Information & Objectivity in QM

0:00 The utility of quantum mechanics. Okay, so now in the following, what I want to do now is just show you that, I want to show you how to construct a universal cloning machine for classical information sources. And then I'll just briefly show you that you probably won't work for quantum information sources, and then I'll talk about the same thing. So here's a CNOT boolean gate, a control not boolean gate. There's two inputs and two outputs, the control and the target. And the control just goes through unchanged. The target is just a bit. The input is 0 or 1 here, 0 or 1 here. The target is flipped if the control is 1. Otherwise it just goes through. So this characterizes the gate. If the control, which is the first, is zero, then there's no change. If the control is one, then this zero goes to one, and this one goes to . Now, it's easy to see that you can make a universal cloning machine just by stringing control not gates together. So here's a classical information source producing using outputs, E1, E2, E3, using probability. The outputs are the classical information source can all be represented as strings of bits. I mean, whatever the outputs are, you can represent, but you can distinguish them and say you can represent them by strings of bits to whatever accuracy you like, and you can, of course, then represent the strings of bits as bits which have, which are sort of sequences this way where they're all zeros and just one one in a different slot so I mean that's not necessary for recording but it makes it easier to connect with this picture here so I output this particular output here I'm going to input into the control part of the control node gate this is three control node gates in sequence where the first one acts on the first bit the second one maximum . So that's the input. The target is always set to, in this machine, to 0. And clearly, this bit will, this goes through my chain, this bit and this bit are unchanged, but this bit is flipped. So I'm here copying the 0 and 0. And the essential thing, then, that makes this possible is that classical states, classical fewer states, these outputs of this information source,

2:30 is represented as the vertices of a symphlet and all the mixed states in the area is represented as a point inside this surface here, this triangular surface, inside the two symphlets. And this is the essential feature of classical theories that is relevant to the concept of classical quantum mechanics. It's not that it's a theory about the position of momentum, I should mention here that a lot of what I've done is very closely related to the work by John Barrett and also the later work by Howard Barnum and Alex Wilkes and Matt Leifer and John Barrett. So this is, in effect, the Bonham-Wilf-Lander-Barrick paper for sit down against the book of the desk. So, okay, so now, now here's why you can't have a universal quantum learning issue. Suppose you've got a quantum information source, which is, you know, just producing two quantum states could be, you know, a sequence of zeros. case of p1 and p2, I put that into now a quantum C knock gate, a control knock gate, which has to be a unitary gate, because this has to work dynamically according to the principles of quantum mechanics. Well, it'll work this way, that if the control is 0 and the target is 0, it will just, if the control is 1 and the target is 0, it will flip the target. but now you have a problem because if the state is a linear superposition then I have to I have to compute the resultant state on the basis of the linear combination so the 0 here and the 0 here will just

5:00 sorry, 0 here and the 0 here will just be 0, 0 but the 1 here So I end up with the state C100 plus C211, in case the state is really a supposition, and I have an entangled state and not a product, which would be side and side. So that won't prove. Of course, that's just a sketch, but that's essentially what analyzes the proof that you can't have enough to get in coding. Now what I want to show is that if simply from the Noclone principle and one other principle, you can derive an uncontrollable measure for students. That is extracting information from an unclonable source is going to change the source of your risk. So the two assumptions are Noclone and there's no universal planning machine. And the other assumption is that every state row, and this row could be a big state, can be constructed from the properties of the measurement outcomes of the fiduciary set. I think this already formation will be complete set of . Now, the fiduciary set, of course, is not unique. And in the case of a qubit, the fiduciary set is just the three spin, the sigma x, sigma y, sigma. If you have the properties for plus and minus then that's sufficient to characterize the state. Note that this proof just depends on these two assumptions, so it does not offend other features of the local states. For a classical system or a classical information source, the fiducial state consists just of a single measure with n possible outcomes for some n, and each distinguishable output corresponding to a pure state can be represented as the various other subjects Okay, so here's the proof. So let's suppose that you have this finite set of fiducial observables. The state rho assigns these probabilities to the outcomes of these measurements. I'm going to write P rho in these brackets for the set of these probabilities.

