient analysis of the measuring apparatus, I do not need separate measurement postulates. All I need is the unitary dynamics of quantum mechanics, the unitary dynamics of amplifiers that I know how to build, and I can do quantum measurement without any stinking quantum measurement problem.”
Simon Saunders: “You also need a partial trace over the environment….”
Denker: “Absolutely. In the limit where my voltmeter is sensitive enough to where I can see this thing, it is absolutely not equivalent.”
Seth Lloyd: “It’s certainly not equivalent to just taking a bunch of projection operators as in the decoherent histories approach. Projection operators in the decoherent histories approach are supposed to correspond to things that you could be able to say about the universe or your system without having this sort of interaction that John was describing.”
Denker: “Postulating a projection operator is very different from postulating unitary dynamics and then tracing over the environment.”
Wojciech Zurek: “The term decoherent histories is perhaps a misnomer although as I’ve seen it used during this meeting it was essentially used in the context of ... you make an appeal to the environment and you look for what sort of projection operators can you stick in there in order to let your history evolve nicely. What Jonathan (Halliwell) talked about and I assume what Todd talked about was very much in this spirit. It’s very different from imposing from the outside consistency conditions [unintelligible] set of the projection operators. If you recognize these projection operators are secondary, as emerging from within, what has happened, that’s one story. If you impose them from the outside it’s a different one.”
Gerard Milburn: “Did you say as the signal to noise ratio goes up in your measurement [unintelligible]?”
Unruh: “Just to reemphasize that the decoherence stuff, et cetera, does not address the fundamental issue that when we look at the world, our experience of the world is of definite things.”
Denker: “Is not.”
Unruh: “It can at best produce for you a set of classical probabilities. Now classical probabilities make sense.... Projection operators refer to individual events.”
Denker: “No, nope nope. Let’s actually go into the lab and we will set up the NMR experiment that corresponds to his thing and we’ll have a blip here, a little blip a little blip.... We were actually doing this experiment once and we were remarking on how damn sharp that line was — until you look at this, the sharpness of that line almost makes you believe in eigenstates. And I believe every word in that, including the almost. This thing is not completely definite. There’s real physics in that width, and the notion that this thing is in an exact eigenanything is an approximation. The idea we have an experience of complete definiteness, I disagree with. It’s almost definite.”
Anton Zeilinger: “Each point of your diagram is definite.”
Unruh: “Each point on the diagram, the height of that voltage curve there, you’ve got a number there, 3.49528. At the next frequency it’s another number, it’s a definite number, it’s not some fuzz. Whereas what quantum mechanics tells you is some fuzz.”
Denker: “Man, when I do experiments, that thing’s fuzzy, I’m sorry.”
Todd Brun: “I’d just like to speak to one very minor thing since I’ve been invoked just now to support both sides of this. I guess the word that I object to is the word impose. The idea that you are taking these projection operators and imposing a condition on the world that the decoherence functional vanishes. This is not my interpretation of what’s going on at all. What I see it as is that you have this system which is the universe or whatever you’re describing which is evolving, and the projection operator is a way of phrasing questions that have well-defined answers. So you are not saying these are my projection operators, I want to get a definite answer out of these, you’re saying, what questions can I ask that have definite answers? So when one is talking about approximate decoherence, the sorts of things that we see, one is sort of making the assumption even that obviously that there are some projection operators onto some set of variables which are somehow close enough to this that when we look at it we say that this gives the answers that we’ve calculated.”
