Locality & Bell Theorems

So my name is Mike Olre. I used to work at Boeing. I don't anymore. I have a new hobby and I'm about to show you some of it here. So for my talk at first I'd like to thank my sponsors. It was very fortunate circumstance. I don't know if you read this, but the Nobel Prize in Physics was awarded this month like two weeks ago to these three guys who figure prominently in the kind of thing I'm talking about. So just in case you thought I wasn't going to talk about anything important, this will show you. Yeah, indeed it is important. They're the ones who did some of the first experiments and the last experiment, Zeilinger in the middle, which verified that, you know, the Bell inequalities are not satisfied in the certain type of experiment. You can read the article in Quantum magazine and you can watch a PBS NOVA episode called Einstein's Quantum Riddle, which came out in 2019, which is kind of interesting, which shows Zeilinger's very exotic experiment where he used the light from quasars 9.8 billion light years away or separated by 9.8 billion light years, just to prove that there wasn't any common cause effect to the detector settings. It's a pretty interesting video, pretty interesting show. So, overview question. Does quantum mechanics admit a classical local hood and variables explanation context? Today there's a simple EPRB experiment. Two observers, Alice and Bob. Two measurements for observer, two outcomes per measurements. The historical approach is typically bottom up. You start with classical assumptions deriving inequality. That's what Bell did and lots of other people have done. My approach is not terribly different, but I think it's more intuitive, at least to me. I'd call it a top down approach. I spent four years studying this and reading and I wrote a paper which appeared in a journal in the archive. The links are there. You can read that if you're interested in more details. What I do is start with a completely generic model that includes both quantum mechanics and several classical alternatives as special cases and then show that the classical alternatives cannot replicate at least some QM statistical predictions. Also show a hierarchical tree where all the classical alternatives stand. With respect to qm, I come up with, well, Mathematica, which was my big help. This all this way doing visualizations and computations. Of course, 3D plots I've never seen before in the literature of correlations. Some have, some have not. I have an alternative characterization of hidden variables, the classical approach for this, which I think is more intuitive, and the concepts of outcome predictability and Measurement predictability. Measurement predictability has to do with faster than light transfer of information. Of course I show it's impossible, but it's a different kind of proof from the usual one. I'm not going to do that today. But you can read about that last one in the paper, Brief History of Bell's Theorem. Epr, Einstein, Podolsky and Rosen of course, wrote a famous paper in 1935, not questioning quantum mechanics, but questioning its completeness. Elements of reality, they said, are not really included. You got to get those in there somehow. They had a thought experiment. Einstein and Bohr debated it starting in 1935. It started to just trail off. Then people sort of thought, well, Bohr won these debates, but nobody was really sure. Bell came out with his famous theorem in 1964, coming up with an inequality. He said, if classical assumptions must satisfy this inequality, if you have an experiment that can show that they're not satisfied, then you'll probably say there's no classical explanation of this kind for quantum mechanics. And several experiments were done. Aspects experiment in 1982 was the first one that confirmed this. But there were still loopholes and that went on and on and on. And in 2018, Zeilinger did his loophole quote, unquote loophole free experiment, which apparently closed the case. But anyway, that's the story. Let's talk about the EPRB experiment. This comes from Aspect's paper, this diagram, the whole idea. It's a wonderful paper. It's called Bell's Theorem. The Naive view of an Experimentalist. This is the setup here. I'm not going to talk about the, the hardware in any detail at all. Just two photons that go entangled, photons that go in two different directions. The Alice's lab on the left and Bob's lab on the right. And they read, they read the measure, the polarization. They get out a minus one or a plus one. That just means it's bivalent. We'll just call it minus one and plus one. And then there's a coincidence monitor that checks whether they both get pluses or minuses, that's sameness. Or they get a minus plus or a plus minus, that's differences. So they count the sames and the different ones. Then this is how the statistics are collected and correlations are computed. As I say, Alice gets a plus or minus one, Bob gets a plus or minus one. They go into the coincidence