Subquantum Mechanics & Information

0:00 Yeah, just in front of the bus or behind the bus will be fine. Thank you very much indeed. Oh, excuse me. Thank you very much. Have a great day. Oh, I can't go over there. Well, where can I? I'm going to tell her, let her sit on the chair of the old man, yeah. No, I may need to flip around early, that's why I was thinking about the things that's very much your...

2:30 What's your excuse for doing this? Um, Mr Souter's industrial relations policy, apart from that evil. you're allowed to be made because it's an informal thing there you are that's what I should have said there's a difference between an informal and a formal thing we were in Brazil what happened is if Anthony would arrive he'd walk you ten minutes away he'd be brought in by his host or something and then five minutes later somebody would poke their head in come in here go out they'd poke their head in It's a bit like fixed point theorem, isn't it, really? Eventually you reach the stable leader. And then you've reached the critical singularity at which you can actually start the seminar. That's the solution they always use in Florence. I went to a seminar in Florence about three years ago where only two people turned up, which was the space for an extremely distinguished speaker. So we just adjourned for the restaurant and spent the whole of the rest of the seminar over lunch. He gave us the seminar over lunch. He was quite a bit funny. Well, we're going to make a start now. Yeah, we're going to do it again. It's going to go on through.

5:00 Thank you. Well, I'm going to be talking about some possible new physics beyond quantum mechanics that one can think about in variables theory. but before I do that I'm going to have to give some background and motivation for why I'm thinking about these things but probably you're all familiar with this background, so maybe I'll just skim through a few things. The background and motivations, while there's one paper I have on quantum pH, I'm also going to be talking about what I might call the non-local flow of sub-quantum information. There's also a paper. And then I'm going to get on to what is really the main topic of the talk, some new stuff, which is not published or on the archive yet, which will involve sub-quantum measurement, eavesdropping on quantum key distributions, and reading all the results of a parallel quantum computation. So, well as most of you probably know I've been going on for years about the idea that the physics that we see is really the physics of a particular

7:30 equilibrium distribution, and the initial idea for this was that it's sort of odd that in modern physics it seems that there is a fundamental non-locality, but you can't get your hands on it and use it to send a signal. Some people might be happy with that, it's just the laws of physics are like that. It always seemed to me that this smacks of the conspiracy, that there's something going on behind the scenes, something on local, and because of this quantum noise, heuristically speaking, you can't actually use it for anything. It seems to me that this couldn't be a law of physics, it's some sort of contingent property of a particular statistical state. So there's this analogy in which, I'm sure most of you have seen before, is it in focus? Reasonably? Yeah. So this analogy, what I'm saying is the physics that we see is analogous to in classical physics a state of thermal equilibrium in which the whole universe has reached a single temperature. And if the whole universe is this famous heat death of the universe when the stars burn out One day, perhaps everything will reach the same temperature. You will have this Boltzmann rule, a particular temperature which tells you probabilities. There will be a universal thermal noise in all systems. And while there would be a huge amount of thermal energy, it wouldn't be possible to turn any of it into work. One couldn't do useful work. You need differences of temperature. This is a heuristic analogy. If you look at our quantum physics, the system with a wave function side of these probabilities and it's supposed to be a universal law you have this uncertainty noise everywhere and it seems that there is a non-locality fundamentally but you can't actually use it for signaling the idea is is that this this side just like that side these are not I mean this is not a fundamental principle you cannot turn thermal energy into work

10:00 It's just a contingent feature of a state in which all systems are at the same temperature. And I claim that this feature that you can't use non-locality for signaling is similar. It's just a contingent feature of this quantum equilibrium state. So the claim is that an analogue of the classical thermodynamic heat depth has actually happened in our universe at the hidden variable level. conspiracy between quantum theory and relativity, that there's non-locality that you can't use it in the signal. It's not a conspiracy in the laws of physics, it's just a particular feature, a particular equilibrium distribution. Now, I started working on the Brogdon theory really as a way of having a concrete model of this scenario, and I tend to more than I believe the details of de Broglie-Bohm theory so everyone here knows basics of de Broglie-Bohm theory there's no need for me to say anything in de Broglie-Bohm theory well there are two particular things one is one can develop a sort of statistical mechanics analogous to classical system which I mean I mean, I've just flashed this at you, there were papers here years ago, and I can't believe how old I am, anyway. But basically, there's an analogue, one can develop an analogue of classical statistical mechanics in which one can justify this p equal psi squared as a sort of equilibrium distribution, analogous to thermal equilibrium. and there's another point which I've talked about many times, but I'll just remind you again in the Broeboehm theory so there are two things I'm claiming, one is that we happen to be in a certain state of statistical equilibrium and the other thing I'm saying is it is for this particular state that you can't get your hands on the non-locality and send the signal and I showed again In my young days, this thing in de Broglie-Bohm theory where I had two boxes, a particle in each, there's an entangled state. Of course, in de Broglie-Bohm theory, there are definite positions for the particles, and if I do something here, the velocity of the particle over there responds instantly.

12:30 but at the ensemble level if I have an ensemble of similar systems with the same wave function and of course a distribution of particle positions some initial distribution which need not equal psi square then a sudden change over here does affect the marginal distribution over there so in other words I do something here each system over there responds but the statistics over there don't respond unless I'm out of quantum equalification. Now, so that's just a brief recap of where I'm coming from. So here's on to something that I've done more recently. So we showed in pilot way, well, see, I said to you that I believe the general picture more than I believe the details of the Brueggeum theory. So one thing that would be nice to prove is, well, this signal locality theorem was proven 10 years ago in pilot wave theory. You get instantaneous signals at the statistical level if and only if you're out of equilibrium. Recently I managed to show that this is true in any deterministic hidden variables theory. So, this picture that it is the quantum noise that masks the non-locality, and this argument that this suggests that we just happen to be in a certain equilibrium state, this argument can be made in general. It doesn't depend on the Breuven theory. I'm going to go through the proof in a second. I'd just like to remark that it's interesting, this historically, that the Breuven theory and played this role. Originally Bell looked at the Breuven theory, saw it was non-local, and asked does any hidden variables theory have to be non-local? And there is a general belief that he proved that's true. It's also the case that the Breuven theory has its property of contextuality, that in general the