7:30 It's a finite set of probabilities, and they completely characterize the state of Rho, which could be pure Rho. And they can be stored and computed up to a rational approximation. So you begin with a state Rho, and you measure ANB sufficiently many times to generate all these probabilities in the Rho, and you get P-Rho, and then you prepare Rho from P-Rho. And since this step is possible just by management, and this step is possible just by preparation, you would end up with two states broken. So you would have a clubbinger state. And that's impossible by this assumption, by the clubbing assumption. So the initial state must be changed irreversibly by step one. That is, the original state has to go out of control of the students. And let me just do this like that. So no puny implies there's no complete dynamical unitary account of measurement. Kahn has a measurement device that functions dynamically in such a way as to identify with certainty the output of an arbitrary information source. Kahn has a device that distinguishes a given output to maybe another output undergoing some kind of dynamical transformation. That gives you a distinguishable way forward without altering the information source. role at least. So what then is a measurement device? Well, we understand a measurement device is something that produces distinguishable measurement outcomes, point of readings or records. And in the case of an information source that produces non-informations, a measurement device must produce distinguishable outputs stochastically over a range of measurement outcomes. That's the same for different outputs of the information source. or else the outputs of the information source would be individually distinguished. And hence clever. So such a measurement device itself acts as a classical information source that produces a range of distinguishable outputs, where only the probabilities can depend on what is being measured. That, then, is what we have to do by a measurement device in quantum mechanics, a classical information source. So now just to the measurement problem, I would say there are two measurement problems. There's a small measurement problem, which is just a problem for accounting for the possibility of classical information sources in a world in which

10:00 cloning is in principle impossible. And that's an internal problem for physics. One way of getting a solution is by decoherence. There is a story by which you can solve that problem. How do you get classical properties out of quantum pensions? The big measurement problem is explaining how do you get individual measurement outcomes, individual measurement outcomes, how they come about dynamically in the measurement process. But the big measurement problem from this, from the perspective of this analysis is the pseudo problem because the solution amounts to a dynamical explanation. It's analogous to Lorentz's attempt to provide a dynamical explanation for, dynamical explanation for length contraction in terms of distortions of bodies. And as Einstein saw, the significance of the surprising discovery that light doesn't ever take light is that there's simply no relation of absolute simultaneity in the world. It's not that there's a relation of absolute simultaneity, but there's some dynamical reason why events are not absolutely simultaneous, but appear simultaneous to them to handle different to the state of motion. And similarly, it's only unrestricted cloning if you've got it as possible in principle that it makes sense to look for a dynamical explanation of our inability to clone now and follow us in the states. Say, in Bohm's theory, with the form of the equation of motion and the fact that the distribution of these variables between two and three. So, Bohm's theory is possible in the early universe it would have been possible to clone Now, in the remaining time, I wanted to talk about a recipe, in effect, for deriving these Lorentzian interpretations of quantum mechanics as a way of showing how they're related and really what they're doing. From these probabilities of, this is probably the out-given state road of the measurement, defined for various eigenvalues of observables,

12:30 whereas here is not necessarily the financial, but usually simply any finite set of observables in a finite dimensional space. We define a classical probability space with just the product measure of these probables. This is a trivial invariable theory, where you simply take all the outcasts and and ignore algebraic relations among these different observables. If we impose additional constraints on the P-Row, this P-Row is now the probability measure, this product probability measure, and it may not exist. And this is, in effect, in Markitavsky's way of formulating the patient's perspective here. If you assume the following, that if these observables commute, Then this p-row coincides with the quantum mechanical bell. Then the Corsion-Campus spatial theorem can be formulated in this way. There is a set of observables such that for every state, there's no probability. And the Bell theorem can be formulated this way. There is a set of observables of this on a product space. And there is a state such that there is no problem. Now, those are the negative results which characterize the quantum states. Here's a positive result. For it, it's a bit more complicated, but it underlies the possibility of building all these, what I call the Renzian two-tapes of the planet. For every viewer state, and every partition of urban space, silver space here, into orthogonal eigenspaces, of some of the human space partitioned into a certain line of spaces which in general can be multidimensional. There is some maximal extension of this observable for which there is a probability space. The other results have to do with the big neural properties. And this is unique if you require invariance with respect to all emorphisms on the subspace structure with all the space that are preserved inside and are. So this set of observables is in some sense characteristic

