Reference frames, relationism, quantum theory / discussion

0:00 I'm going to start the talk today by telling you about a controversy that's arisen in various different contexts in the quantum physics literature. And it's something that I don't think philosophers have spent that much attention on. And I think it's a really interesting problem. So part of the reason I'm telling you about it is just so that perhaps you might be interested in it. But I'm going to tell you my own perspectives on the problem. And then the second half of the talk is going to be building up from there. So the controversy is about whether eigen-spaces of additively conserved quantities can have coherence between them or not. Let me give you a specific example so that we know what we're talking about. So for me, the nicest example occurs in the context of quantum optics. so this it really is the modern form of the debate started in my opinion with this nice paper from 1997 by Klaus Moomer entitled optical coherence a convenient fit so here's what he pointed out the standard assumption for what quantum state of the electromagnetic field is generated by laser is that it's a global coherent state the projector of the state alpha. Alpha is a coherent superposition of number states with these weights, but importantly, alpha here is a complex number, so this has some phase, an e to the i of phi n, where phi is the phase of alpha. So this is a coherent superposition of number states. That's the key. So if you have a cavity and you've got a gain medium and you pump it, you expect laser light to leak out of this cavity, and everybody in quantum optics assumes this is the quantum state that emerges. And if you do calculations with that, you find you can predict the results in all sorts of experiments. It works incredibly well. And so the idea is that this coherence really is there, that it is a coherence of resistance of number states. There's no reason to doubt that. And then Molmer in 97 said, hang on, maybe that was a bit too quick. Let's see what happens if we quantize medium. And furthermore, he makes a couple of assumptions. Assume that the atoms in the

2:30 gain medium start out in an incoherent mixture of energy eigenstates. So a thermal state, that's an uncontroversial assumption. People typically would buy that for the gain medium. And secondly, apply energy conservation. So if you do that, and you focus, for example, on the state of the atoms that's of a particular energy, you find that the atoms in the electromagnetic field evolve in the following way. If a particular atom is excited and the field has N photons in it, then there's some amplitude for the atom becoming de-excited, so this indicates the ground state, and the field gains a photon. And that's a coherent process. so for that particular case if you ask what's the quantum state of the field what you have to do is take the reduced density operator over the field degree of freedom and you get the following an incoherent superposition of n photons and n plus 1 photons now if you do a proper model with the whole gain medium in a thermal state what you find is that you just have basically an incoherent states of this form. And at the end of the day, when you look at the reduced density operator for the field, you find it's still an incoherent superposition of number states, because if you sum up incoherently a bunch of things that have no coherence between number states, then you still have no coherence between number states. The particular distribution is the Poissonian distribution, so it actually matches the distribution for the coherent states, but it's not a coherent superposition. So that's the difference between this state and this state. And so Molmer says, well, despite the fact that we've been using these coherent states in quantum optics, when you analyze the situation carefully, and you ask what is the quantum state of the field, you find there's no coherence. So, concludes Molmer, this coherence is a fiction. Maybe it's a mathematical convenience, maybe it works well for calculations, but it's not really there. uh so in in 97 people read this paper in the corner box community and thought wow that's really strange um i can't be right and uh you know people didn't really know what to do with it um and so various people got interested uh and uh at the time i i was chatting with a bunch of

5:00 people about this problem and there was lots of uh arguments that could be made one way or the And I won't take you through all the arguments, just maybe one argument, which is the question of whether you can settle the issue by experiment. Can you determine whether the coherence is there or not by doing some measurement? So here's an experiment that would suggest that you could figure out whether it is coherent in that state or not. It's just a homodine measurement. So the idea is you feed in your system, your electromagnetic field in its quantum state, in this port of a beam splitter. On the other port, you just feed a classical electromagnetic field. And in the office community, that's called a local oscillator. And you have a phase shifter here, and you look at what photocurrent you get in these two detectors as a function of the phase shift. And you look for interference in the difference in the currents as a function of that phase shift. And now if you do the quantum mechanics for this, you find that the difference in those currents is given by this expression. Trace of rho, so rho is the quantum state of this signal, times this combination of the annihilation and the creation operator where beta is just the value of the electromagnetic field here. and so what this does to you is that because this is an annihilation operator here only if the density operator has off-diagonal elements on the number basis will this be non-zero and the question of whether we have a coherent superposition of a number of states or an incoherent superposition is just the question of whether there are any off-diagonal elements and so for the coherent state you do have off-angle elements and you'll find something non-zero so you will see an interference pattern as you vary that by case. But were it incoherent, you wouldn't see one. So that seems decisive. Just do this experiment and look and actually this experiment has been done and you do see interference. So that would seem to suggest that the standard way of doing this is right that we have coherence. But it's not that easy because there's a response that the proponent of coherence as fiction