15:00 outcomes of quantum measurements don't reflect the value of something that existed beforehand. This quotient specter tells us that this is true of any hidden variables theory. It seems that there is now another property of pilot wave theory, this signal locality theorem, that is true of any or at least any deterministic hidden variables theory. And one wonders what other properties I suspect that there may be other things which are universal. Anyway, let me now just go through this proof. It's a very simple proof, really. Just sort of set up Bell's theorem as he originally formulated it. So two spin-half particles of A and B. I can make measurements along arbitrary axes. I'm going to take the axes in the xz plane, so the axes are just specified by an angle of each wing. I have the singlet stage, and I measure spin components, units of h par over 2, so values plus or minus 1 at each wing. Okay, we have this usual quantum correlation between a and b. Now, hidden variables theory, deterministic hidden variables theory Well, if you try and make a local one, you say that the outcomes are determined by Given the settings at A and B, and given the hidden variables lambda The outcomes should be determined And if you assume equations are local, the outcome there doesn't depend on the angle over here and so on um and if you assume well now you assume of course fundamentally the theory is about individual systems if it's deterministic hidden variables theory it says given certain settings given lambda there are some equations that determine the outcomes and that's the whole theory really fundamentally but it is then assumed that for a certain distribution of lambda, which is usually called rho of lambda, and I just added a little subscript EQ for equilibrium, a certain distribution of lambda reproduces the quantum statistics of the ensemble level, okay, so this idea, this expectation value, can it reproduce the quantum correlation, and Bell's theorem shows

17:30 that no, you have to have some non-local equations of this form, where the outcome of A depends on the angle of B, and vice versa. I mean, well, or at least one of these. I mean, it could be that the non-locality is just one way. But we won't worry about that. So now I want to prove to you, let us say let us say I've got now a deterministic hidden variables theory of this form. It might be pilot wave theory. It might be something completely different. But whatever it is, I want to prove, let's assume, so what is the assumption? The assumption, there are these equations, and there is a distribution of lambda, an equilibrium, I just call it an equilibrium distribution, it's just a distribution of lambda that does reproduce the quantum correlation. Let's say you've got that theory. What I'm going to prove now is that if you give me this certain distribution of lambdas that reproduces quantum mechanics, theoretically, I can always contemplate a different distribution. I could just pick out certain values of lambda. Just mathematically, one could contemplate, within the context of the given theory, a distribution that is not rho equilibrium. How are you prepared in practice is another matter, but just hypothetically. And what I want to say is, if you're given a theory like this, then out of equilibrium, you will get instantaneous signals at the statistical level. And the proof is very simple. If we, if you consider, so we consider an ensemble of experiments where the settings are fixed, theta A and theta B, and I've got a certain distribution of lambda. So there's an ensemble of experiments, the same settings. Lambda generally differs from one run to the next. But the whole ensemble satisfies the quantum statistics. Now, for each run, there's a specific lambda that determines an outcome at A. Let's think about the outcome over there. okay given the settings given lambda the outcome over there is determined now some values of lambda are going to give

20:00 sigma raise plus one some will give sigma minus one question is what happens if i change the angle now here at b what happens well i mean the angle is really almost trivial but just to formulate it carefully just think about the set of possible values of lambda. You could partition it in two different ways. One way is to say that with the original settings, these are the lambdas that give plus over there. These are the lambdas that give minus over there. With a new setting, shift the setting here, these are the lambdas that now give plus over there. These are the lambdas that give minus over there. Now, it cannot be that these partitions are the same, right? It's a trivial thing. It can't be that for an arbitrary shift in angle, the lambdas that give plus over there still give plus, and the lambdas that give minus over there still give minus, because that would mean you have a local theory. There would be no dependence on this. So there must be some non-empty, some intersection here, which, I mean, just in words, it's a trivial statement, I'm just saying simply that under a shift in angle here there must be some runs where a minus flips to plus and a plus flips to minus Now, of the equilibrium ensemble so this set here, this intersection is just the set of lambda that give minus originally and give plus with a new setting so this integral is the fraction of the equilibrium ensemble that makes the transition from minus to plus, when I do this shift. Similarly, there's a fraction that does the transition from plus to minus. Okay? So now, and here is really the key point that makes the proof, is a very simple point. One thinks now, well, with the original settings, assuming this theory produces quantum mechanics, 50% have to give plus and 50% have to give minus over there. With the new setting, again, there has to be 50% plus, 50% minus over there.

22:30 So what does that tell you? That tells you this condition of detailed balancing, that the fraction of the ensemble that has flipped from minus to plus must equal. the fraction that flipped from plus to minus to keep the 50-50 ratio the same. So, for example, just as a sort of simple geometrical picture, I mean, if, say, for instance, this is the set of possible values of lambda, and say that the equilibrium distribution is uniform on this circle, and let us say with the original settings, this half of lambda gives minus over there, this half gives plus with a new setting I rotate the angle a bit let's say this changes and now this half gives minus, that half gives plus so what happens? This area in here is flipped from minus to plus this area in here is flipped from plus to minus the areas are equal, equilibrium measures uniform so the 50-50 is maintained now the point is but in this theory so these transition sets these are the sets of lambda that flip from minus plus or plus to minus in equilibrium they must have equal measure to maintain this balance but it is clear that these sets are determined by the underlying deterministic theory those equations, sigma is a function of theta theta d lambda determine these sets okay the actual distribution of hidden variables is a completely independent mathematical physical entity so in other words for instance in this example if I had a row of lambda that is not uniform then clearly in general the measures of these sets will not balance so what you will have is that the ratio, for instance I might have a non-equilibrium ensemble where with the original settings, over there I get 60% plus, 40% minus. When I flip the angle here, maybe it will change to 70% plus, 30% minus. There will be a signal at the statistical level.