15:00 of the state you start off with and the preferred observable that you choose. So what does this go through? What means extension? You extend the Hilbert space? No, no, I don't extend the Hilbert space, but I start off with one observable, and then And I can add other observables without having it in such a way that the total set of observables can be associated with the classical property space. If I add any more, then they won't work. This is a maximal set of observables. Now, what observables are in it? Well, it contains all the maximal observables whose spectral measures comprise one, all the one-dimensional projection operators onto the orthogonal projections of psi onto the non-null eigenspaces of the partition, and the one-dimensional projection operators onto any orthogonal basis in the orthogonal complement of the span of all these bits. And all the non-maximal observables that are functions of these. So it contains all the observables whose eigen-spaces are spanned by the rays that find in one and two, and it contains non-commuting observables. Nonetheless, you can find a property space. And you can find the property space this way. it's going to have some set with some sigma algebra and some measure function on this depending on psi, where the elements of x are just the projection operators onto these psi-r, which are the two-value homomorphisms or the one-zero maps on the lattice of subspaces Now, just quickly, let's see what this has to do with measurement. Schematically, measurement is just this. You start with a state, which is in the instant position. This represents the state of the measuring instrument. This represents the state of the environment. So you end up in a rectangle state of this form. where these environmental states very rapidly approach the functionality because of the nature of the interaction between the environment and the measuring industry. This will be the case if all these pointer projection operators, the ORI of the pointers of the measuring industry,

17:30 And that entails that the correlation between the measured observable and the point of observable is preserved under the interaction of ancient Syria. That's all. Now, so that means that the environment would determine through the form of the interaction of the third point of observable. And there's a theorem which says that this thing is unique, this representation. I mean, if you think the right side of this form, then you can't write any other form that looks like that. And by the construction of the theorem that I had before, these are all normal states. The observables in here include all the observables whose spectral measures contain the projection operators onto these states, the states with the pointer. So this set contains the measure of observable S and it contains the pointer observable R. So you could use this idea to get an interpretation of quantum mechanics that will resolve the measurement problem for this state and this side of the load. For this stage side, there exists a probability space where the elements of x are the projection operators onto basically these pointed states. And these are just the worlds of the Everett interpretation in decoherence-based visions. Now, we could, if we want, introduce a stochastic dynamics for these worlds. Of course, you don't have to do the Everett interpretation, but you could. That's the Schrodinger equation in this finite vision. Here I've just written again onto the R basis, so that we function onto the R basis. From this you can get a continuity equation for the probability density, that would be the equation for the probability density, where these JNNs now represent its probability current matrix proportional to the net flow of probability from Rn to Rn, defined in this way.

20:00 it's really just and suppose there are discontinuous jumps in these R values these point of values governed by transition probabilities where this denotes the probability of a jump from R n to R m in time et then you get a master equation for these probabilities and it's sufficient for consistency with a continuity equation And now there are many solutions to this equation because there are n squared elements of t and fewer equations. Bell, in a paper, suggests this particular choice of solution where you relate tn's to jn's this way. And you can show that in the continuum limit for this particular solution, when these R positions, when R is positioned in configuration space, and the distances between these, the latin are both ends and ends, the finites of the lattice as the distances shrink. In the continuum limit, this equation just turns out to be Bohm's equation. So, you have a relationship then between the, you know, the Everett interpretation and Volkens. And Deutsch says Volkens are Everettens in a chronic state of denial, Val says Everett's Volkens are decrees. Both theories are Lorentzian, and Everett can be seen as just an alternative from the perspective is in an alternative way of writing with all properties without the machinery of the Bohmian trajectories on the sketch that I gave you, they would be stochastic trajectories. You can use them as a way of getting the probabilities, or you can give them an analysis of the probabilities. The Bohmians take determinants and determinate particle trajectories as a priori in the sketch. The final properties of superposition and entanglement then reflect our ignorance of the given variables, and this is the next reference to the sugar dynamics of the problem in theory if the hidden variables are meant for their quality of the quantum distribution. The Viridian's take the timidness of the depth of this, as I find it by a superposition, is that in terms of multiplicity. That is, all possibilities are realized and probabilities arise from the decision theory

22:30 of analysis. But in both cases, it's only as if the world is irreducibly probabilistic. It only seems so because of some physical mechanism, which according to the theory itself, we have no independent access to the critical access to. And I call it these . It's the . And this is . So, I think I'll just end with it. All the work. Okay, I just want to make some sort of comments which affect the questions about the, you know, I'm not quite sure what the intent of it, I mean I'm not sure what the picture you're proposing is that the cross wants to Einstein's opposed to SR. I take the point that the known cloning principle plays a sort of role analogous to Einstein's right posture. but then of course it's what the right postulate and the principle of relativity I mean so that the apparently paradoxical thing is how to define the right postulate and Einstein makes it all intelligible by deriving the learnings transformations and giving us a new space-time structure that we can understand and see how these two things can sit together now from the no coding principle I guess you can derive that theories can come as a standard So I guess one thing I'd like to know is, in the SR case, you can say, here's what seems paradoxical, how do we combine them, relative to the principle of the right possible? you were saying that in quantum mechanics you can understand

