Further thoughts on non-contextuality and the finite precision loophole

0:00 The board on non-contextuality and the finite perception report. Thank you. Thank you. We're having technical difficulty. And so Adrian, I think, is going to attempt to begin, But you're going to have difficulty projecting, so if you're in the rear and you can't hear, you're on your own. You're just going to have to move up. There are FDCs in front, and then after a while, after Jeff gets a technician,

2:30 and then you can go and move backwards to . So without further ado, maybe in catch. Well, I want to review and enlarge on some work which I carried out two or three years ago now with Rock. And I have to say, it was just a delight to get us with Rock. which is one of the smoothest collaborations I've ever had. So, as other people have already mentioned, Rob's enthusiasm for, if anything, was overwhelming and was also carried along on the surge of that and he was also a pro. The paper just more or less materialized. We had some email correspondence and then suddenly there was a draft which didn't really need very much further doing to it. It was a very, very enjoyable collaboration and very great to have a chance to work with them. So, our interest was provoked by, of course, Koshin and Speckers' famous theorem and then by some more recent work, which I'll describe in a minute by David Meyer, on Koshin and Speckers' theorem and whether or not it applies to real-world measurements, real-world experiments. So let me start with Koshin and Speck as it can, to review that. I've tried quite a lot of these transparencies. I've sort of extracted what nearly, they're quotation, what are they speaking quotations? I just compressed the text a little bit here and there. So I promise you I haven't changed the sense of anything that any of you all have said. So Koshin and Speck were interested, of course, in the question of whether quantum phenomena could be explained by hidden variables. The aim of this paper, as they say, is to give proof that non-existent hidden hidden variables. And their first goal was then to say what that means, what

5:00 is a precise, necessary condition to exist hidden hidden variables. As they say, proposals for classically reinterpreted quantum mechanics, you tend to introduce phase space of hidden dual states as statistical mechanics. And then, when these attempts succeed, they do so in the sense that the quantum technical expectations equals the student expectations of the classical variables. And they went to point out that there are deeper questions of issue with it, but more than itself. As they say, this doesn't take into account the algebraic structure of quantum observables, which form a partial algebra on the field of space. if we restrict the sum of product operations in the algebra to design B-defined only when we have community operators. A necessary addition for hidden variables for a classical algebra is that the algebra should be embeddable in the communicative algebra. And they argue in their paper, or they argue in their paper, that there's just finite partial algebra of quantum observables, for which there is no such class that may be made. There was a strong quantological focus there, but also a very direct physical interpretation of . And I've excited a couple of . The physical description may be understood intuitively, quite independently of the former operations they just considered. We can consider, and then they get a concrete experiment with an atom of halterhelion in a metastable, lowest energy state. We can apply the electric field in any one of the specified number of directions. And they argue the classical account of the results of that experiment must predict the energy changes in every possible one of those directions. And for any possible classical interpretation, there exists an experiment that you could do. There exists a direction in the specific set in which you could apply the field such that the classical predictions are contradicted by the experimental model.

7:30 I'm a little worried now. I'm all thinking this thing. So, all the helium, let's say, it's a triplet state with a three-dimensional space. It's a metastable state, so one can let it just sit there and carry out experiments on it. And we're applying an electric field, which has wrongly symmetry. about three different perpendicular symmetries about three different perpendicular axes for the X, Y, and Z. And this electric field returns the original Hamiltonian with this perturbation term Hs, which has different coefficients associated with the S-squares for the three different axes, A, B, and C are different. You can then measure the energy a sum of two of those terms, A plus B, or B plus C, or C plus A, for which you can infer the values of Sx squared, Sy squared, and Sz squared to be some permutation of 1, 1, and 0. So for instance, if you get A plus B, you infer that Sx squared is 1, and Sy squared is 1, and Sx squared is 0. So, concrete experiment here. Cochrane's back famously gave a construction of 117 different axes and say that if one attempts to ascribe truth values, values of 0 or 1 to the Sx squared, Sx squared, Sx squared, Sx squared, Z squared, will all be done from 17. then somewhere among that 117, one reaches a contradiction between invariable prediction and the experimental evidence. So, thus the invariable prediction contradicts the prediction of quantum mechanics and experiment. So in the terms that we've become more used to,