25:00 So, you know, these transition sets, they're fixed by the underlying theory, are independent So that's it. I mean, very simple proof. The action of B in general, of course, there will be some special non-equilibrium distributions that happen to have equal measures on those sets, but in general, you will cause an imbalance. So that's the theorem. It is true for any deterministic hidden variables theory certain distribution hides the non-locality from viewers statistical level and then in general the deviation from that distribution will make it visible at the statistical level which seems to me to strengthen this this scenario that we happen to be stuck in the certain equilibrium state just briefly remark in while the paper where I've got this proof there's some other stuff about, um, one can look at, if one looks at, this is the fraction of the ensemble that flips from minus to plus when I do this, plus the fraction that flips from plus to minus when I do this. So it's the fraction of the ensemble for which the outcome over there changes, irrespective whether it's plus, minus, or minus to plus. You can interpret this as the average number of bits of information per single pair that is transmitted non-locally from B to A. If one thinks about, imagine some demon who says he knows lambda, and he knows that if he flips the angle here, it's going to affect the outcome over there, and he decides whether to flip or not, to send to his body over there a signal. For the equilibrium ensemble, there's a certain fraction for which there is a change in response to this flip. So this is sort of the average number of bits of information that are propagating, as it were, right? It's a fraction of the ensemble for which there is a flip over there. Now Bell's And similarly you can find beta As a sort of, I call it, degree of non-locality

27:30 From B to A A fraction of the ensemble that flips over there One can define something in the other direction I flip the angle over there What fraction here changes This sum I mean Bell's theorem says You can't, alpha plus beta Cannot be zero for all angles You have a local theorem And I wanted to derive some lower bound on alpha plus beta. I managed to do that only by assuming certain symmetries, as in this paper. Assuming certain symmetries, you get results like when I flip here by 90 degrees, at least 25% of the systems over there have to change their disposition, that kind of thing. but it's, I'm trying, I suspect that if one looks at the average overall angle one could get at a lower bound without any symmetry assumptions but I'm still working on that anyway, so conclusions so far conclusions I draw anyway from this okay, so rho is psi squared is an equilibrium distribution analogous to thermal equilibrium in classical mechanics there are instantaneous signals at the statistical level if and only if you're out of equilibrium I mean this is a sort of physicist if and only if I mean there are some special non-equilibrium distributions where those two sets happen to have the same edge and get a signal but we won't worry about it so I mean the message is that there's a lot of new physics hidden behind this quantum equilibrium noise. Now, one might think, and here I'm always going to start skimming again, because I've talked about this before here, but when I gave this talk in Calcutta, if I'd stood up and said, imagine we have matter that violates quantum mechanics and look what we can do with it, they would have thought I was completely crazy, so I had to try it. of course I am a bit crazy but anyway but is this new physics forever inaccessible right I mean you're talking about these things okay we're stuck in equilibrium but if we're forever stuck in equilibrium you can never see these signals you can never use it isn't it all just metaphysical so and so on well there is this possibility

30:00 which I'll just very briefly remark on perhaps which seems natural mind, at least in this scenario, that now we're in this special equilibrium distribution where things balance and you can't use the non-acality in practice. But perhaps the universe started out in a non-equilibrium state and there was a relaxation to equilibrium soon after the Big Bang. And there are two possible effects, observable effects, this could have, which we've been working on for some time. One is Relationary theories of cosmology, at very early times there was a quantum, a vacuum state of a certain scalar field, maybe the Higgs field, the inflaton is usually called, and it is believed that the quantum fluctuations of this field generated primordial energy density perturbations that seeded the gravitational condensation of galaxies at later times. And that the imprint of these early perturbations is visible on the microwave background in the form of variations in temperature across the sky. Now, inflation predicts a certain spectrum for the statistics of these temperature fluctuations that are being measured. which so far agrees very well and the way it's derived is the spectrum is derived by assuming quantum vacuum fluctuations at every time in this scalar field now of course the point is if the quantum probability is P is psi squared was not P equals psi squared maybe if you had say the same quantum state but a different formula for the probabilities this immediately has a knock-on effect on the predicted spectrum and so that is one way which this might have some observable consequences. Another way is that if this relaxation occurred at very early times, then particles are decoupled very early, and it is believed there are such particles, some supersymmetric particle, gravitons, or all kinds of possibilities. They may have decoupled sufficiently early before they had time to relax to equilibrium and could still be floating around now in space

32:30 and people are looking for things like dark matter which is often thought to be made up of some sort of exotic particle that decoupled very early and now just has gravitational influences. People are searching for this matter. So this matter, it might still be in a state of quantum non-equilibrium with probabilities different from psi squared. Now, should I, um, um, yeah, there's no real need to, well, maybe I could just say that this, so there's this sort of idea that one might have a very distant source from the early universe of some particles freely propagating. the green line is their wave function spreading and the red line is rho, which is different from psi squared on small scales what it turns out in pilot wave theory, I won't bother you, I've talked about this before but as this packet spreads, the length scale on which you have these ripples also gets expanded and so even if these ripples are on tiny scales, because of the large distance and this packet is spread essentially freely, you can get an expansion, and you could have some sort of blurring of the interference pattern. So, but the basic message I want to put across is there is some plausible possibility of, that there might exist in the universe today, matter that is in a in a sense a different form of matter or in a different phase if you like to think of it there's a different phase of matter matter where the probabilities differ from the quantum probabilities so now I get on to what is more more really what was the subject of this talk is what could one do with such matter? I mean, the matter itself, let us say it was, you've got a big cloud of hydrogen atoms in the ground state. If the end of psi squared looks like this, you might have the actual distribution of electrons is different from psi squared. So you measure it, and it violates quantum theory. But what could one actually do with this matter?

35:00 well, so non-equilibrium matter might be left over from the very early universe I'm going to show that one could use it to send signals faster than light which is not what we've already seen but I just want to go through exactly how one would do it you can use it to perform sub-quantum measurements what I'm saying is let's say I have a box where I've got some of this matter what could I do with it I've got ordinary systems here in quantum equilibrium. I could perform measurements on these ordinary systems using this material as a sort of apparatus, if you like. But the first thing I want to talk about is, well, okay, you've got this matter that may be out of equilibrium, but how do you determine its distribution? What does it mean? How do you know what distribution it has, and how would you actually use it? the idea is I have let's say I have a big cloud of matter of atoms that I suspect are out of equilibrium what I do is I take a random sample let us say they're all Heisen atoms in the ground state I take a random sample I measure the positions of all the electrons and I compare the the measured distribution with what is expected from quantum theory. And if there's a significant deviation one can calculate well, it's just the usual sort of sampling theory. If I choose a random sample of the population and I ask them whether they vote in Labour or Tory You know, I've chosen 10,000 people And 70% say they voted labor And by statistical analysis Of course it's possible that that was just a fluke Maybe only 50% voted labor I've just got a skewed sample But from the central limit theorem One can calculate the probability That that is just a fluke So for example, let me give you a concrete example I've got a large cloud of hydrogen atoms in the ground state. In this simple case, each electron in pilot wave theory is at rest. This doesn't actually matter in this thing. One can just do the measurements quickly.