7:30 is the following. If you ask where did this local oscillator come from, and you trace it back in your experiment, what you find is that it came from exactly the same source as your signal. So further back, there was some common source that was put at a beam splitter, and maybe this thing was highly reflective, so that most of the signal went here, and just only a little bit of the signal came through here. And then you phase shift to the signal relative to the local oscillator. You need to do this in order for these things to be essentially phase-locked. So, Molmer and other proponents of coherence as fiction can say, well, really, your common source came out of the laser. It's an incoherent superposition of number states. So this is, say, the state of the common source. If you have vacuum in the other port, that's your initial state. You hit the beam splitter, every particular term in here the n0 term for example becomes an entangled state so you get a superposition of different numbers going along this path and going along this path of this form and your phase shift just changes the phase of different ways of distributing a fixed number of photons between the two paths and if you average now over this Poissonian distribution you find that your overall state states, which can also be written in terms of these coherence states of these two modes. All the details are not so important. The key is that if you take the reduced density operator over just the system, so you take this and you trace over this guy, what you find is you get a state that has no coherence between different number eigenstates. And yet, if you ask, well, what's the probability of getting certain quote-occurrence, and you do the math, you find it's And essentially, this asks what the relative phase between these two modes are. And because that relative phase is well-defined, you do find interference. So the coherence-as-fiction camp can explain the experimental evidence. And they still maintain, and it's still true in their analysis, that the reduced density operator, the electromagnetic field, has no coherence between the Boregian states. Good question. special about this to optics, or is it really an instance of a general question

10:00 about state preparation and the angle of the preparation device? Yeah, it is a general question, and I'll quickly point to other cases where it's a particular problem. But I'm just using optics to illustrate it a little bit. So there's many arguments that you can make. You can ask, well, are there other experiments that settle the issue? And And I and some co-authors have written a long paper about the backs and forths in this argument. And I don't want to go into all those details, except to say that it's interesting. And it seems, after you do a lot of analysis, that both sides of the argument can explain all the experimental data. And so you start to wonder, well, is this really an issue that needs to be settled? There's some mistake that's being made, a presumption in the argument. So let me tell you a bit about the broader context. So after Molmer's paper, there were a lot of papers written on this issue. That's just to give you a representative sample. But you can actually find the same debate occurring elsewhere in the literature completely independently. So, for example, back in the mid-90s, before this paperback won't work, the debate had already occurred in the context of discussions of non-locality of a single photon. In the superconductivity literature, people were puzzled about the following thing, which is the Bardeen-Cooper-Schrefer ground state of a superconductor is a coherent superposition of different numbers of Cooper pairs, and a Cooper pair has a charge 2. superposition of 2, 4, 6, and so on different charges, and yet there was this general belief that there should be a superposition, a superselection rule for charge. You can't have a coherent superposition of different charge so that led some people to say the BCS ground state is a fiction it's great as a mathematical convenience, but it's not the real quantum state of a superconductor, because the coherence between different charge ionizations. Similarly, in the context of BEC, those last time condensation, people wondered whether you could have a coherent superposition of different numbers of atoms

12:30 in your condensate, because the atom number superselection rule would suggest that you can't. But on the other hand, people liked to introduce an order parameter as the expectation value of the quantum field operator, and for that to be non-zero, you have to have a coherent superposition of different numbers of atoms. And so there was good reason to think that that assumption made sense and worked well in calculations. And then finally, if you go way back to the 60s, you'll find a similar discussion occurring between these guys, or the proponents of superselection rules, and this very nice where they argued that even a super-selection rule for, say, proton versus neutron, so this is ultimately a charge super-selection rule, can be circumvented. So I heard on Susskind argue you can prepare a coherent superposition of a proton and neutron if you have a map of mesons to work with, whereas others would argue that you never can. So anyways, there's all these places where this issue arises. if you have an additively conserved quantity like charge or atom number or photon number, can you or can you not prepare coherent superpositions of the quantity? Why do you say photon number is conserved? Why do you say photon number is conserved? Well, I guess it needn't be if you have nonlinear optical elements. That's true. So maybe I should just drop it and serve. the traditional debate occurred for conservative bodies but as I showed earlier, even in the context of optics, there's a question. Can you portray these coherent states or not? And I don't believe that non-conservation of photons in the world really help you excel. Other questions? is anybody who is a philosopher Has it seen this debate in some form or another? Okay. So I think it's interesting, worth looking at. But I'm going to tell you what, and this incidentally is the paper I mentioned, that discusses all these sort of backs and forths on the issue, and it's written as a dialogue, so it's a pretty easy read.