10:00 discussing and the relationship problems, they are examining the question of whether a lot of quantum mechanics can be simulated by non-contextual hidden variables. By which we mean, is there set of pin variables which have the property that for any quantum state their truth value 0 or 1 assigned to all the possible projections defined independently of the any other operators that are involved along with those particular predictions or quantum measure. In other words is there about T, truth value, from the set of all predictors to 0 to 1, such that, well, if you have any set of orthogonal predictions, which I have to the end of it, it should describe truth value 1 to precisely one of them, and truth value 0, for less than the rest of it, and so this equality should come to be this one else. And what Cauchy and Statistic Theorem shows is that there is no such true value in D greater or equal. And that's a very satisfying and elegant result, which is actually something. And the consensus was it establishes a technical or personal perspective. I think that was pretty much the consensus until David Meyer wrote a very pretty, thoughtful physics review letter in 1999 with a provocative title, Finite Precision Measurement Nullified the Commission's Effective Theorem. So David's point was, well, firstly, if we're actually doing this experiment again, or not or any other experiment, we have a practical problem. But we can't, to input precision, specify the predictor that we actually want to carry out measurements or so on. So perhaps, we argue, we should include that possibility in a quotient spectra-like analysis and see whether that conclusion still holds when one takes into account the finite precision And he said, well, no, it doesn't. The Kochen Speck theorem is indeed valid.

12:30 And his reason, which was at least the reason that you won't pause the thought, is that these true functions can be well-defined on dense subsets, at least for the protections in R3, the case for which Kochen Speck of the original proof was set out. and in particular he found a very, very pretty result that he proved for quite a bit of reasons by giving an explicit construction of a true value coloring, this description of zeros and ones to all the rational vectors in the intersection of the spheres all the rational vectors lying on sphere S2 from Median Archery and that's that's where Rob and I began to take an interest in this we extended that result our constructions are not at all beautiful they're horrible things really but they do a certain job they describe truth values to dense subsets of the sets of all projections not just in 3D but in any number of dimensions and also to then subsets of the sets of all positive operator value decomposition of the identity, again, in any other dimensions. So the point of doing it both ways is, whichever way you do it, somebody can come along and say, well, you've done this with projective measurement, but in fact, the most fundamental thing, the thing that really we only carry out in nature, are not projective measurements, but positive operator value measurements. So your theorem doesn't really apply. So that way around, people can say, well, they'll actually, you know, positive operating value measurements are all very well and good, but they're only sort of effective descriptions of what's really going on. It just means that you've considered two forms of system. And really, the fundamental thing, I believe, about quantum theory, fundamentally possible to be framed in terms of projection, and all measurements are fundamentally projected. So you need to do, you need to cover both cases to satisfy everybody. Now, this word here, nullifies, has caused a great deal of angst and debate and worry

15:00 into controversy, and, well, it was David's choice, but I certainly went along with it at the time. Perhaps when we think, in hindsight, we should spend a little longer saying exactly what we've been doing, because it certainly is capable of more than one interpretation. And what we didn't say in that, if it wasn't obvious, look what I should stress now. We said we weren't trying to say that there was anything logically inconsistent about We are trying to say, with second definition, that when you take into cap finite precision, the force, the effectiveness of the direct implication of contradiction, is counteractive. And I should stress something else. So I think we did try to address this, but again, perhaps we're involved in not spelling it out as sharply as we could. This wasn't supposed to be a great attack on Kersh's Beckham or a suggestion that the theorem was on any less interesting than it had been previously or any less physically relevant. The game we were interested in, the point we were trying to make, is that there is a loophole, and it's a logically possible loophole, and it was clear to have a watertight result. One should be interested in logically possible loopholes, however physically important we are, and I think our models are a stronger. So, let me enlarge on that. The first thing was never to advocate non-contextual hidden variables or any sort of hidden variables. And there's very good reason not to, I think, which is that you believe in standard relativistic causality. Even if you find you can get around the quotient spectrum by using classical diapositive, You can't get rid of Bell's theorem, and you can't get rid of all the experiments of confirmed quantum quantum mechanics. So the aim wasn't that. The aim was just to get things straight. Which arguments against the classical idea of quantum mechanics are watertight?