37:30 It doesn't have to be static, but just as a simple example. So there's a static distribution of electron positions which may or may not equal psi squared. If the cloud is in equilibrium, this is the predicted distribution, but perhaps it's not in equilibrium. do you to test this you draw a random sample of atoms from the cloud and you measure the positions of the electrons so I now have a sample a large sample but a sample small compared to the total number of atoms I use this to make statistical inferences about the parent distribution particularly with an estimate the likelihood that rho is actually psi squared. Now, for example, one could look at the sample mean. If I look at the sample mean, now, the parent distribution, I thought some parent distribution rho, now, there's a theorem that if the parent distribution has a mean mu and the variance sigma squared, then if I take a random sample, for a large sample, this random variable, the sample mean, has an approximately normal distribution with mean mu and variance sigma squared over n prime, the central limit theorem. So in other words, this mean is likely to equal the parent mean, but in general there are small deviations to Gaussian distribution. one can calculate the probability of the sample mean differing from the parent mean. So if your sample mean differs from the parent mean in a way that is the odds, what is the probability of getting that result with an equilibrium parent distribution? And if that probability is 10 to the minus 200 or something, then you're going to draw the conclusion that something is wrong here. Another way of looking at it, one often thinks in terms of these things that are called likelihoods. This is strictly the probability of getting that sample mean given an equilibrium parent distribution. This usually interpreted as the likelihood of this distribution, given the sample mean that you see they're not really probability.

40:00 Anyway, one could think about chi squared z. There is actually nothing, this is really all trivial stuff, just like, you know, in chemistry I have in some matter, I take a random sample, analyze it, and you can be pretty sure that the random sample the distribution of the sample as a whole it's a trivial thing, the only novel thing is that I'm talking about well, it's just a different sort of context but anyway, so now so let's say I don't take my random sample I've got estimated the likely distribution so now I've got all the other atoms that I haven't touched yet I haven't disturbed them I know almost certainly what their distribution is and let's assume it's not okay so I have some known quantum non-equilibrium distribution what can one do with it now we know that thermal non-equilibrium and chemical non- equilibrium is very useful what could one do with quantum non-equilibrium well for a start and I won't bother getting details here because I've talked enough about the signaling but if there were these cloud of atoms I could pairs, right, and prepare each in an entangled state. Given the initial probability distribution for the single atoms and the details of the preparation process, one thinks each pair, there's an evolution from an initial product state, and I switch on some potential between them and do something to generate this entangled state. so I know the evolution of the wave function I know the de Broglie-Bohr velocity field I can calculate the evolution of the joint distribution from this product of known factors to something which in general will not be psi squared I'm just integrating the continuity equation I mean all I'm saying is given this non-equilibrium cloud and I know the distribution I can pick pairs, entangle them and I can end up in a situation where I know the entangled state and I know the joint probability for the pairs. So once I've got that, I've got this situation like my two boxes where I've got a distribution of pairs that is not psi squared

42:30 but now I know what the distribution is so I know what the marginal distribution is over there. So when I do my shift here, if the marginal over there changes then the guy over there knows he's got a signal. one could use this matter to send non-local signals another thing one could do sub-quantum measurement again I assume these particles these non equilibrium particles let's call them apparatus particles okay assume they have a known wave function and the known non-equilibrium distribution known from what I've talked about, the sound plane and what it is, some weird particles left over from the early universe, whatever it is, but we've got them. Now, if, let's first look at the case where this non-equilibrium distribution is very narrow, stringing out of a very narrow Gaussian or delta function either, it is very narrow, if it's arbitrarily narrow I've got these particles this is a very narrow distribution but of course the wave function is not narrow it's the rho that's narrow I'm now going to use these here I've got some ordinary matter in quantum equilibrium better say this ordinary matter it's an ensemble of particles with wave function sine or and the usual distribution rho is sine squared I'm going to use these particles to measure the positions of these particles here without disturbing their wave function, right? Break the uncertainty principle. So, simple example. I switch on an interaction between my non-equilibrium particles and the equilibrium particles. This is the usual sort of von Neumann ideal measurement of the position of this particle where the position of my non-equilibrium particles is acting as a pointer, right? non-equilibrium probably have position y and their positions are acting as pointers so I have this simple Schrodinger equation I mean this is a sort of slightly odd

45:00 I'm neglecting the rest of the Hamiltonian because we'll see the Broebbin theory that results from this is a bit odd you don't get velocities equal to gradient of phase but it doesn't matter it's just a simple example the point is From that Schrodinger equation, you have this continuity equation for psi squared. Okay, the hidden variable, x dot is the velocity of the equilibrium particles, y dot is the velocity of the non-equilibrium particles, must satisfy this. So I get these guidance equations, which sort of look a bit funny. It's just x dot is 0, y dot is Ax. And I've got these de Broebbels trajectories. What is it saying? the equilibrium part, they just remain at rest and the pointer particles move in a way that depends on what the initial position of the equilibrium particle is. So now, let us say I have an initial wave function which is known. The equilibrium particle has wave function sin0, non-equilibrium particles have wave function g0. This evolves into this. okay, we need to check from the Schrodinger equation that predicts this evolution I mean, this is an entangled state now, of course it's not a product of X and there's an X in the X in, let's look at the limit where, of very short times or a very weak coupling A is this coupling constant in Hamiltonian in the limit, where that is very tiny, the wave function is essentially what it was at time zero, okay to arbitrary accuracy, as this is made arbitrarily small. Okay, so the sine-orps of these equilibrium problems is undisturbed to arbitrary accuracy. And yet, no matter how small A-T is, at the hidden variable level, the point of particle has shifted a tiny bit, a tiny bit, but nevertheless a finite amount, depends on the actual position of the equilibrium particle. so the hidden variable thing here contains information about the value of the initial position of the particle so now, and this information will be visible to us, provided