15:00 But I want to tell you what I think is the solution to this. I think, given the conference, the topic of the conference, many of you will probably be very ready to accept the solution. So maybe it's not particularly controversial. But in my mind, the problem with the whole debate is that it presumes that quantum states only contain information about the intrinsic properties of the system. So these two camps are saying, look, either the quantum state of the system is this or it's that. There must be some matter-of-fact about what this system's quantum state is. but it seems to me that so it strikes me that the answer is that that's just wrong that quantum states also contain information about the extrinsic properties of the system specifically the relation of that system to other systems and so whether or not you have coherences depends on the external system to which you're comparing and there are many choices of what system you might want to compare to. Okay so let me be more specific about what sort of solution I have in mind and so I'm going to move instead of having my optical example or I'm going to move to an example where I hope things will be a lot clearer, which is angular momentum and spin decrease of freedom. So what does it mean to say that a particular spin-and-a-half system has spin-up along the z-axis? Well, it means that it's up relative to some other system. So you might have, for example, some gyroscopes in your lab which define for you what up is along the z-axis. And when you say the spin is up, you mean relative to those gyroscopes in your lab. So typically we don't think that something like spin is an intrinsic property. When I say zed up, I always mean relative to something. And so the point is that in this case, it's relative to a Cartesian reference frame that defines a triad of axes in space. and so that's how we usually model spin we say I assign a density operator

17:30 on some Hilbert space and what that density operator describes is the orientation of that spin-and-a-half system relative to some chosen reference frame I'll call that the external reference frame paradigm of description but there's another paradigm of description I could have chosen to use there is nothing that prevents me from taking in my lab, which are defining the side axis, and treating them as dynamical degrees of freedom. So introduce a Hilbert space for them. There it is, H sub R. And now write down a quantum state for the gyroscopes in my lab and the system I'm interested in. So all of this I'm going to say is internal to the quantum description. And by saying that I'm assigning a quantum state to this pair, then I'm acknowledging that I can prepare state to be pointing in some direction itself. But if that's to mean anything, then there has to be some other reference frame in the background which defines what up for that quantum state would be. So the idea is that you internalize this reference frame. There's another reference frame in the background that now allows you to talk about whether this quantum state is a pointing along this analysis. Okay, so that's the idea. Now note that if I take the reduced density operator of this state on just the system, so let me call that sigma sub S, what that describes is now, so remember this state describes the relation between this pair and its background reference frame. The reduced state is just the relation between S and its background reference frame. So I've got two quantum states system. But because of the way I set it up, it's clear that they actually describe different things. This quantum state describes the relation of the system to R. This describes the relation of the system to R prime. And because R and R prime don't necessarily have to be aligned, those two quantum states don't have to be the same. So that's a way in which we can resolve the problems I talked about earlier. So in the quantum optics example, this would be my electromagnetic field.

20:00 And so with respect to one sort of reference frame, it could be that it's a coherent state. And with respect to a different sort of reference frame, it might be an incoherent state. That's the way out that I see. Okay, now let me be more specific about how you get coherence and incoherence for two different kinds of reference frame. so imagine that I write down a quantum state for the internal reference frame R which I'll just write as a projector onto the state E and I'll leave it undefined for now I'll come to it later but I write down a quantum state for it and now I imagine that instead of having this external reference frame aligned with the internal if these guys are aligned describe this system relative to R prime in exactly the same way that I described it relative to R. So I assign the system the state rho, just as it was here. And this guy is some state which says, basically, he's pointing up in the same axis. But if I imagine that R prime is rotated relative to R, then the state of R s is going to be rotated. So omega will be some element of SO3, which describes the relation between R prime and R and S will be rotated in exactly the same way because both of these guys are described relative to the same reference rate. But now suppose that I don't know the relative orientation between R prime and R. Okay, so let me just denote that schematically by a little question mark here. Well, in that case I have to now average over all possible rotations. Did I just forget to put that average? I think I did. There should have been an average over omega here. I'm sorry. So there should be an integral over all possible elements of SO3, because that's how you describe lack of knowledge in quantum mechanics. You just take an incoherent mixture of all those possibilities. So that would be the quantum state of these two systems relative to a frame that wasn't correlated with this guy. So let me now just write that in this shorthand. So script G, every time this script G is going to show up a lot on the top, it's going to be a collective rotation, right?