17:30 I think Bell's theorem and the experiments pretty much are. And which are not quite watertight, which rely on some assumptions. and the claim that's often been made and is still being made and certainly obviously debate worth having about is that it gives an argument quite independent of Bell's theorem against classical seemingly and an argument which doesn't rely on any hypothesis about locality and which is susceptible to independent experimental tests that don't don't need light-like, sorry, space-like acceleration of systems and which ones are carrying out independently generated measurements. So, that's all we've claimed. Our argument is, well, literally speaking, logically speaking, that isn't true, and our model illustrates that. Because of the finite precision loophole, there is no experiment which can exclude full stop, end the story, without any further So, again, this is from the abstract of the paper that I wrote. No physical measurement could be performed with infinite precision, I think everybody agrees with that. The claim is this leads to a loophole in the stat of no-goal arguments because not pertains to certain variables. because every such argument relies on choosing special sets, for instance, the 117-ray construction of observables with outcomes that you can show and that doesn't exclude the hypothesis. The measurements that you can actually carry out correspond not to every possible quantum observable but merely to some dense subset of all the theoretically possible observables. And if that were the case, one could find more contextual invariable models that simulate quantum mechanics. How do we know that? Because we expressively construct such models in the paint, and we do it with connection and positive operating by humans.

20:00 Well, that was the claim, and I don't intend to revise it here, but there's certainly been a lot of discussion since. I think there are so many people who have had the pleasure of listening to cricket commentary in the UK in particular. But if you have, you've noticed when they talk about batsmen whose strokes are especially athletic, they say of them that when he hits the ball, it stays hit. so to speak, retrodictive and ambiguity about the fact of the matter of the world. And the question is whether that's true about nullifying the Christian spectrum here. When we nullified it, or I believe we nullified it, how did it stay nullified? And a whole series of critics have come along to see that it's not the case. one of them has even written a paper called nullification so I want now to go through a list of some of the some of the points that have been raised against our constructions and our arguments and and the first the first is this issue of While the trick of looking at some dense subset of a set of all possible observables is an allowable trick or not. So certainly if you follow the initial spectrum of the written original paper, they would claim implicitly not. They say if a physicist believes in in variables, he should be able to predict in theory the measured value of every quantum observable, no is, no buts, no subsets, everything. Is that a moral quotation or a literal quotation? Now, if we can replace that phrase, every quantum observable, by a dense subset of quantum observables, then certainly in a mathematical fact, the conscious fact of theorem no longer holds, and the contradiction they claimed no longer arises. And one question that certainly caused quite a lot of controversy is whether or not that A is a legitimate substitution, whether or not it's reasonable to do that. Can we weaken Christian Specker's requirement just a little bit,

22:30 given that by that decision is all we can actually obtain in the real world? So, this has been raised by more than one person, certainly by Asher Perez and also by Marcus Appleby. So he'd issued, well, what he would describe as a challenge to those on Rob's and my side of the argument. So he says sort of wearily, so he's included emails. You should just turn that off. Should I just... So, as I said, finding a decisive argument will convince these three authors to be elusive. It's always possible to find loopholes. All I can do is challenge them to find a loopholes in the following. And the following is this, whatever the causes of imposition are, if in fact we don't measure exactly n dot j, where n is some axis that we were trying to pick out, then we have to give some physical interpretation. What is it? Do we measure some other nearby operator at this point, n dot j plus some sum of unknown, presumably small coefficients, bilinear in the j terms? If it's that, as Asher points out, that can't generally be reduced to another component of JIT. It's not linearized JIT. So, any lack of precision, well, that's just the, it simply means the positive operator actually implemented only approximates the operator of the experiments intended, and the actual operator may have no classical interpretation at all. Well, I have more than one response to that. One response is this distinction between projections and positive operators is not as important as Escher seems to have thought. Since we gave models both for projections and for positive operators, if you want a discussion

25:00 The second response is, I think the statement it's always possible to find loopholes is a misunderstanding of the situation. There is one precisely defined loopholes to do with bio-position which we've been trying And the third point is, well, okay, suppose we restrict rejections. Still, still there's a point being missed here. And I have here an embryonic. So this thing has some relation to the quantum mechanics. So this is supposed to be an illustration of three-dimensional, or exceeding three dimensions, and every triple of drawing pins that have the same color, so for instance, these three green ones, and I'm sure they're not precisely, but that's an idea. So there's the red ones, and there's the yellow ones, and there's the white ones, and so on. And to complete the model, you need to describe truth values. You need to write something like 0, 0, 1 on the three white And obviously there's a limited number of drawings into that available, but obviously this thing is supposed to be messed up well, a good deal more. But let's just work with these for the most. So the point is, Asher would say, okay, we come along, and we try and measure this axis, and this axis, and this axis, which haven't actually hit any of the drawing things at all. So what do you, with your, with your, with your model, variable model, said about that? Well, what I'd say is, as part of the work, hidden variable model, there's some little library that decides which nearby set points are actually going to be measured.