47:30 the non-equilibrium distribution pi or y is sufficiently narrow, in fact one can show I've got this initial joint distribution for X and Y which is equilibrium for the X particles and this arbitrary non-equilibrium for the Y particles and I assume this is very narrow but anyway, in general, I start off with this the continuity equation predicts this evolution of the joint probability density now, if the non-equilibrium density is localized the 0, some range of y, width, w, it's concentrated in here, 0 outside there, then the joint probability is non-0 only if x is in this certain range, okay? So, if I now measure y, the y-particle position, I can deduce that x is equal to y over 80 plus or minus, so there's this uncertainty, and it just corresponds essentially to the of this non-equilibrium it's a simple thing really so the point is, no matter how small, of course I'm saying AT is very small because I want to the wave function here to be disturbed hardly at all, okay, but no matter how small AT is if W is sufficiently small this error margin can be made as small as you like so an arbitrarily narrow non-equilibrium distribution enables me to measure position of an equilibrium particle without disturbing its wave function couple of remarks I mean one could think about the sequence of such measurements so one could determine the particle trajectory of an ordinary particle without disturbing its wave function I talked about this extreme case where the non-equilibrium distribution is arbitrarily narrow, what happens for a finite width, when I've got this finite width, if the finite width, sorry, if the width of this non-equilibrium distribution is smaller than the position spread of the wave function, you get an improvement over the quantum measurement

50:00 in the sense that I can say, for instance, particle is on the left side or the right side of the initial package. In quantum theory, we'll just say, well, it's 50-50, I have no idea. Whereas here, you could get an improvement on the odds. I could say, well, it's 80%, it's on the left. You know, and I will have a statistical advantage over quantum measures. On the other hand, if the non-equilibrium, it could have a spread that is larger than psi squared. It could have psi squared and my non-equilibrium has a bigger spread in that case the measure would be even worse I have even less information so anyway yeah I mean if you before you go to the next yes yes I didn't understand why are the yes because the Hamiltonian you have what I did is I took the total Hamiltonian should be I'm going to have to be careful here because I've gone and written these damn slides with non-permanent ink and if I touch them I get this much this interaction Hamiltonian which, okay, see this is just one if I take that to be the total Hamiltonian this is the Schrodinger um... you... how can I put it... the point is, to say that x dot is given by the gradient phase is something that comes out if I have the usual sorts of Hamiltonians that you deal with, P squared over 2m plus a potential, that kind of structure if you look at that Schrodinger equation and then you get the continuity equation from that you find that grad S is the natural velocity field appearing in the continuity equation but if you have a Hamiltonian with a different kind of structure the total Hamiltonian looks like this looks like that and the continuity equation the size squared simply looks

52:30 like this right so now I mean the fundamental idea of pilot wave theory in general is that quantum theory says there is a local flow of probability in configuration space, that suggests there are actual variables there that are evolving. And so one identifies a natural simplex velocity field, which in general, it doesn't have to be grad S. In fact, if you look at, for instance, another example, if I take a Hamiltonian, which is a von Neumann measurement of momentum, so I've got a coupling constant multiplied by the momentum of X times the momentum of Y if you look at that Hamiltonian and you go through this you find that X dot is given by the derivative of s with respect to Y and Y dot is given by the derivative of s with respect to X it's um I mean in that case actually in that case the momentum canonically conjugate to x is actually y dot. You get this formula reversal. So I mean, there's no... I guess I'm just confused here because I thought you were allowing for... I thought that you were sticking to the usual guidance equations, allowing for a non-standard distribution, non-equilibrium. The thing is that this is just a simple model that one can calculate with. One could do this in, say, the more standard theory with standard Hamiltonian, where you have the usual guidance equations, the usual Schrodinger equation, and one has non-equilibrium. It's just that, I mean, if you're talking about wave packets spread and it's just a bit more messy, who's chosen this example because by neglecting the rest of the Hamiltonian, the wave packets don't spread. It's just a simple model. I see. It's not necessary to... Sorry, I should have underlined it.

55:00 Oh, okay. Sorry. So, I mean, one can think of ordinary non-relativistic quantum mechanics, particles moving in a potential. I have the usual pilot wave dynamics. If I had, here is my particle, let's say it's a harmonic oscillator or something, in quantum equilibrium, here I have some free particles. They're out of equilibrium. They have a very narrow distribution. I'm just saying if I couple these, then if this distribution is sufficiently narrow, I will be able to see, even with a very small coupling, so small that this wave function is hardly disturbed, at the hidden variable level there is nevertheless a small motion here, depending on where this position is. And if this distribution is sufficiently narrow, you'll be able to see it. Just here is a very simple example where one can calculate exactly, analytically, the whole interaction and show it's just a convenience. Could you share that slide again, because I just missed the end of it, how we can still see how the points have passed, we'll still tell us what's going on, despite the original way of action. Yeah. Um, so, uh, um, so, so, so, so, this, this, this, this, here. And then you can, I mean, you can calculate exactly the evolution of the initial joint distribution. and quantum goes like that you get this business here but you know, no, it's got nothing to do with any quantum interaction the one way of looking at it, if you like, is this interaction Hamiltonian, you could say, I'm considering such short times that I can neglect the rest of the Hamiltonian it's just a toy model get the same I mean I think the gist of it is it's a it's a general point really so it's possible so so I've got this business anyway that I can do I can

57:30 measure I can use my non-quantum matter if it's distribution is sufficiently to beat quantum measurements to measure the trajectory without disturbing wave functions on it. So now, what can one do? Well, the first thing one can do, in standard quantum mechanics, it's not possible in general to distinguish non-orthogonal quantum states. This is usual standard proof of why it's not possible to do that. Here, if I could monitor the trajectory without disturbing the wave function, In general, two different wave functions, even if they're non-orthogonal, the trajectories will differ. So I can distinguish non-orthogonal quantum states. Why is that useful? Well, if I want to eavesdrop on quantum key distribution... So, quantum key distribution. I've got Alice and Bob who want to share a sequence of bits, a key. The sequence of bits mustn't be known to anyone else, okay? These bits are supposedly generated by some random quantum outcome. And there's an eavesdropper who's trying to read what these bits are without anyone knowing. so there's three famous protocols here BB84, EPR and B92 are known to be secure against classical or quantum attacks if Eve possessed this non-quantum matter she would be able to eavesdrop let me just show how that B is quite straightforward first look at the BB84 and B92 protocols both rely on the impossibility of distinguishing non-orthogonal states without disturbing them. Okay? So, I mean, I don't know if you're both familiar and there's no point in really going through these protocols, but well, let's say the BB84 in Alice sends a random sequence got a spin half and it can be plus Z, minus Z, plus X, or minus X sends a random sequence of these bits to Bob Bob randomly measures spin along Z or along X.