Jonathan Halliwell: “To reiterate the sort of thing that Todd has said, and just try to make some sort of clear statement about what decoherent histories is and what it’s trying to do, and how these projections actually arise. Where it starts from is that it’s a formulation of quantum mechanics that’s designed for genuinely closed systems, such as the entire universe, and it does not assume the existence of any kind of classical domain, and it does not have notions of measurement, just fundamental motions and theories. What replaces those notions of classical domain and measurement is an emphasis on classical logic, just Boolean logic. So instead of assuming classical domains as in the Copenhagen interpretation, it tries to say let’s try and find those situations in quantum mechanics which we can actually talk about, which we can relate to each other using the ordinary logic of everyday language. Now mathematically there is a connection between Boolean logic and probability theory, so from there, saying that we’re going to deal with classical logic you can go very quickly to this idea of probabilities of histories because you want to be able to say things like is there a logical connection between something that is measured and some property of the system in the past. Can we just on the grounds of pure logic deduce, given a measurement of the universe now, the past history of the universe?
“[Unintelligible] say nothing about things happening [unintelligible], we’re just saying can we make probability context of quantum mechanics using the Hilbert [unintelligible] space formalism of quantum mechanics, can we make logical connections between different candidate events? Then given that we can then focus on probabilistic histories, you can ask the question what is the mathematical expression that gives those probabilities? And it was argued by Omnes, just by putting forth a list of reasonable requirements, those probabilities should satisfy that one is led more or less uniquely to this trace formula that the probability of a history is given by the usual density matrix with a string of projections operating on it. These projections, as Todd was saying, essentially just characterize the different types of properties that the system may exhibit at different moments of time. It is characterized from within the Hilbert space formalism of quantum mechanics. Statements like a particle was in this range at a certain time and another range at a later time and so on, so these just enter in a purely mathematical way, it just represents classical Boolean logic.
“Ultimately at the end of the day one actually has to make a correspondence between one of those projections and some actual event in the universe. One makes the correspondence between the final projection and some piece of present data for example, and then conditions on that quantum event to try and find things which have probability one in the past. Which I think essentially is what Bill was saying.”
Asher Peres: “Jonathan, you said a particle was in this domain or was in another domain, what do you mean, WAS?”
Halliwell: “It was misuse of language.... You don’t think of it as an event that actually happened.... You can phrase it all in a much more roundabout way.”
Unruh: “The first thing is that the decoherence formula, that expression for the probabilities of the various histories, that’s just basically standard quantum mechanics. There’s absolutely no difference from that from straight quantum mechanics. The place in which the consistent histories or the decoherent histories people make a change is that they state there are only certain histories that are in some sense legal histories. And those histories are the kinds of histories where the individual events in the histories have some sort of, one can argue that they have some sort of objective significance whether or not you determine them in my language or measure them in other people’s language. They state that all we are only going to talk about those things which have that kind of an objectivity where it doesn’t matter whether I go in and measure it or not. Now that’s a huge additional assumption. Without that assumption you’ve just got absolutely everyday standard quantum mechanics that we were all taught in grade 1, and with that extra bit of stuff you’ve got a new theory that doesn’t make me feel very happy.”
Halliwell: “There’s more in that sense, but there’s less than the Copenhagen interpretation in the sense that you drop any kind of assumptions about classical domains and measurements.”
Unruh: “We learned long ago that that was Bohr’s attempt in my language to associate this determination with measurement. That was his attempt to marry those two things together. I don’t think that one can actually ever ultimately do that.”
Denker: “Just for the record I want to emphasize that that little resistor I drew there was not classical, no implication that there was anything classical in that. I believe there’s only one universe, it’s fully quantum mechanical. Including that [unintelligible] environment….