27:30 You can think of this as though you've got a magnetic field that's not precisely lying. You try and push it in there and the magnet just flips over. So let's see, it flips. So these white, they're by white ones. That work isn't being done by quantum mechanics. That work's being done by given variable one. So, these are the measurements we're trying to carry out. The model says, uh-huh, those, and then he looks at the truth values and picks them out. One, zero, zero. So everything that actually says is true, but it misses the point. The work can be done within the observable models. And that, I think, that's the most fundamental important of the responses. Now, the second worry that Barker Zappelby and others have expressed about our worries are more than this. is that they're not really classical at all, or they might not be reasonable to describe them as classical at all. Because if one has a model which describes truth values to a dense subset of the set of measurements, and if you look at the details of the model, he points out that this is certainly right, in every little neighborhood there, no matter how small, you have subsets of 0 and subsets of 1 just co-existing with each other, there's a complete discontinuity there on every scale. And he says, whatever it is, it is in classical. And one could argue that, but again, I think it misses an earlier point, which is important And that only point is, you don't, if you're actually constructing an architecture with a variable model for any particular experiment that anybody has in mind, or even for the whole experimental programming for the next 10 years, you don't need to use that full-length subset. You only need a finite uncovering to deal with the actual level of experimentally precision that's going to actually be realized in the particular experiment that's in mind. So you can pick out a finite sub-subset. do something really like the tennis ball and the drawing pin, with some final collection of drawing pins that cover the tennis ball.

30:00 And you assign the true values not just to the points where the pin went in, but the same true value to everything that's covered by the hands of the drawing in here. So, there's a little problem that coverings of things that you can flesh that out into a patchwork of neighborhoods that generally do cover 0 and R and overlap in that sort of part. So, what we don't then have is discontinuity on every scale. What we still do have, of course, is discontinuity. You have true value zero here, one here, and zero here, and one here. And, of course, generally when you're crossing from one drawing into the next, quite likely the true value changes, crosses from zero here to one here, let's say. But, well, that's a discontinuity, certainly. But we never needed, you know, we needed none of those discussions to realize that true value is going to be discontinued. But some of them have, there are some value 0 and some value 1. We never needed to push the spectrum or anything else to tell us that at some point we'd have to cross over from 0 and 1. So if this continuity alone were an objection, then nobody would have ever taken any interest even in the push and spectrum 0. That can't be the criticism. So the criticism must be to do with the radicalities that we think of those . And that's not a necessary feature of the point. But even if it were, your point would remain. That wasn't an assumption in the non-radicality of it. It wasn't an assumption. So if you point out that you need an extra assumption, it would still be... Even if it were, I agree. So there's still a point to be made, there's still a discussion to be had. But I think probably some people get by that necessarily. And then there's a third line of criticism, which I found really interesting. And that claims not that our models are non-classical, but that they are actually not longer textual, And the notion of contextuality in quantum theory could be misunderstood until the year 2000s came along and rethought the question.

32:30 And so what Simon and collaborators and Lars and Arjun is that we really need an operational definition of contextuality. This is the correct way to think about quantum contextuality, not contextuality. So they point out that, and again, I agree, you're right. If you're going to carry out a quantum experiment, then at least a reasonable model, an accurate model for many purposes, is one in which you have a set of dials, and you dial in some number from a finite list to . So you dial in N1, N2, N3, these are all integers, and these are supposed to specify, at least to a good approximation, some triple of projections that you're interested in simultaneously measuring. But the fact of the latter is the beta you have, as far as your input is presented, they're not directly in projections that you think you're specified, but these are N1, N2, N3. And their suggestion is that contextuality and non-contextuality should be considered in terms of these normals. So the experiment fixes these. The experiment gets an output which is interpreted as a true value for a set of projections. And what you can then ask whether those outputs depend contextually or non-contextuality contextually and what Simon and collaborators and Larsen showed is that, indeed, you can sort of modify the efficient spectrum theorem and demonstrate that it, as long as you're in position, is not too bad, the outcome does depend on the textually and the end of the set of multiple devices. So their conclusion from that result is that quantum contextuality is, in fact, it's So just to flesh out what they say a little more, they say there will be some experimental