1:00:00 In each instance, they publicly announce whether Alice sent one of these or one of these without saying which one it was, but the Z or an X. And Bob announces whether he measured along Z or X, but he doesn't say what the outcomes are. They discard instances where the choices differ. In other words, they only keep it when it was both Z or both X. and these results, so they will coincide with each wing if they've sent you well, at each wing, I mean, if Alice has sent one of these and Bob has measured along Z, then he gets a plus or a minus Alice knows which one she sent, Bob knows what outcome he got so they've each got a common string of bits similarly, the B92 is that an eavesdropper, Eve, if she's trying to tell which state Alice is sending these states, she tries to see which state is it she's sending, make some measurements on these states, you will create some sort of disturbance that will upset this correlation, which if you test a random sample, Alice and Bob will become aware that someone's been disturbing Similarly, B92 shows a similar thing if I have a random sequence of non-orthogonal quantum states that Alice sends and Bob randomly measures either of these projectors it's a similar kind of thing The basic point is that the eavesdropper trying to distinguish these two non-orthogonal quantum states will inevitably disturb them but this will show up Now, just a trivial remark, that Eve, with a sufficiently narrow distribution, would be able to distinguish, to arbitrary accuracy, without disturbing these states, could look at the trajectory and tell which state it is. okay so E will know which states Alice sends and will therefore know Alice's key and Bob's key without anyone without anyone suspecting a thing what about the EPR protocol Eckhart

1:02:30 I think it's interesting the way Eckhart praises it completeness of quantum theory as he puts it, no hidden elements of reality there's this quote the eavesdropper cannot elicit any information ah, sorry, I should say you have pairs of EPR particles in the singlet state, one goes to Alice one goes to Bob they perform measurements along random axes, X or Z they announce which axes they use but not the results they discard the instances where the axes And if there's no eavesdropping, the remaining instances where they measure along the same axes will be perfectly anti-correlated. And you could test a large random sample to check that there's no disturbance and so on. And Eckhart says, so the eavesdropper cannot elicit any information from the particles while in transit because there is no information encoded there. In other words, the sort of information, these bits come into existence when the measurements are performed. As long as quantum theory is not refuted as a complete theory the system is secure. So now as you will know in pilot wave theory can measure the particle positions without disturbing the wave functions then in say a Stern-Gerlach when I measure the spin, if I know the position of the particle in the wave packet I can predict the outcome. So RE will while these balls are coming out, will measure the particle positions without disturbing the wave functions and can then predict the outcomes of spin measurements that Alice and Bob get and so can predict the key that is generated at both wings. So, non-quantum matter would be used for espionage as well as for sending instantaneous signals. what else could it be used for it could also be used in quantum computation there was this original intuition you have a state vector representing the state of the computer and it can be in a superposition of in effect

1:05:00 it's like it's having a superposition classical computers in different states, and I have these parallel classical-like computations going on in different branches, of course, unfortunately, you can't then access all the results. They're all there, in a sense, in the state vector. When you make a measurement, there's a collapse, and so on. But in quantum computation, by cleverly superposing branches, interference effects, being clever one can do some things that you couldn't do classically or would be much harder to do classically but let us say the full power of the original intuition it's somehow frustrating because in a sense there's all this parallel quantum computation going on but you can't get your hands on all of it it turns out if you had non-equilibrium matter if here's my quantum computer and it's done all these different parallel I would be able to read the results of all these different computations. Let's give a simple example. Let's say I have a single particle as a trajectory guided by a certain wave function. We put m and h bar equal to 1. Let us say the particle ends up in a superposition of n energy eigenstates. these could be the states of an atom or a particle in a box or whatever the point is there is a set S of quantum numbers let's say, I don't know, energy states going from N is 0, N is 1, N is 2 up to N is infinity, an infinite tower of energy level now the results of a computation could be encoded in the eigenvalue or quantum number. So say I've got a superposition of the 15th energy level, the 231st energy level, the 2891st energy level. These numbers are the outcomes of computations, okay? And they're all there in this superposition, right? Now, if I measure the energy in quantum mechanics, well, I just get one outcome, and I say, well, I thought I can only access one of these results. Can I access all of them? Well, the point is, if these energy eigenfunctions overlap in position space,

1:07:30 the trajectory of the particle, in a sense, reads all of these functions, reads it. If you look at, basically, by looking at the trajectory, The trajectory depends on all the energy states in the superposition And if I take n pairs of values of position and velocity at n times I can deduce the values of energy in the superposition Let me just give a simple example Say if I just have two states, for instance so I've got a particle in a box and I've got two well I know what the energy eigenfunctions are for the system but I've got a superposition of two energy levels I don't know which energy levels they are Ea and Eb but some superposition the velocity here given by this horrible looking formula but the point is if I knew the value of x and x dot at two different times I could solve this equation for A and B, and deduce what the two energy levels are. So, if I had an exponentially large number, I mean, here now, if I was thinking about, well, I can extract, for this superposition of many states, knowledge about what states are in the superposition I can extract all the results of the quantum computation if there's a situation where the number of these states is exponentially large then you get the famous exponential speed up in general it's not just for some funny particular situations so my final conclusion is that again we are we are stuck in this equilibrium state, there's this quantum noise like a fog that hides what is going on underneath from us. The point of

1:10:00 this particular talk is to show that this is true, that there are immense resources are hidden from us by this noise. Resources for non-local for espionage and for parallel computation now we're unable to get our hands on these resources for as long as we're trapped in this heat-dead state so it might be important to search for this quantum non-equilibrium for particles from the very early universe or whatever, if we could get our hands on this matter there will be some interesting applications here's just a few references and that's the end that's it, thanks very much applause applause Well, I'm a bit worried about the thermal analogy, I suppose, but perhaps for detailed reasons, I won't go into it. It's only a heuristic analogy. Well, but nonetheless, if you take the heuristic analogy and push it, then it's true that in classical physics, measure something, the best way to do it is to cool your detector to near absolute zero, who wants to make it that parallel low temperature. So locally there is a low temperature here, and the object you're measuring, well maybe cold will not be, whatever, so you switch or whatever. um but of course that is a um a thermal a thermal state in which you've got a low temperature here and high temperature overalls in non-equilibrium but it's a very degrading sort of state i mean the moment you turn off the electricity maintaining that system in a refrigerator it's going to come back to for living very quickly. And it seems to me that that is a situation you would have to expect of a non-evolution state, that it would degrade. Now, second point, and I'm very comfortable sort of pushing the analogy in the same way.