22:30 So it rotates both R and S by the same amount. So think of it as amount that acts on the density operators for this guy. It's also sometimes called a twirling operation, which is a good way of thinking about it. It rotates everything the same way. So that's the quantum state of this pair. It's just the twirl version of E cross rho. And now note that that state is rotationally invariant, because if I average overall rotations, what I'm left with is rotationally invariant. And you can work out that any rotation invariant state has no coherence between eigenstates of total angular momentum. So this is the analog to what I had in the quantum optics case, where I had no coherence between the number of eigenstates. have no coherence between the different eigen spaces of J squared. And so the idea is that if I look at the system alone okay, so if I oh yes? Is that state? The twirl state? Yes. Do we have to understand that in relation to the external framework? Well actually you're right that there's a sense in which I don't because now that I've twirled it even make reference to R prime anymore, because it doesn't have any orientation space. But in a sense, there's still an implicit use of R prime, because you're writing down a state in this Hilbert space of R cross S. You have this big Hilbert space, and you're writing a particular quantum state in it, which has no orientation, but the Hilbert space you're using allows for states that have an orientation. So by using such a big Hilbert space, you're saying, it's possible that I could have this system oriented in So I still think, for that reason, that it's appropriate to think states like this as defined relative to some external frame. But this particular state happens to be rotation invariant. But we'll come back to that. Yes, Steve? What do you do if G is compact? Well, then you have some mathematical troubles. And all the groups I'm going to be considering here, actually most of the talk is just SU2, are going to be compact groups. non-compact groups. I suspect most of this will go through, but I don't know for certain. There's been some work done on the case of non-compact groups, but most of it's been done as a public-compact groups. Other questions?

25:00 what I want to do now is just focus on what's the density operator for S, if this is the quantum state of the pair, and so all I have to do is take the reduced density operator on the one system. And when you do that calculation, you find it's just the twirled version of rho. So this twirling operation applies just to the system. It's just an average of all rotation. And once again, this state has no coherence between different eigenvalues of J squared. And so it's like your incoherent superposition of number states. So this is exactly the analog of what we saw in the optics case, that the the density operator for system S relative to one frame can have coherence. So this guy could be a coherent superposition of different eigenvalues of J squared, whereas this one has no coherence. So that's the particular way you could resolve this problem in the context of any element. So I think that's a reasonable response. From this perspective, the people, the proponents of coherence as fact, we're insisting that reference frames that are correlated with your system, that are somehow aligned with your system should be treated externally so they're insisting that we don't introduce quantum degrees of freedom for these guys and therefore our systems will be coherent whereas the proponents of coherence as fiction are essentially insisting that if you have a reference frame that's correlated with your system like the game medium of the laser like the local oscillator in the context of the interference experiment, then you ought to assign it some quantum degrees of freedom, put it inside, and when you do that, you'll find that the reduced density operator for the system is always incoherent. So that's basically what was happening in that debate. Okay, so now I want to start talking about measurements and whether there really is no difference between these two sorts of paradigms. So the overarching question here is, is it the case that we can really use either one of these descriptions? Are they equivalent? So to do that, we should look at measurements. So I'm just going to denote a measurement as a positive operator-valued measure. So quantum information theorists use this a lot.

27:30 It's just a set of operators, each of which is positive, and that's sum to identity. If you're not familiar, then just think of this as a projector value measure. So these could be the projectors in a spectral resolution of some Hermitian operator. You won't lose much by thinking about them that way. So the question is, well, anytime I do a measurement on a system, say I measure whether it's up or down along the z-axis, well, clearly that involves comparison with a reference frame. What defines up along z? So clearly, these POVMs relate the system to the reference frame. And so the question is whether I can model measurements as basically relations between this pair of systems, so an internalized reference frame, and my background external reference frame. So is there some E tilde, which is now a POVM on a larger covered space, which, when combined with the density operator on a larger covered space, gives me the same statistics as I would get here. So the statistics, you know, the probability of getting outcome K over here is just trace of the density operator times this POVM element, whereas over here it will be trace over now the larger Hilbert space of R and S of this density operator on the larger space times the POVM element. And we want to know whether we can do this in such a way that the POVM elements are themselves rotationally invariant. just like the state. So that's the question. Can we find some way of representing the measurements on the larger space such that I get exactly the same predictions? Okay, well, the answer's going to turn out to be yes, and let me try to convince you of that. So let me just focus on the measurements now. Here's a way you can do it. So imagine you start by just measuring R, and you're going to measure it in a particular way. There's something called a covariant POVM, which is just one of these positive operator value measures where every element is associated with the group element. So omega labels elements of SU2. And the POVM elements are just some fiducial element