35:00 procedure trying to measure S n squared, S n squared along S n is actually possible. And what we say is the observer sets the, so to speak, control switch to N. In an operational sense, the measure of physical observable is entirely defined by the switch position. In non-contextual hidden variable theories, it seems that the predetermined result does not depend on the context of the result. In particular, which other observer was measured to simultaneously, but only on the switch position and the hidden variables. The idea is it was quick, you had switch setting n1, n2, n3, and that was the truth values t1, t2, t3. And you were to change, or have alternatively chosen, setting n1, n2' n3'. If you have a non-contextual invariable theory, it must be the case that you still get the truth value t1 for n1. So, well that's a good question, certainly an interesting discussion we have there, should Should we define quantitative textuality operationally? But I think some logically prior question we should ask is, can we define quantitative textuality operationally? And I want to look at this in the . Not a moral quote, but a little quote. This quote of this. In an operational sense, the measure of physical observable is entirely divided by . So is that actually consistent? And I claim the answer is no, it's not. The fact of the matter is that whatever device you set up, even if it has these darns, it's just following them more precisely, if you change the settings of a couple of the other dials, whether or not you change it slightly or not, and however well you've tried to design the apparatus, you will slightly change the quantum operator A1. that you believed you were measuring and had set to be measured by the first dial. And you do that because the dials themselves have gravitational fields and electrostatic fields, and rotating them a little, sort of the numerals on them, say, N2' and so then 2 will slightly change those

37:30 gravitational fields and electrostatic fields. And of course, it's a tiny effect, but it's there, and you'll never get the ability to eliminate it. And in principle, and live long enough to even in practice, verify this experiment, statistics of the outcomes for the first operator will genuinely be slightly different depending whether the settings are in one M2 or in one M2 prime or M3 prime. So if the statistics are slightly different, the observable cannot be, it cannot be the case that you're measuring exactly the same observable. The observable must be depending on the setting. And one might come back and say, aha, well, maybe so, but that's simply an evidence form of non-continuitability. That in itself is a signature of quantum non-continuitability. But I think that's wrong. I think we can't possibly interpret that as a signature of quantum non-continuitability because exactly the same observation could be made in detail with this. It's still the case. If you rotate without the limit changes, the rotation will be slightly changing whatever it is you want to measure it. So, to sum up the counter objection, finite precision makes any operation of the definition of simultaneously measured quantities necessarily no perfect definition. The quantity that you actually measure depends, for this reason, on the measurement context. Even before we get into a discussion about truth values. Can I ask a second one? The graph here in red says it's actually the textual circle which is defined by the truth. And you argue that this quote attempts to define quantum contextuality in terms of description of this one. But the term of quantum contextuality is not in the right quote. So the question whether this, assuming that this refers to the right quote, is the right definition, is not really answered by your argument. Well, I think I'm trying to sort of summarize our argument.

40:00 I believe I represented it completely correctly. Does the earlier quote help? There will be a certain experimental procedure that we say that the observer sets the control switch to an inoperational sense. The measure of the physical observable is highly defined by the switch position. That was the red quote. In non-contextual hidden variable theory, it's assumed to predetermine results, not depending on the context of measurement, in particular which other . So I don't want to suggest that Simon collaborated in Larsen's results aren't interesting, I think they are. extremely interesting what it seems to be they show is indeed that something somewhere in the workings of this whole experimental configuration, the apparatus plus the quantum system is a mechanism which produces outcome data, the truth values T1, T2, T3 which data aren't determined solely by the corresponding settings plus hidden variables so it is not the case that T1, T2, T3 depends on lambda, t2, n2, lambda, t3, lambda, n3, lambda, n2. Now, as I've already said, classical physics rather suggest that you shouldn't expect that inequality to hold. Sorry, that equality to hold. And your intuition would be that it should not hold in the point of the term. But it's interesting that, quite independent of those observations, you can prove it as a mathematical theorem But what you can't prove, and what they haven't proved, is that the responsibility for this belongs to the quantum system, the thing we were actually interested in the first place, because we already have the problem with the classical apparatus and the gravitational theory of this contextuality that's necessarily inside there. And you would need to pin it down onto the quantum system alone in order to exclude classical explanations of quantum context offenders, and I believe I'm not going to illustrate. You just can't do that.

42:30 It's apparent, I suppose, that my own position hasn't really shifted very much, although I think many of the critters in the past have been very interesting. But a wider moral that we draw is that whatever Kosh and Spakka exactly does and does not tell us is less well defined and there's been less consensus on that than seen and seen apparent five years ago. in fact, because the theorem actually is meant sometimes subtly or sometimes even sharply different things to different people. And that's already been the interesting thing. And if the debate, the papers in the final edition debates are keeping something else, they've helped us highlight those differences which seem not previously. So I think it's helpful and fitting to give the last word to Rob here. There's another criticism which is still not logical criticism so much as, I don't know, almost a moral criticism, it could be leveled against our models in David Myers' earlier paper, which is that somehow, you know, we may logically sort of sneakily have a strict matter of fact won the argument, but we do it by missing the entire point of the Koshin's Becker theorem and the reason why anybody would be interested in the first place. And this is the point that David Mermin argues, at least for some while, with charm and force in email correspondence with us. And his point, representing at least part of his point, was that one of the amazing things about quantum theory is the probability for commuting observables It's dependent of whether you, of how you carry out a measurement. All sorts of different experiments to carry out a particular measurement of three different structures. And probably distributions are supposed to be the same.