Saunders: “...The point about putting the brick in front of one of the slits in your history approach, as I understand it, that whole operation of placing that brick, the description of the brick itself, the various wires that you’re going to have ... all of that will be described in terms of a sequence of projection operators which are merely stating what are the properties, perhaps the components of the quantum mechanical state. The whole apparatus of projection operators in decoherent histories theory is not being used as [unintelligible] anything about the dynamics, it’s a method for transferring properties of the state, in other words associating the state or coordinating the state with subsets from the spectrum of the various kinds of dynamical variable. The way something different from unitarity comes in the unitary dynamics in particular is entirely when you start to throw away various components of the universal state, or you can use Bill’s terminology when you say that something definite has happened, something which is recognizable according to our experience. Now that [unintelligible] decoherent histories approach it depends on how you phrase it. Some people ... would hold that only one history is actually stochastically developing in time, and in the frame of the decoherent histories approach that would indeed be to invoke the projection postulate, now you use projections in a rather different way. But in your own approach, if you’re going to take a partial trace, OK fine, but now to interpret the impure state that you get out as a result of that partially traced [unintelligible] state in terms of the description of an ensemble maybe, or at least such that one or another thing has actually happened, given that interpretation, the universal state that you will then work with following taking the partial trace and supposing that something has happened, the state that you will then work with thereafter will not be unitarily related to the state that you began with.... It’s indicating a breakdown or a failure of unitarity. That is put it to you how are you going to take that partial trace and interpret it to mean that something has happened, consistent with unitarity. I see no option [unintelligible]”
Denker: “Well, I do the calculation pretty much as you have described. I write down the unitary operator that describes my voltmeter, and I get out of it — there is a point where I turn a crank and take a partial trace, and the thing that falls out of the partial trace is a number with dimensions of voltage on it. And then I interpret that by saying this is the voltage. And the voltage is big enough that it’s classical and I know what it means and I just don’t—”
Unruh: “In the usual language you say that’s what the expectation value is. You sum out over all these probabilities times the value of the voltage in each one of the probabilities and get the expectation value.”
Denker: “Yeah, and the action behind this expectation value is big enough that the stationary phase approximation is good and the watchyacallit theorem, the voltage that comes out of my voltmeter is big enough to be classical and I run it to a stripchart recorder and I show it to Wigner and Wigner shows it to his friend and it’s done.”
Unruh: “You now go into the same lab, you run the same voltage and you get a different value, you do the same experiment in exactly the same way you get a different value for that voltage. How do you interpret that different value, because your theory gives you exactly the same answer in both cases.”
Denker: “That’s not a quantum mechanics problem.”
Unruh: “Sure it is. ... It’s the quantum noise, in your language, that caused that difference....”
Zeilinger: “... I should confess that I am probably one of the few surviving Copenhagenists in this thing. I don’t see any reason why I should adopt another interpretation. Because in the lab we have classical stuff, we have stuff which we describe with our everyday language, and definite events happen, period. There’s no way around it. And quantum mechanics is never going to [unintelligible] as long as the formalism of a certain interpretation is isomorphic with the same as quantum mechanics formulation. If I talk about a different formulation ... as long as I have something which is isomorphic I will never be able to explain in [unintelligible] language why events happen. I can have beautiful things like Wojciech’s beautiful demonstrations and then other people’s that you get this coupling to the environment, you get this nearly purely diagonal density matrix, which makes sense, which is in the right places and so on and so on. But that still does not explain why events happen. Because even if I had the density matrix I will never get an explanation by the specific result that we obtain in one round of the experiment, in another round of the experiment I get another result. And this is quite different from classical probability. People usually say OK this is just like classical probability and so on. It is NOT the same, because I start from identically prepared initial states and I get different final results. I get sometimes this detector clicks, sometimes this one, sometimes this one. And it’s never explained, this difference, in quantum mechanics.... So in my opinion there will never be a solution to the measurement problem.”
Samuel Braunstein: “I don’t want to get into any kind of interpretation stuff, mostly because I’m in the young generation and the young generation tend to ignore those problems. But I want to really thank John Denker for that brief, pretty little presentation and in particular because it gave me a new way of thinking about another problem, which is eavesdropping in quantum cryptography; when the eavesdropper is reading some information, she’s in a sense dissipating a little bit of quantum state, of the information in the quantum state, and that invariably is going to lead to some fluctuations on the other end, and that’s a very beautiful way of looking at it.”