30:00 rotated by omega. e I told you, I showed earlier that was in the definition of space but I still haven't told you what e is but I want to avoid that, I want to put that off even further but the point here is merely that you can choose this in such a way that if you average these guys over all rotations, you do get the identity operator such that this really is a valid possible measurement intuitively just think of it as the best you can do in quantum mechanics to decide what the orientation of this reference frame is. If this is a finite dimensional system, then you can't estimate exactly what direction it's pointing in. That's impossible. But you can do a measurement that gives you a good guess. And this is that measurement. So you do the measurement. You get a particular omega. And now after you've obtained an omega, what you do is you go up to the system, and you don't measure this DOBM. You measure the rotated version of it. Because essentially what you've found is, to your external frame, you've said, well, my best guess is that R is pointing at some axis, and it's got some configuration. So if you wanted to measure up along your Z axis here, right, so if you wanted to measure up along Z for R, and you've just discovered that relative to R prime, R is pointing off in some crazy direction, and you say, well, I want to measure up along that direction. My best guess as to what that direction is. So you just rotate this POVM to that direction, your best guess of that direction, and you measure it along that axis. And then you just, I shouldn't say G, I should say omega, you forget the outcome of the first measurement, and so you can average over it, and this is your resulting POVM element. So it's just what you measured on the first guy times the measurement you did on the second. Labeled by K. So that's just a fancy way of trying to simulate what you're doing here by measuring the pair. And again, because this is a twirling operation, I can just abbreviate it this way. So it looks a lot like the state I had for the pair of systems. That's the measurement that I'm going to imagine. And then the question is, if I combine this density operator

32:30 POVM, do I get the same statistics as I would here? So you're probably guessing how this is going to work out. I don't want to get too much of the mathematics because it's boring. Maybe I should just skip this calculation. But for the non-believers, if you throw in this density operator and this POVM and you substitute in these twirling operations explicitly, you change the dummy variables and you do a bunch of manipulation And what you get at the end is that you first trace over R, and you're left just with a trace over S. And essentially, all of this rotation stuff can be turned into a map acting on a row. So I'm just going to call that map script D. So it's an integral over rotations of rho weighted by this weighting factor here. Right, that's what we get at the end. So let me just summarize it. Trace of some decoherence map on rho times ek traced over s. And so it's not the same as what I had in the external reference frame paradigm because I have this extra map acting on rho. But now here's the key point. If this quantum system here was of unbounded size, so it would be arbitrarily high dimension, then I can make this weighting factor here go to a delta function centered on the identity element. So think about what this is. This says take your quantum state E, rotate it, and then see what the overlap with E is here. And so if these E's live in a very high dimensional space, such that even an infinitesimal rotation takes you to an orthogonal state, then this is essentially a delta function centered on omega corresponding to identity. And so if I do the integral of that delta function, I just pick out the unitary, which corresponds to no rotation at all, and so I'm just left with this map being the identity map. So if I have really large reference frames, then I do get exactly what I had in the external reference frame paradigm. so I do get agreement in that case but if my Hilbert space is finite dimensional and my group is a lead group, a continuous group then there's no way to make this into a delta function because I have a continuum of different rotations

35:00 but only a finite number of orthogonal states and so this thing will always fall off with omega and so I'll always have some decoherence this map won't be identity So it looks as if this is something in favor of coherence as fiction. The fictionist could say, oh, look, obviously you have to always treat your reference frames internally as quantum systems because unless they're infinite in size, which they never are in practice, you will miss this critical feature, which is that you don't quite get this expression for your probabilities. You get this one, where D is some non-trivial math. So it looks as if maybe the fictionist is winning here. But actually, what you realize is that this is wrong. To say that by treating my reference frame externally, I necessarily have probabilities that look like this is simply mistaken. Any time I actually do this in the lab, the reference frames I'm using, the gyroscopes in my lab, are always finite. They're never infinite in extent. And so even though I choose not to assign them degrees of freedom in Hilbert space, I choose not to model them that way, it's nonetheless true that I can never prepare a pure state relative to those gyroscopes. You never actually get pure states in the lab. If you want to know how well can you do with a finite reference frame, Well, you just look at this picture here, you know, treated quantum mechanically, and you see that, you know, you get this decoherence. So clearly, it must be that even if you choose to treat it externally, that decoherence is still there. So the point here is that when you look at this expression, it refers only to the Hilbert space of system S. And so there's actually, at the end of the day, once I've calculated what this decoherence map is, space, I can move to this picture. I can say, well, I'm only going to describe states and measurements on the system alone, but by virtue of the fact that I know my reference frame is finite, I have a necessary decoherence mechanism, which I always have to add in. So the resolution of this is that, yes, you can treat your reference frames externally

37:30 if you wish, but you have to bear in mind that if your reference frames are of a finite any extent, then there's this unavoidable decoherence mechanism. If you want to calculate what it looks like, well then you might have to go and do a model of your reference frame. But once you've done that model once, thereafter, if you have this map, you don't have to use a big space anymore. Just use the small space and keep that decoherence map around, and you'll be able to do all your calculations. So that's the idea. Bandwidth size reference frames, can be modeled externally, if you wish, but you just have to assume that there's an effective decoherence mechanism on your states. Questions about that? The argument says you can't get the delta function. That's on the final dimension of space. That's right. you're only modeling the symmetry is freedom which is giving me a final dimensionality is that a reference you're saying perhaps the degrees of freedom that you need here are ones that have the appropriate symmetry group so my reference frame has to be a gyroscope so it has to have an orientation space Even though the emotional degree of freedom might be an infinite dimensional hover space, it's irrelevant where the gyroscope is. Where it won't help me align a spin relative to it. It's what its orientations in space look like. So, in fact, your point is well taken, that if I had chosen to model it with the degree of freedom that lived in an infinite dimensional hover space, then maybe I could get around this. I think you're also assuming that You're assuming there's no interaction between rotation degrees of freedom and translation. Yeah, that's what allows you to do this argument, yeah. Yeah, and that's not as clear as related. But I don't think it's relevant to the debate. I mean, you know, the idealized, that idealization