monitor, out comes a plus or minus one. Then they're categorized by bin, four bins, one bin for each measurement combination. Alice has two Bob has two. So altogether they have four. So there's four bins. Then you compute a correlation for each bin. This is what this is right here. It's the number of positives minus the number of negatives over the sum of the two. And notice that makes sense. If NK minus is zero, that is you got no different ones, you all got same ones. Then you're going to get NK plus over NK plus, which is one, a correlation of one. Just what you'd expect. If you get all minuses, then the nk plus goes away. You're left with minus nk minus over nk minus, which is minus 1. Anti correlated perfect. If there's just as many matches or mismatches, then a plus equals NK minus. You get zero in the numerator correlation zero. So that makes sense as a correlation measure. So that's what you're talking about in the experiment, computing the correlations. Then after you get those correlations, you compute these S functions. And now this is aspects notation. He calls these S functions. I'll call them that too. They're basically the sum of the correlations where you put a minus sign in each of the four possible places. Then you compare these four numbers that you get out the S functions to minus 2 and 2. If they're in between minus 2 and 2, you say the CHS inequalities are satisfied and we'll see. As we see, most classical alternatives require these to be satisfied. Quantum mechanics violates them. The theory anyway, and ultimately experiment. You'll notice Clauser, the first name in the chsh, is the guy, one of the guys who just won the Nobel Prize on my first slide. So he's an important guy. And then as I said, the Zeilinger experiment kind of closed the case on the experiments in 2018. So now I want to talk about how I model it, how I think about it. What I tried to do was mimic what QM does. This is the photon polarization. This is just the classic quantum physics. I shouldn't use classic. This is quantum physics. Predictions for the outcome. The probabilities of getting a minus one, a minus one, minus one and one given these measurement settings. So there's 16 possible probabilities. This table over here, just emphasize all this does is emphasize that these are conditional probabilities. Now let's compute the correlations and S functions. This is pretty easy to do. It's straightforward, it's well known. These are the correlations you get. Notice if you do the spin entangled, the spin experiment, you get theta1 instead of 2 theta1, but it's off by 2. But in the photon polarization you get 2 times. But anyway, the end result is that the bounds now are not minus two and two, but you have a multiplicity factor of square root of two in there. Those are called the Serialson bounds. Is that how to pronounce it? Anybody know if I'm pronouncing it right? Anyway, I'll call it Searlson bounds. And those R4 quantum mechanics. And this is all dependent upon these measurement angles satisfying some linear constraint like this. If you look at the way Alice and Bob's measurements are and you assume they're coplanar, it's very easy to see geometrically. You have to have a linear relationship between the difference angles. Remember, the thetas are not the measurement angles, but the differences between the measurement angles. Okay, so let's move on to. Come on sometimes my page down. Why does the page down work? There we go. Ok, what is. Sorry, what is a model instance? I'm going to talk about model instances a lot in this talk all I mean by a model instance. I'm going to define some more model types later on. You also already saw the QM model type. A model instance is just where you give it specific values for the parameters. So if you give those values 67 and a half degrees, 22 and a half and so on, you get these numbers for the conditional probabilities. That's called a model instance. That's what I'm going to mean by a model instance. This is the one, the famous one that maximally deviates from the CHSH inequalities. You get one of these S functions which is two times the square root of two, which is as big as it can be. So that's a bad one there for classical interpretations. So what I do is I define a generic model type based on the QM idea. I just posit 16 parameters, gamma one through gamma 16. But I require that each row in this table sums to one, because this is going to be one complete probability space based on the context of Bob's analysis measurements. And again I emphasize that over here, both the conditional probabilities explicitly written out, then the correlations are like this and the S functions are like this. Now obviously each one of these includes all gammas which sum to four and they've got plus and minus signs in there. So they have to be somewhere between -4 and 4. So obviously some of them will have correlations which are small, some with are too large to satisfy quantum mechanics. So we're going to have all