45:00 An experiment verified back. And that's true even if you, even if you do these things sequentially. Even if you measure P1 first and then P2 and P3 later. It's still true for the theory, says the probability of this region to be just the same as they were And of course you can then do a delayed choice version after which you measure P1 first and then decide only later whether I can measure P2 and P3 or some other projectors P2 prime P3 prime also can be with P1 and so on So David's point was that the only real reason why anybody, given the successful mechanics, why anybody would take non-protective invariables seriously was that it seemed to offer such a natural explanation for that sort of thing. It seemed the most natural explanation for the constancy of these probabilities, independent in the experimental situation is that they're revealing pre-existing truth values and his worry was that when we'll, in particular when we'll consider sequential measurements, models like ours have no, are able to count the reporting results but are not able to do it convincingly. They produce ad hoc explanations which rely on the battery. from P1 you look at the truth value and then the truth values everywhere else across this thing change in order to give you a set of truth values that will reduce contradictions for excessive measurements so that's it, the idea that you carry a measurement and then the truth values everywhere else change as a result of that measurement and yet somehow they aspire to agree with common statistics worry data particularly and Rob's response to that point some other points in the discussion we'll set up here, well, we have a long email, but this particular one was said on Christmas Day, 1999 It's a beautiful in many ways it's insightful, it's modest it's historically incredibly well informed logically as an irrefutable and important .

47:30 So I just wanted to add a few points to the other correspondence. Well, it hasn't changed. And then what he calls a few rhetorical points. So I think we should be careful about talking about violating rules of the game, because it depends on the game you play. Social speckers themselves hardly used the term . They were sort of established in the And then there's this beautiful blend that starts with broader rhetoric. So it starts with rhetoric, and then rhetoric with a touch more substance. The independence of the sophisticated observable x in the way it's measured, sequentially or otherwise, is indeed suggested that x have a predetermined form, x dual, non-sequentially the third value. but we also learned the lesson from Bell's critique of the alignment that one cannot always infer that property which holds in the mean must hold in each individual case whether or not it does hold in the middle of the question I think Bell would have taken this point to apply equally well to David's use of sophisticated authoritarianism for a single case of authoritarianism and even more so to David's inference of single case I mean he goes on to point out that in fact you can find concrete results through concrete bone theory in which precisely this sort of currently conspiratorial explanation works out and works out actually, more naturally than what everything he did. All rhetoric aside now, well, then Bob points out but that is not going to point out one further loop of a specific sequential experiment. You may be thinking that you want to measure X and then Y's A and Y's prime and D's prime, and hoping that X, Y's A and X, Y's prime and D's prime, both constitute a parallel to S. But you can't guarantee that, and particularly you can't guarantee that in sequential measurements. Now, if your assumption is that projected measurements are kind of natural, and that's this particular discussion, sort of, but the data might run more properly. Then there will be some systems, not three-dimensional, that will be larger, for which you can guarantee that when you carry the measurement, it will be defined by some projected decompositions.

50:00 So if you carry out a single-time projected measurements and the system has to be three-dimensional, then indeed you can guarantee that x, y, z is compatible. But as soon as you move from single-time measurements all bets are off. You can't rule out the fact that Y and Z and something else that isn't X but is nearly X are actually the pattern set. And similar to Y and Z and something else, different again, be nearly X. And again, that little wiggle room is all that you need So, well, you can read the rest of these details here in your later. The final paragraph, though, is also a portrait and rather lovely. None of this is saying that this is remotely plausible, remotely satisfying, as an explanation of how the world reveals, that's what he thinks it really sticks. None of it is trying to detract from the beauty and the importance of social structure articles. And none, certainly not taking it quite seriously as explanation of the real world. But there is an insistence that you do achieve what they set out. So let me leave it there. Thanks very much. One of the criticisms that you didn't mention was Bale. So, do you want to check out what the... Well, as I understood the criticism, it was though you have demonstrated that you can color the rational... though you can color the rational vectors, if you try to imagine that your ignorance about which one you will find in measurement, in other words, there's some, to use, from second phrases, epistemic distribution, probability distribution. In other words, quantum mechanics doesn't specify which truth value you will find, and it captures your ignorance of that for the probability distribution.