25:00 sort of weird paradoxical nature when you were going back to discussions about as a conflict of no cloning with certain ways of thinking but you didn't actually say how that worked well first of all it's not going to be the case that no cloning itself is going to give you quantum mechanics because no cloning is characteristic of not only quantum mechanics but these super quantum theories and all those theories satisfy no cloning and no signaling so if you want to really exclude the theories which are super quantum when people are looking for principles like that. But the analogy, to go back to the analogy between quantum mechanics and relativity from this point of view, there is something apparently irreconcilable between the light postulate and the relativity principle. and what I say shows you that actually they can be reconciled what's puzzling is the light postulate given the relativity principle that you have light shouldn't behave like this, light should behave like light should be able to behave like, that is if I'm put on a flashlight somebody runs past me very fast with another flashlight they put on at the same instant the speed should and relativity theory resolves that puzzle. So the analogous thing in the case of quantum mechanics is what is really puzzling is, well, in one way you could say what's really puzzling is the fact that in a situation where you have no signal, you can still win this game with more than 75% probability. And if you take classical theories as characterized by this simplex structure for their states, then it is also impossible to understand how you can possibly win the game with greater than 75%.

27:30 So what's involved in that theory is the possibility of cloning. It's the tension between cloning and no signaling, which has to be resolved in this case. And so, what quantum mechanics does is, in making the transition from quantum mechanics to classical mechanics, set up in this sort of analogy to relativity theory. One is, I think, doing what Einstein was doing with the principles. something where it was this quote, a very nice quote, at the beginning of the first paper Einstein that Michael Friedman had about elevating the postulate to a principle. Similarly here, one elevates, if you like, the no cloning to a postulate and then takes that as underlying the transition by itself, not from classical chronomechanics, but from classical and non-classical. Grace is next. So, Jeff, as you know, I'm not entirely persuaded. I always want to know it's better. Part of the reasons that Oliver points to, and part of the reasons we discussed, let me make two comments. The first one is on the Everett picture. So it's not going to be the case that David Deutsch and Weiss and so on is they're going to agree with your characterization. It's not the case that determinants is a priori. That's the old-style way of doing many worlds. Say it again to give the last bit. It's not the case there. But determinants is a priori for the Everettians. That's the old-style way of doing many worlds, where the worlds are introduced as primitive components. Their view of the primitive component is not represented by the wave function. That's just a particular kind of complex physical field. And worlds and determinants experiences are emergent from that, so it's not a priori. characterization. The more general point I wanted to make was about the YQM and the climate concern. Okay, so you wanted to address climate concern. No one understands quantum

30:00 mechanics by pointing to ways in which you understand why quantum mechanics by giving principles up to the moment and just see how it fits into some sort of general theoretical multilateral structure you can find intelligible. Now that only address there are two aspects of the lack of understanding that I think kind of you're worrying about. One is we don't have a good grasp of where quantum theory comes from sort of theoretically what principles drive it naturally and intuitive principles have a relativity drive it. That's one aspect of understanding. The other crucially important aspect of understanding is what's the world like if the theory is true? Okay, so we lack both of those Typically that's where we start off with a failure to understand the quantum planet. Are your approaches to answering why the quantum are never going to deal with that first question, that first lack of understanding, never going to touch on the second one? What's the world like if the theory is true? What's the ontology of the world? Well, the second question, what the world like if the quantum theory is true, is I think sort of a very slippery kind of question. I mean, it leads you to this sort of answer where you step back outside the theory and say, well, look, this is what makes sense. So unless we kind of get the quantum theory to fit this, we're not really going to understand what's going on. And then you slip in elements like determinism or determinism. I mean, you say there's a different way of looking at everything. But, so, inevitably, I think the answer that you give when, to, how could, you know, what was it, what was it going to be like in order for quantum mechanics to be true, is going to be some answer which takes the puzzling features of quantum mechanics in a sort of hands-off sense. takes the, let me just step back for a minute and then answer that slightly different. The underlying thing that has to be understood about quantum mechanics is the indeterminiveness, I'm sorry, is the fact that it seems to be an irreducibly probabilistic theory. I mean, it's the uncertainty principle that way. I mean, you have to explain the fact that there's, why is it the case that you can sort of pin down the position but then the momentum fluctuates. And associated with that stochasticness is the peculiar correlations