40:00 I don't think it's going to change any of this discussion. Well, okay. I mean, maybe that's right. Just wondering, could you give a sense of what the other side of the debate looks like? I mean, it seems like this is a nice demonstration of how you can have effective pure states out of tracing out what we've referenced, right? But I don't see how anybody could have thought that via a reparation, via an interaction with an external system, you could ever truly get a pure state. I mean, you're always going to have any non-tributive coupling between S and R and trace out R, and S is going to be something that you don't care if you're trying. It's not going to be your state. Well, I think the issue is slightly different. Okay. See, what I'm saying here is not... The fictionist argument was that the state of S has no coherences at all. because it's the reduced density operator of something that's correlated with R, and the whole thing has no coherence between different values of J squared, so it has no coherence between different values of J squared. All that's being argued here is that, yes, you can have coherence between given values of J squared, you just have a little bit of incoherence by virtue of the fact that your reference frame is finite. So it's not a perfect approximation so you get a little bit of incoherence because of that. Now, so I mean, so the reason there's no inconsistency here. So what I was saying earlier was that a fully incoherent state is only when you try to describe this relative to some reference frame that's totally uncorrelated with it. Then, yeah, it's a completely incoherent state. But this is is the more pertinent description. Describe the system relative to the recognition that is correlated with it. And then it looks coherent. And similarly here, if you just say, well, this is my state for the pair, and this is my measurement on the pair, what do the statistics look like? Again, I get the appearance of coherence.

42:30 So the coherence is now on this state, and that's the place where it's relevant. Rob, yes? This is related to Steve's question. It's a two-part question. The first thing is a little bit of interference. you have the estimates that in fact it looks almost like a coherent statement so if you're getting to that I'm going to show you how much incoherent and then I think the intuition which Steve has which I think is a good intuition so tell us where it goes wrong one could think of it as the orientation of S and then somehow it might be might have several possible orientations and augmented by our so there is indeed entanglement so that we are and from some kind of internal immorality in perspective one might say okay this should appear as the appearance of coherence is that we are focusing on on one of the components, but this entanglement gives us, from an external perspective, gives us a very incoherent state for the system, and that incoherent state sounds a bit like You are arguing that if you use an internal reference frame

45:00 you get an interherent state in principle where's the difference between those two things where does that it's just measuring so what you say is fine there's nothing wrong with that description it's a valid description of what's going on perhaps this will help the point I'm trying to make here I'm making is that these are two descriptions and both of them are valid. We shouldn't think of them as competing with one another. So in the one description, these two are indeed entangled. And this guy is incoherent. And the situation describes is S has some particular orientation relative to R. So they are in some well-defined orientation relative to one another. But that very same situation can also be described this way, by saying the density operator of S is some coherent state. So either you have coherence for S alone, or entanglement between R and S, and both those things are descriptions of R and S being in some well-defined relation to one another. In one case, and the only difference in those two descriptions is whether you as a physicist choose to treat R as part of the quantum formalism or out of the formalism. That's just a choice of how to model things. It's not a difference in the physics. It's just a choice of physicists that we make. And what I was trying to argue here is that even though you might think, well, you have to make the choice where you put it inside the quantum system in order to take account of, say, the finite size of your reference range, you don't. You can also deal with that case by treating the reference range on the outside. So all along, it's just a choice you make. You can use either one we should stop arguing about, which is the right choice. Neither is the right choice. Either one's adequate. So let me move on. example and show you what the map, that decoherence map looks like. So suppose that your reference frame is built out of a whole bunch of spin-and-out particles, any of them. And so I want to treat it internally and figure out what the decoherence map looks like. Now, so one way I can do it is by preparing a bunch of them spin up, a bunch of them spin along Y, a bunch of them spin along X, and that could simulate my three axes of the Cartesian frame. It turns out though that sort of product state is not a particularly good state as a representative of a Cartesian frame. So to tell you what the best sort of state is, I'm going to have to remind you a bit of angular momentum coupling theory. Suppose I have two spin-and-a-half systems.