52:30 But though you can color these things, if you attempt to write down probability distributions over the outcomes, you will find that they are necessarily inconsistent with the quantum probability for any given, for any quantum state. Well, so does that have any force in the review or does it not? I knew that answer. But let me re-elaborate with that one. So if that was stable, we'll make the single shot measurements. It would just be simply plus. No, he does make it for single shot measurements. Well, okay. But he may have all of our producers. If he makes it for single shot measurements, I thought that the point, at least the point was to do with sequence of measurements. You carry out a measurement of something, you get a truth value, and now you carry out the measurement of another triple, and the transition probabilities from the first truth value, the second truth value, are not, in general, consistent with quantity. That gets to the point that when you carry out the first measure of the gap, the first triple, and find the first truth value, all the other truth values, in general, change. You change the quantum state, but you also change the quantum state. Now here seems that the original set of truth values carried on being there unchanged, not measurement that's very healthy with the measurement. Well, that wouldn't work for us. That would mean for us to be able to do it. All right. Adrian, you mentioned that the Kuschnitz-Becker theorem has many readings. And this brings to mind, really, even at the moment of its birth, there was a very strong distinction between the way Kuschnitz-Becker wrote the theorem theorem and the way Bell would have read it. In fact, Bell, as you know, in 1966, had his own version of the Christian Spectre theory, which really should be called the Bell Christian Spectre theory. He got exactly the same results, except for his argument between the continuum, which I was not as the street said. It was a continuum version of the Christian Spectre theory, much simpler.

55:00 Good question. Well, it proves much simpler. Now, what's really striking in Bell's paper and the Christian Spectre theory, parallel, they do many, many similar things. But it's in the interpretation of the results of some point. For Bell, he is presenting the Bell version of the person's vector as simply yet another failed attempt to show that his invariable periods don't exist. And of course he always has the back of his minor bone period. So his whole paper is a review of these attempts to show that the variable doesn't exist, he's failing that. He actually says, well, I can give you an even more interesting, intricate proof that doesn't make all the mistakes, for example, of one knowing the name of others. And even still, because all of you rule about an architecture was invariable, and there are grounds for thinking that they're not necessarily . Kirshen and Specker, on the other hand, as you pointed out in the way, they're transparent. they claim that they are refuting hidden variables and again, Bob correctly says that they don't refer to the distinction between contextual and non-contextual and unless my reading is completely wrong it seems that they think they are ruling out the world in other words, they're providing the definitive proof against hidden variables whereas Bell producing essentially the same result simply saying all of these things just basically show up So, I wish I knew more about this story, but did Koshin, certainly if you read the original Koshin's paper, that's what you get, because the question is... Koshin actually admits it, but he said he originally had it wrong. That was what I was about to ask, did there be no change in the lives of elders and young citizens? But the revising which I think that she would still maintain is that no I think that she does not exist in her existence. Okay, that was next. Coach Harvey said much of what I was going to say. But I'll add a little bit more history. Bell didn't make any great claims for his proof of the impossibility of a non-contextual different variable theory, because he said it was just a corollary of Gleason's theory. But he also said something personal.

57:30 I knew that I either had to understand Gleason's proof or make a proof of my own. I knew it was easier to make a proof of my own. The second point is, it's not surprising that Coach and Inspector don't use a term like contextual or non-contextual because they hadn't thought of a new family of hidden variable theory. Whereas, there's your main point, Bell thought of a new loophole. There's a new possibility for hidden variable theory. He didn't have a name for it, but he thought of it. Therefore, after Bell, it's reasonable to start making distinction between contextual and non-contextual. I mean, the curious thing, if I can make a further remark, the question-respective theorem is an attempt to show that a non-contextual theory is mathematically inconsistent. It doesn't have anything to do with probability. So you don't need to feel performant. The bond theory, on the other hand, is manifestly self-consistent. Don't blame them so much. Most physicists for the next 20 years, if not more, continue constantly cited Cochin and Speckers proving that Bohm was impossible. I was thinking a little bit about your argument for the inconsistency of operationally defining contextuality as shown by even classical and your point was that this argument would run in a classical context as well Well, I more or less agreed with that, and yet I thought that a kind of abstract model of classical measurement with certain assumptions, happy, optimistic assumptions, must be in play when we, on a daily basis, ignore the line of thought you had in mind. So, roughly speaking, if we've got enough control of the triggering initial state of the unmeasured system and some limitations about the variance of the observable A,