32:30 in the statistics. So you have statistics and peculiar statistical correlations and that's associated with the probability of winning this game more than 75% of the time. So that has to be explained, or understood in some sense. But something radical has to be said about that. What these various Lorentzian interpretations do is to, in effect, give some kind of structure and some dynamical account of how that comes about, which involves features which are essentially classical in some sense related to the classical picture that I gave before. So, for example, Bohm wants to show, and this is very clever, that actually you can have trajectories, deterministic trajectories, and get the stochastic features of quantum mechanics, including the correlations, out of a theory where where the hidden variables, that is the actual positions of the particles, are necessarily hidden if the distribution starts off in the equilibrium distribution to the borne distribution. And the last part of the talk, what I've tried to do was to show that irrespective of how you look at Everettian interpretations, they can be related to Bohmian interpretations and it seems to me that they're just a different way of looking at the same thing but that's where the significance is different in the new way of doing it Michael Friedman was next okay I guess I want to raise three things that are all the way to what has just happened first I agree with you about the what is it like if the world is true thing is a strange question that smuggles another thing because, I mean, we know by Tarsky that quantum mechanics is truly the only of quantum mechanics. So if that's all you're asking, then you can answer it very simply. And if you're asking something else, I think you're asking what is it like for some classical point of view. So that's, I agree with you on that one. But with respect to relativity, perhaps the difference is that Einstein doesn't just have those two postulates. Then there's also a, you know, quote-unquote understanding of what's going on in terms of the relativity of simultaneity,

35:00 something that we can, you know, it's a geometrical picture of space-time, we can then think of classical mechanics in an analogous geometrical way, and so we have this kind of nice way of interpreting what those possibles come to that is geometrical. I don't think that adds, it's not necessarily classical, but it subsumes the two under this geometrical. And the third thing is about Everett. I guess it seems strange to me when you said Everett is a Lorentzian dynamical addition because, you know, original pure Everett, the only dynamics is Schrodinger dynamics. That's the whole point, right? It's just the state of the whole universe evolving. And there's no collapse, really. Instead, there are these relative states. And I guess this dynamical explanation comes in when you want to add through decoherence, some story that adds to Everett about, because somehow they don't think the relative states is satisfactory understanding of things. They have to add something else so that there's really real, something that's more like real collapse. Yeah, well, the way I see Everett is it's true, you take the wave function then you have an analysis of what goes on in measurement and in terms of the way that it's done by let's say the Oxford school you bring in decoherence and so essentially you can write the wave function as a linear supposition think it's a wave function of sort of humps which are picked out by the process of decoherence. So now, what you would, I mean, this is now a sort of fork in the road. You can simply say, well, I've got these different humps, so to speak, or different elements of the wave function. Now you can, if you like, introduce the stochastic dynamics where you say, well, one of these is the actual one. I mean, this is not done in every division, but you can do this. One of these is the actual one. introduces stochastic dynamics which will jump around in such a way that you get the right quantum statistics just from the stochastic dynamics. And you can relate this to deterministic dynamics if you want,

37:30 and that gives you exactly both theory. So you can treat the stochastic dynamics or the deterministic version of it as simply a story about how the quantum probabilities come about. alternative you can you you can you can get the flash of insight and say hey look I don't really need these two trajectories at all I mean I can just give a I can just assume that all these possibilities you know exist simultaneously and a a all I have to do now is account for the probabilities then you account for the probabilities on the decision theoretic analysis my point here was that that is simply a different way of getting the probabilities. Now, you avoid all this sort of mechanism, but it's entirely equivalent to introducing this mechanism for the probabilities. There are just two different ways of ending up with the probabilities. Either you say that there are these trajectories, or you say that everything is there, and you just give a decision-driven analysis of the problems. Okay, so we have any more questions in very little time. We'll take brief remarks from several people, and then if you could answer, try to answer in one big answer. That's the only way, unfortunately. Michel is next. Yes, I'd like to... I like very much your analogy, of course, between two conceptions of quantum mechanics and the Lorenz-Einstein difference. I want to make another analogy. Here, between Einstein-Lorenz, on the one hand, and the difference between the transcendental and the empirical holistic view of proposition in the style of scientific propositions. Because I think in Einstein and in transcendental views, something, for instance here, principles such as the no-cloning principle, or in the other case, a set of stent propositions, are taken as the hinge around which everything else revolves. Principles and basic propositions are taken as presuppositions for everything else. Instead, in holistic empiricist readings and the Lorentz view, all the propositions are on the same footing. For instance all are to be explained