47:30 They can couple by I think J1 minus J2 all the way up to J1 plus J2. So they couple to spin 0 and spin 1. If I have four spin-and-a-half systems, well, in the first pair coupled to 0 and 1, and the second pair couples to 0 and 1. And now here, I have a 0 coupling with a 0, which goes to 0. 0 couples to 1, which gives me 1. 1 couples to 0, which gives me 1. And then 1 couples to 1, which gives me 0, 1, or 2. And so I get 2 copies of the spin 0, 3 copies of the spin 1, and a single copy of spin 2. So I have multiplicities of the coupled spins. I can do the same thing for six spins, and I find higher multiplicities, and the total spin goes higher. So I can do this for n spins if I like, and what you find is at the end of the day, your quantum states for the n spin-and-a-half particles are labeled by j, the angular momentum quantum number of 0, 1, 2, 3, m, which ranges from minus j to j, and alpha, which is my multiplicity index, which tells me where I am in these five-dimensional, nine-dimensional, five-dimensional, et cetera, spaces. And now, here's the critical thing, which is that I can define a tensor product structure by the following ephemorphism. I take my M degree of freedom and my alpha degree of freedom, and I think of them as living in two different Hilvers spaces. and so I can think of this total over space of the m's and a half in this way it's just a space wherein the jm states live that's these guys and a multiplicity space wherein these quantum states live so now that might be unfamiliar but it's sort of a critical point so let me give you a classical analogue of what's going on here So I'm going to give a definition of a virtual bit. If I have two bits, so I can think of them as A and B, which take values and set 0, 1 each, then I can write down the state of those two bits as just A comma B. But I have another way of writing down the state of those two bits, which is A comma C, where C is the parity of the two bits. So this is just the sum mod 2. and this, from this description state, I can always recover this one

50:00 but here this C is a virtual bit, it tells me something about the relationship between A and B and I can do something similar in quantum theory I can talk about physical systems virtual versus virtual systems so if I have two qubits, two quantum two level systems and they're living in hybrid spaces H A and H B then I can always that Hilbert space as hc cross hd, where these guys now are virtual subsystems. And let me give you a particular nice example. So a and b are my physical cubits, and say I prepare them in one of the bell states here. They're all four entangled states. The bell states are often denoted this way. These guys are the phi states, this is plus and minus, and these are the psi states, plus and minus. And now, just introduce an isomorphism, where I have now two degrees of freedom. One and the other tells me whether I'm plus or minus. And so I can think of these as now two Hilbert spaces. They're both two-dimensional and they're virtual. They're virtual degrees of freedom. And so that's exactly what's going on with this new way of writing the Hilbert space. I'm thinking of n qubits and then these Hilbert spaces describe things like the relations among those qubits and the collective degrees of freedom of those qubits. So these guys actually are the relational degrees of freedom and these guys are the collective degrees of freedom. Okay, so all of that is just to be able to define for you what state of the reference frame I want to use. Here's something, what the dimensionalities of these spaces look like in one particular example. The state I want to use is the following form. So it's basically maximally entangled between these two spaces I've just introduced here. And if you look at these dimensionalities in this example, you'll find that these spaces are always much bigger, except for the top one. So it's always possible to find a set of states in these Hilbert spaces that are at least equal in number to the states you find in the first Hilbert space. So if you exclude the top one, you can always write down these maximum-tangled states. Anyhow, so all that was just to be able to define the state for you. This state has great properties. It's essentially the best way you have of sending information about a Cartesian frame. This state is most sensitive to rotations in space.

52:30 And it comes up all over the place in the literature on this sort of stuff. And so it's the one I'm going to use for my particular example. So I'm just going to take you back to that old slide. I have my state for the pair, my measurement on the pair, and I want to calculate now what this decoherence map looks like if, for E, I use that state that I just pulled you out of here. So you work it out. You calculate what this weighting factor is, and you find it's not a delta function. It has some width. And I won't go through the details, but when you do the integral, you get a really nice result, very simple, which is that your decoherence map is mostly the identity map plus a little bit of twirling. And this is something that's completely decohering the cupid. So the amount of decoherence you have goes like 1 over jr, which I didn't define correctly for you, but it's essentially proportional to the number of spin-a-half particles you have in your reference frame. So it tells you how big the reference frame is. And as jr goes to infinity, this term goes to 0, and you're just left with the identity map. we do see that your decoherence goes away if your reference frame becomes large enough. Five more minutes. Five more minutes. Okay, so let me skip to the conclusions. I'm just going to... There's a whole bunch of slides to skip. Okay. And if in questions you want to ask me some related questions, I can go back to these other slides. So the main lessons are, one, the quantum states describe extrinsic as well as intrinsic properties, that is, relational rather than absolute degrees of freedom. An external reference frame paradigm description is just as good as one where you treat the reference frames internally, and you can even model bounded size reference frame effects within such a paradigm and those effects basically look like