1:00:00 imagine perhaps the boundaries of its level surfaces being subject only to minor perturbation. And in thinking about this daily practice and the kind of toy model of measurement that would sustain it, I realized that I was thinking of an integer, discrete-valued, observable, so that its level surfaces would partition up the phase space of the microsystem neatly. So here's the question. That took me back to the Appleby thing about discontinuity versus everywhere discontinuity. So do you see a connection there that in order to run your argument against the impossibility of an operational definition of non-contextuality, you do actually have to envisage classical contexts in which you have non-continuous observables. We certainly have non-continuous observables in the true practice in the true practice of one to the other. You're right also that if you run that argument they don't have to be radically discontinuous but they have to be discontinuous on very, very small scale. So you were trying to flesh out just how just how You need to go through the future, but I think the question is the critical. Right. I mean, I did endorse what you said to both these critics, but I just thought there was a connection there, essentially. Yeah. Yeah. So this kind of work, in some sense, is not very, very reversed. Yeah. It's a slight error. On the other hand, if you look at Bell type they seem to be quite reversed. Yeah. And yet, again, often Bell type theorems are sort of

1:02:30 important, at least in some sense, as proving So is it easy to see how Bell's theorems could be so robust, while at the same time Well, I think the difference is between an inequality that has, well, I'm not sure if it's helpful enough, but in the Bell rule, it's a change case, you have an inequality It has a spirit of a wiggle room for it. And the expectation values of little quality. So little expectation values may change a little bit, but they won't be able to push you the other side. Why would they probably let the GHS know? Oh, well, GHS would be a different story, I think. Yeah, I think indeed on that position, especially as GHS said, you couldn't really reverse it. I think that I think that Wayne Mermin made the point. There's a difference. It's stronger and weaker in one sense, stronger in the sense that it doesn't depend on the state of the political system, it's just a statement about the cerebral since in the Bell and in GHC You choose a particular state, meaning a particular probability, then it's very robust. It remains robust also in this case. Because these are just one and zero probabilities in this case, but they are robust. If you move a little bit to the side, you get 0.99. And there are not a single case violation. If you want to show that there is a real violation, you have to take a long experiment with many repetitions in the same state. What comes to mind at this stage is a paper like a pent-up cone. I hope you're aware of it. He gives a quotient spectator result using bell logics. That's to say, he takes an electron in an atom and looks at the correlation between the orbital angular momentum and the spin. So it's local. It's entirely just one system.

1:05:00 But you're looking at different degrees of freedom in one system, and applies the bell here. And it shows that a non-contextual theory can't be consistent with one prediction. Now that presumably is robust, but of course it relies on more, it relies on the state. Ooh, so I'm... I don't know that I've read that paper carefully, but I don't see how you can appeal to locality, which is the crucial thing you would do. No, you don't appeal to locality because they're not protections. Let's see how we are in the road. Have you appealed to the texturism when you've been talking about Have you appealed to Bell's theorem No, no. But I mean, formally the non-contextualism in this case and the locality in the Bell theorem are exactly the same. Ah, yeah. I see. So then there's nothing in the Bell theorem that says the two particles have to be separated. I mean, there's nothing in the Bell theorem that says The question then is, can you, how do you exclude the fire position new property, or in the Bell case you exclude it by the fact that you know you're acting locally over here, and therefore the project you're carrying out must be something of the form P-10, sir. and not some non-local protector that's pretty close to one of the other reasons so locality enforces and if you don't enforce that then you have this slightly room No, I was just on that. Because if you take a new . Then maybe even . There is a chance that the . So, for example, you might say this apparatus is over here. But actually, those apparatus is themselves have some wave .

1:07:30 center that actually has some spread exponential sprays after the separation so this one might not be in the same place with some very very very small authority but that might just be enough to give you the wiggle room to to get around uh belston but but i think the ranger is to say that could be the case because belston is so robust um but i just wanted to let's talk about that Just to follow up on these things, I mean, there are calculations that you can do if you take into account the full wave function rather than just the spin burden. And you take into account spatial locations. In fact, what you're measuring is spin up here, spin something over there. And then as soon as you do that, I think what Lucian said is exactly right. You get little room that you didn't have before. theorem so that given sufficient distance, you can erase the development. I mean, if you satisfy the development, you can erase the violation because of the special time. So I think you've got wiggle rooms there as well. It doesn't show up in these experiments in which you're measuring nearby parts meet something in the order of media, but it would show up . Okay, if that's it, we should thank Adrian. And we'll now take a 20-minute break and be back by 4.10. Thanks.