Zurek: “I want to make a couple of points which are related to what was said by various participants. Let me start with Anton. I think if one wants to follow through the program that John (Halliwell) has outlined and that has come to be known as decoherence process or decoherence program, one needs to recognize that one has to work with the wave function of the whole universe. In other words, there’s no cop-outs, you have to give up Copenhagen interpretation. Whether it’s going to get you where you want to get Anton, I don’t know. But let me try, OK? So in other words if one does have the whole wave function there, it’s clear that all of the other things which are not supposed to happen, or which we don’t perceive, are really happening. All of the other alternatives of measurements, somewhere they’re in this wave function — I’m saying what Everett said what, 40 years ago now. So the issue, which I think has to be stressed, is that the problem is not to explain why there is a single universe there, really physically, but it’s a more limited question, why do we perceive one? A single one. And I think there decoherence does help. Decoherence process makes it impossible to, for instance, remember superpositions of things or put neurons in superpositions of different perceptions. And this will make — if you include yourself within that wave function, you will be able to understand why you can’t see anything else, if you think of yourself as a computer, now I don’t know if you are willing to do that. So the issue is that it helps you draw that boundary between quantum and classical that Bohr wanted to draw, or put it differently in Everett language, it helps you define what the branches are. Now I think it’s very important to start defining these branches not by putting in projection operators from the outside, and this is the point of this transparency, but by recognizing them from within. And to do that there’s no other way but to start, yes, with a closed universe, but then recognize at some point that in order to state that problem of measurement, we have to divvy up this universe into subsystems. Once you have divvied up the universe into subsystems in order to pose that question, we have the right to use that division to answer that question. I think a very good demonstration of how the right sort of projection operators emerge from within the process that John Denker has reminded us of, is for instance in a harmonic oscillator with — harmonic oscillator is coupled weakly to the environment, it turns out that the states which are most stable and which will end up being classical states are decoherent states. They are the most stable ones, and then having obtained decoherent states, you can start looking at histories. They are going to be the histories given [unintelligible] by classical dynamics with a bit of luck....
“The point is that just by looking at the Schrodinger equation and splitting the universe into subsystems in a natural way, you can get the right sort of projection operator rather than impose them. Now imposing them from the outside as is done in the consistent histories approach, poses a danger. For instance, one can put in projection operators which will make it difficult or impossible to put sensible projection operators further down the road and satisfy consistency. Especially perfect consistency. So I think in a sense there are two programs there. One of them, the consistent histories approach which goes back to Griffiths, Omnes, Gell-Mann and Hartle to some degree, which recognizes certain conditions for mathematical additivity of probabilities of histories. And that was an approach, with mathematics. Then there is another approach which starts with a proclamation of a closed system, but then recognizes that what we are treating is actually a collection of subsystems, and which tries to fish out from within that approach, the right sorts of projection operators which give us classical reality. And I think most of the people which commented on including Todd and Jonathan firmly sit on the boundary between the two approaches.”
Lloyd: “I think this discussion shows that quantum measurement is quite a horse. You can beat it for 50 years and it still isn’t dead yet.”
Gershenfeld: “Asked from a naive perspective of not understanding the details, in this discussion I’m not sure I’ve heard anything falsifiable.”
Charles Bennett: “You got the basic point of it.”
Gershenfeld: “In any one situation, people use different words but come to the same answer. Is that right?”
Zeilinger: “What is just said, this coupling to the environment, it’s very important work, because it shows how a classical world is possible, but it does not give you a classical reality. There’s still something which is left over, which you cannot explain, which I mentioned before. Let me say one thing. I don’t want to give a false impression. You know I consider myself a Copenhagenist because I think it’s the most economic interpretation. But I think all the interpretations are important because for two reasons. Number 1, even if they are isomorphic in terms of predictions, they might lead our intuition in a different way. So we might invent different experiments with interpretation A or with interpretation B. And the second reason why I think it’s important to have different interpretations is that I still feel that someday we might understand, in John’s (Wheeler’s) words, why the quantum. And we have not the foggiest idea, I think, which interpretation will finally help us….”
©2014 by Tom Siegfried
Follow me on Twitter: @tom_siegfried
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