40:00 dynamically in one case and all can be submitted to an empirical test in the other point so I would like to ask to understand which one of these two sets of propositions can be said to be right and I think in some way they are both right but in a very different the transcendental view and the Einsteinian view of theories are right at the scale of the theory itself, whereas the other position is true historically, because you can always imagine that one day another theory comes and takes, for instance, the proposition, the postulate or the principle of former theory as something to be explained dynamically. So historically, it's true that you can change, Yet, this new theory will also have a certain principle, which is to be taken outside the cycle of dynamical justification. So I think empiricists and Laurentian are mixing up the two standpoints, the internal, the intraparadigmatic, and the historical point of view. So that was my remark. Thanks. But some of my comment was that some of the dissatisfaction I hear seems to have come from the fact that the no-cloning principle is probably just a preliminary step in explaining why the world isn't classical, because you can't really derive quantum mechanics from it. if you had some more, and that's partially because that's a negative principle, that you can't really derive specific numerical consequences from that, like to take out quantum mechanics from these other super quantum theories. So I'd like to hear what the state of the art is. I guess I'm going further and really deriving that is strong, stronger principle. So Jeff, one way to put it is that you're arguing in favor of a principle theory understanding of quantum phenomena, of a constructive theory, of understanding of quantum phenomena. But it seems to me that although that's worthwhile having such an understanding, it doesn't

42:30 free you from seeking a constructive understanding unless perhaps you're an instrumentalist. It's content with experimentalism. And I just wanted to point out that even in the context of relativity, I think that's the case. I think any realist would say, I'm not happy with just having an understanding of how the bronzing clocks behave in terms of space-time curvature. I'd also like an assurance that if I think of those bronzing clocks as collections of microscopic constituents, then I can also make sense of their behavior. And so it seems to me any realist in the context of quantum theory you should say, yes, I'm happy with the principle of understanding, but I'm not yet happy until you can show me that thinking of those apparatuses as collections of atoms also makes sense. And that seems to me to be what the measurement problem is. Okay. Okay, I'll try and answer quickly. And I'll sort of give some general answer, which I think is sort of for the spirit of all the questions. Of course, one doesn't want to say, when you come up with principles like this, that this is sort of like the last word, that somehow the principles, by formulating a theory in terms of certain principles, there is no further work to be done, and there's no constructive story. It's not the case if one wants to say, well, here's sort of a principle theory account of relativity or quantum mechanics. and it follows from that that there's no constructive account possible. Rather, I would say, of course, it could always be false. And, of course, quantum theory should always be false. And we expect it to be shown to be false and some other theory to come up. But what seems to me to be highly unlikely is that these Lorentzian, what I call Lorentzian approaches in quantum mechanics, or Lorentz's approach in the case of relativity is going to turn out to be to have in it the clues of a sort of successive theory to relativity or the successive theory to quantum mechanics. It seems that what these particular reconstructions, if you like, of relativity or quantum mechanics are doing are something that from the perspective of the principal theory account is simply, how should I put it?

45:00 I mean, well, it's not on the right track, and exactly why we don't have to expand these remarks. But it seems to me, let me just sort of say this and stop with this, that it's highly unlikely that, let's say, Bohm's theory, which was invented simply as a way of showing that, look, you could interpret quantum mechanics this way, is likely to sort of provide the clues for, let's say, a future theory of quantum gravity. It rather seems to be doing something else entirely. And so these sort of Lorentzian approaches, I think, are backward steps relative to the theories in quantum mechanics, and any future advance, which there may well be, is likely to be just something very, very different. Now, with regard to your question, what should I say here? Well, the aim of people sort of working on these approaches of trying to get quantum mechanics from principles of the sort, would be to come up with a principle on the basis of which you can actually characterize the theory. And it's not been done yet, and as I said, you would need additional principles to no cloning and no signaling. But the idea is that then you would be able to characterize the structure of quantum mechanics, and in terms of that structure, get numbers out. I mean, you wouldn't want to get numbers And that, I think, is the sort of thing that Howard Barnum and co-workers are working on. And we'll probably hear more about that from Howard. I can inject a comment. There's a whole bunch of different reconstructions that people have been doing, each of them with some successes and some drawbacks. Thank you, Jeff. Thanks again.