55:00 the fact of ecoherence for the relational use of freedom what's the significance of these sorts of techniques one thing is that whether a super selection rule applies or not is a matter of convention so just as I said earlier unmodel things is a choice that the physicist makes. Similarly, whether a super selection rule applies to charge or atom number or what have you is really a consequence of whether you've chosen to work in a paradigm description where the reference frame is a quantum system. If it is, then you face a super selection rule. If it's not, you don't. And so, in particular, you can prepare funny states like a superposition of one atom of species one and of atoms, species one and species two, bound in a molecule using a Bose-Einstein condensate and treating it externally. So treat it as a classical reference frame. Relative to it, you can really have these funny coherent superpositions of different numbers of atoms. And so in this paper, we basically proposed an experiment that hopefully we could do in the next few years to prepare these kinds of coherent superpositions. And it's basically like these other sorts of interference experiments. Secondly, I think that these sorts of issues might be important for relational theories of gravity. In particular, it strikes me that perhaps the issue of whether we should have a background-dependent or background-independent theory might also be just a conventional choice and not a fundamental distinction. So I like to think of background-independence as a case where the spacer of temporal degrees of freedom suffer back action. And so what I've argued earlier is that the back action that a reference frame suffers can still be treated by not assigning that thing a dynamical degree of freedom. Treat it as external, that the back action basically appears as some sort of anomalous decoherence map. And so similarly here, similarly you could imagine that background independence treated or back reaction could be treated in a background dependent framework and a related point is that there's this nice paper

57:30 by Hartnock Cowfair that suggests that maybe gauge potentials arrive due to back action of a valid side reference frame and so roughly in cartoons if you treat your reference frame like the wall here internally then it suffers some sort of recoil if you treat it externally, then that recoil appears like an anomalous acceleration of the ball, which is like the decoherence map I showed you earlier. And so Harnav and Kaffer have this interesting idea that maybe that anomalous acceleration appears at the gauge potential. So I think there's some interesting work to be done here. There's some related ideas in the literature. And then, finally, I have some views on why I think relational theories might potentially provide an explanation of contextuality and non-locality. And I saw that there's a few people at this conference who are going to talk about related issues. I'm not sure I'll agree with those particular approaches, but I think we agree that this is an interesting possibility. And with that, I'll leave it there. Thanks for your attention. Thank you very much for this very interesting talk. as well. We have exhausted a lot of time already for the discussion, but maybe one of the questions. I was interested in the asymmetry between the two different ways you treated reference frames. From the internal perspective, we had two things that were called reference frames. The external one from the internal perspective was not one of the quantum system. It was sort of averaged over, or each potential orientation to average. and that struck me as rather interesting I'm not quite sure what to make of it one might have expected all of the relevant reference frame to be treated in the same way either modelling the viponic system or regarding them as just elements of space-time structure which brings no longer suppose you take everything that served as a reference frame a Cartesian reference frame and you internalise it Hilbert space. The overall Hilbert space you're left with, I could still, within that Hilbert space, prepare a system that was spinning up along the Z axis. That Hilbert space allows me to do that.

1:00:00 And when I realize that, I say, oh, well, I thought I'd internalize all my reference frames, but I still have this quantum state that refers to the direction Z up. But I've got no reference frame on the outside that defines Z up. So either I say, well, you know, internalize everything and there's something out there that defines up. Or I could say, my Hilbert space is too big. I've got to move to a smaller Hilbert space, a relational Hilbert space. And part of the bit of the talk that I had to jump over was a bit that says, basically, in the large Hilbert space, HR across HS, you can ask where are the relational degrees of freedom? Where's the virtual subsystem where relation between R and S lives? And you can use some techniques from quantum information theory and identify exactly where that virtual subsystem is. And so it seems to me the right way to write down a model of the universe where you internalize everything is instead of assigning a Hilbert space to all those systems, you just defined a Hilbert space for the relational degrees of freedom only. So it's a little bit like the Barbara Bertone model of gravity where instead of get rid of a whole bunch of state space and focus just on the relational thing. And so I think you should do the same thing, focus part of the rest we don't I'm wondering about the relation with with the analogy with would bother thought the yes and I think in power thought the are if you have the total angular momentum of the universe exactly zero in absolutist terms, but then you get exact equivalence between this sub-theory or Newtonian theory and both the theory. Right. Do you have something analogous in your speech? I mean I suppose coming back to the question that the answer I just gave I could imagine somebody saying look you have the Hilbert space for everything in the universe and you know that it can be prepared in an eigenstate spinning up along the z direction for example or the angular momentum points in the z direction

1:02:30 and rather than saying oh you forgot that there's some reference frame in the background or You could rather say, no, they are spinning up relative to absolute space. And so there's an absolute space that defines true up.