1:12:30 Well, okay, I mean, just through that, I mean, at least in the pilot wave case, we know it is possible you can isolate particles. If there's no interaction Hamiltonian, these particles, the equilibrium, if they're in a simple, there's no complicated interaction, they'll just remain there, say particles in the ground state of hydrogen. Right, so that's also a pretty classical system, that if I ever have a system that is refrigerated to minus 10 degrees, whether it's translated or not, it's always going to be there, but of course it's going to exponentially come close to the same temperature as everywhere else the same thing is true if you're going to use communication if you're going to use heat to communicate right you have you have two objects suppose a temperature higher than the rest of space, then, and those are T, perhaps a negative, in the moment they're restricted to a given region, each one, in the moment you release the restriction on those, as soon as well, as soon as you, as soon as you let it rocket, then you're going to have exponential tails on both of these, and you're going to be able to detect instantly using the heat, but of And of course you're going to have exponentially bad distance, with distance, you're going to have exponentially less capability. Well, but that's the thing, if you're going to use the analogy with thermal states at all, then it seems to me you have to answer those reservations. I mean, there are differences. I mean, the analogy is just not really used in any of the actual technical applications. Well, except that when you made... The only real point that I'm saying is that the quantum statistics that we see are a certain equilibrium distribution with special properties in the same way that thermal equilibrium classically as a special distribution with special property. There's no, you know, not using any arguments from thermal physics here. We're just using the hidden variable steering structure. But there is an analogy in the specific modeling structure.

1:15:00 You had, your detection system had relatively small oscillations. and that is precisely analogous, or even more precisely, it's supposedly analogous. In pilot wave theory, if I have, say, particles in the ground state of a box, and this is the wave function, and if the particle is predicted to be at rest, so if there's a narrow distribution, it will just stay narrow if this system is isolated. it might be that in other hidden variables in general a non-local hidden variables theory might have the property that particles are always interacting non-locally with everything else in the universe and it's impossible to isolate then you would have these kinds of problems maybe I've got this non-equilibrium but it evaporates very quickly because of this non-local interaction pilot wave theory has the property that if I manage to isolate a system in the sense that it has a wave function that is separated off from the rest of the universe then if it's a simple system that one or two degrees of freedom has some simple dynamics and there's no reason why it should relax but now you're saying that is isolated from the rest of the universe but if you don't have a means for isolating at the quantum level But we do, I mean we do this every day to a certain approximation. I mean you prepare an atom in a certain quantum state for a certain period of time, of course there are some interactions. Well, it's specific. I mean, in order to maintain a quantum state in a given state, you do have to make sure that the environment you keep it in, it does not interact with... Yeah, sure, sure. And in the universe, of course, that's not something that's going to be arranged for a quantum state. Ah, you're saying, how could I have non-equilibrium left over from the early universe? Well, yes, I mean, it seems to be that someone will arrange for it to go isolated. Well, the thing is, particles, for instance, say take relic photons in a microwave background.

1:17:30 I mean, they interacted very strongly with matter before atoms formed, before recombination. Right. But they've been propagating virtually freely. If you look at the Thompson cross-section, the mean free path for a microwave photon in the universe now, with a mean density of 10 to the minus 6 electrons with cubic centimeters, you find that the mean free path is of the order of the observed size of the universe. or larger by a couple of orders of magnitude so the probability of the photon the microwave photon having scattered since decoupling may be 1% otherwise there wouldn't be, what we see wouldn't be an accurate snapshot of the early thing. Now if you think about something like neutrinos, it's much more a relic neutrino today and I want to work this out a little bit, something like a mean free path of 10 to the 50 times the observed size of the universe you get these incredible so if one thinks of partitals that decoupled even earlier, gravitons or some funny supersymmetric neutralino or something, you get these immense numbers of course you may say well that's wonderful, on the one hand weakly, apart from the initial period, offers this hope that they're just floating around out of equilibrium. But if they don't interact with anything, how are you actually going to use them? So one possible answer is that some of these particles, at later times, they might decay. And there are theories of dark matter. You have some exotic particle X that decouples very early, but later on it has a lifetime which have a certain predicted frequency and people are searching for these decayed photons. But they have decayed with a sufficient high rate for them to be useful that they would not actually have got this far into the time of the universe. I mean they would have already decayed if they were decaying fast enough to be useful Well, there would have to be a sufficient number that have decayed to give an observable signal in these photons. And presumably, I mean, if the parent particles are out of equilibrium,

1:20:00 you expect the decayed particle to have some non-equilibrium too. So... But not for long, because they are in traffic. Well, I mean, there are many regions of space that are virtually empty. As long as these, say, it's some infrared photon or something, through some dense gas cloud or something, you know, if you look in the right place. I mean, these are considerations, of course, but I think that they're not. They're quite straightforwardly, you can circumvent these things. We have to look at specific systems. Of course. So, I mean, it's not easy. I think Chloe had a question What's the sense in which it's equilibrium and what's the relaxation that comes in? Well, it's equilibrium in the sense that once you've got that distribution the dynamics predicts that it stays that way but what about if I start out with non-equilibrium what is what sense will it relax well so let me just review in classical statistical mechanics there are different approaches in classical statistical mechanics there's one approach which says i have um a complicated isolated system maybe a box of gas or something and i have an ensemble thereof Simon, I have an ensemble with a certain density in phase space. And what the idea is that, in this coarse graining approach classically, is that if I have a density that is not uniform on the energy surface,

1:22:30 the dynamics will make this density spread out and develop complicated filaments in a way that when you coarse grain the coarse grain level it becomes uniform and you can develop an analog of that in this pilot wave theory that I have a density on configuration space and the argument is that this density obeys the same continuity equation that psi squared obeys So basically, it's like having two different fluid densities that are being stirred by the same velocity field. And on a coarse-grained level, they become indistinguishable. That is the analogue of how one understands the approach to equilibrium. There is an analogue to the classical theory. Now, that doesn't mean that every system approaches equilibrium. if I have a simple system, an isolated particle or something it might just remain at rest, it might not but one can argue that for a complicated system one can understand how this equilibrium relaxation can occur but once you've got it for a complicated system you can show that extracting a single particle you will have equilibrium too so that's in a nutshell how that works But, I mean, if you want to look at this paper, this paper here, I've got some numerical simulations as well in this paper for a simple example. Particles in a box. What's so special about more size curves? Why, why is, if you have more size p, s? Well, I mean, the velocity field, the velocity field that drives P, that pushes the probability around, is the same velocity field that happens to appear in the continuity equation of psi squared. So there is a sort of, mathematically, the velocity field is tied to psi squared as a relation there. I'm afraid you've missed all the new stuff standing around this.

1:25:00 No, I know. I mean, I'll explain. This idea of finding the results, all of the results in parallel computation and contra-computation, And presumably, it gets harder as we have more elements in the superposition. It gets harder to extract what those, to work out what they actually are from the trajectories, we get more and more of that. So if we actually factor that process into the computational process, does that rule out any gains we might get from using a constant system to compute? question, which, I mean, just for Simon's minute, I'm just talking about if I have a single particle that's guided by a superposition of energy states, and I don't know what the energy states are, but the energy eigenvalues encode the output of parallel quantum computations. If I can measure the trajectory, which I would be able to do if I had matter out of equilibrium I could actually figure out what all these these eigen function what the energy eigenvalues are so this example here say particles in a box two unknown energy levels here's this guidance equation if I know the position the velocity at two different times I can solve this equation gray and d so what if I had an exponentially large number of energy levels in this thing, well then I've also got an exponentially large number of equations to solve. I would have to have it at an exponentially large number of different times and so on. I haven't thought about that, really, but I think one answer is that, look, these outcomes, I mean, these energy levels, V-A-N-E-V, unknown energy levels, they could encode the results of an arbitrarily long computation, okay?

1:27:30 there could be some incredibly huge computation of which this is the answer so the work needed, ok there is some work needed to solve this equation but this could be negligible I mean this is just a fixed thing whereas these the information as it were in these energy values could be arbitrarily large so I think that that will be But then couldn't you get the same amount of measurements as the classical measurement of it? The same amount of classical measurements that you need on the people that you can use? What, a classical measurement of energy? Sorry, if I measure the energy I'm just going to find out one of these values and I won't know what the other one was. Yeah, but if you measure, you can see if you do two measurements on average, then you have to write it down. And again, on consumerism, you need two measurements, yes, if you don't write it down. So, well, but hang on, but then I need to, I need to run, I mean, that's like saying quantum computation, if I have a final state that is a superposition of different parallel computations, well, yes, I could, I could read all the, the, the outputs of all the branches just by, I measure it once, and then I run the whole thing again, but then I'm having to run the computation n times. I mean, here I could, you know, this is the output of the computation. Someone knows superposition. I've run the computer and here is this superposition. Sure, I could just measure the energy and I get one or the other. I mean, I run it again and I get one or the other. I run it, but then you're having The point is the quantum computer, I run it once and I'm able to read the results of all the parallel outputs. There is a bit of a tension there, isn't there, because the point about the number of parallel outputs that you want to read will then require a couple number of position and velocity measurements.

1:30:00 your response to that was to say well, any one of them could encode a very long computation what's distinctive about quantum computation is precisely the degree of parallelism so the fact that one No, I'm saying, no, all of them I mean, all of these in the superposition could be the result of extremely long computation very difficult problems absolutely, but what's distinctive is the parallelism, it's not the fact that each element could be the result of a wrong computation. So, if you're going to be accessing the parallelism, as it were, if you're going to be getting information about every single component in the superposition, that's what you need to do to really capitalise on the next state of that quantum computation. So you really do need to make as many positions and loss dimensions as there are. Exactly, exactly. But what I'm saying is the work, when you make those measurements and then you solve these equations for the energy eigenvalues, I mean, this, the work needed to solve these equations is a certain, I mean, it has a certain complexity. So, for instance, in this example, it is a certain amount of work to solve for A and B. What I'm saying is that the computational work that is encoded in E-A and E-B could be arbitrarily larger. So I'm saying that in general, to worry about the work that I do here, in practice, that would be a worry. But nevertheless, for a certain amount of work here, I can, in principle, get. It's only actually worth doing this work at this later stage, if the quantitation is a very big advantage. Oh, advantageous, of course. You'll pay in a second. Let's quantify it in a way that probably doesn't have much sense. And you'll see the problem, so the answer to that would be that we have to quantify it differently. If the amount of information that you can extract is n times the amount of information in each component where there are other components. So if the amount is what? is, suppose in each component you've got an amount of information, call it whatever.

1:32:30 So the total amount of information that you're trying to access is just the product. Now, the amount of information or complexity of the calculation that you've got to solve, call that Q, for each component. Now, there's NQ. There's NQ, so you have to put an NQ amount of work to extract N, P. Yeah, no, quite, P can be much greater than Q. Right, but that's not, what seems to be distinct about quantum computation is the fact that the amount of information scales as N, and N, in principle, goes to a countable infinity. So, but that's not helping you, that's not what you're able to get, because the amount of work you'll have to do scales as N as well. So, sure, I mean, NQ will be what you put, and NP is what you're getting out. I mean, the ratio is just P over Q. Do you see what I'm getting at? Yeah, but P... Okay, I mean, this is not quantum computation. No, no, I know. It's doing something different. Absolutely, absolutely. I mean, I'm saying I can extract all the results of a parallel computation, no matter how complicated these things may have been Right, but I think the most distinctive about the computation is the fact that you can as well utilize the computations done in every component of the state which in principle is infinite Yeah, that's true But you can't do that without infinite doing infinite amount of computational work itself So you can't infinite amount of information. Well, not infinite, no. No, but equally you can't, you know, you can't extract much of it, as it were, because it doesn't really depend on it being a literal infinity. I mean, that's exciting. I've got a large number of branching superpowers. It doesn't even have to be a large number. I would say it's a large number. Well, if it isn't a large number, then... then, you know, it's a large number. It's a large number. I mean, look, the point is that, I mean, as I said earlier, the initial intuition in quantum computation, right, at the beginning,

1:35:00 Somehow I've got different computations going on at the same time and they're all there. Unfortunately, I can't access them all because you collapse the state vector, but there are some interesting things I can do in certain special situations. The point is, if you have this non-equilibrium matter which you would use to measure, you could actually access all of them.