kinds of different models in there. Let's specialize a little. Well, first we need some definitions. Remote context independence means Alice's measurement does not depend upon what Bob's measurement choice is. His can be any of his V or V prime, it doesn't matter. Alice gets the same probability for hers and same for Bob. In other words, Alice's outcome doesn't depend on Bob's measurement and vice versa. Remote outcome independence means that the joint double probability is equal to the individual outcome probabilities based on their joint measurement. That's what it means to be conditionally independent. It's not a real independence. Absolute independence, it's a conditional independence. Factorizability is even stronger. The joint conditional probability is the probability of Alice's outcome given her measurement alone. And that assumes RCI right there. You can't have that without assuming remote outcome independence. So that's implicit. But anyway, this is a very powerful assumption there. As a matter of fact, you can show easily the factorization condition is equivalent to RCI plus roi. Non contextuality is tricky and I'm going to overstate it a little bit. For the purposes of my paper and the purpose of presentation, I'm going to say it means that the measurement process consists of definite pre existing values just waiting to be measured observed. That's what the measurement process is. That's the really classic idea of an experiment in classical physics. Does it apply to quantum mechanics? We'll see. There are consequences to that assumption. And I got a. You can read more about the definition of non contextuality in the Stanford Encyclopedia. They have a nice discussion there. Okay, now, outcome determinism. This is something I discovered using Mathematica. I could never have done this without asking to just do a brute force search. But it turns out that the outcome deterministic model instances fall into several different classes. The most important class are these 16 which satisfy remote context independence. There are others which satisfy CHSH, but not RCI, and 128 of them not only violate CHSH, but even the search and bounds. But the important thing are these 16 that satisfy RCI. Those are very crucial. And the reason they're crucial is that these 16. Well, first of all, let me show you what they are. These are the 16. The columns show the model instance. I couldn't put 16 tables up here, so I compressed them into columns. The colors differentiate the rows in the standard table. So the first pink is the first row, blues the second row, green's the third row. You'll notice no matter which one you choose, there's exactly one, one in each color and three zeros in each color for every model. That's true for these RCI plus RD model instances. Their correlations, not too surprisingly, are all ones or minus ones. They're perfectly correlated or anti correlated. All their s functions are 2 and minus 2. They not only satisfy the CHS inequalities, but they meet them exactly. Okay, tight bounds. Now for the non contextual multi. This is something I've never seen in the literature. I sort of came up with this with by other means I can't go into right now, it takes too long to discuss. But anyway, if you start with any probability mass function that is 16 numbers non negative sum to 1, and you put them in these four rows like this, you'll notice there's exactly one row K out of the 16 in each row in different orders. But they're all in the all represented in each row, so they must add to one since I'm assuming they're a pmf. These are the correlate non contextual correlations. But the really nice thing interesting thing is you get a sum and pluses and minuses of the row ks the 16, all 16, which must be minus 1, between minus 1 and plus 1 obviously, but multiplied by 2, which means that the limits are minus 2 and plus 2. So you just had just did a simple, simple, simple proof of the CHS derivation of the CHSH inequalities for the non contextual model type as trivial proof. And I like that, it's very, very nice. The factorizable model type is one that Bell discussed in his 1976 paper. It was his second paper on the subject. And this has to do with factorizability. I call these conditional probabilities single conditional probabilities that divide up Alice's into alpha one through alpha four and Bob's into beta one through beta four. They obviously if you look at them really closely, you see alpha 1 plus alpha 3 has to be equal to 1, alpha 2 plus alpha 4 has to be equal to 1. Same for the betas. This is what the conditional probabilities look like, this is what the correlations look like. And these are pretty messy S functions. And you could spend some time and prove that those must be between minus two and two. But there's a much more elegant proof which I discovered with Mathematica. Again beautiful how just solved this thing for me. Just define the Lund contextual parameters to be equal to these alpha and beta products of four alpha two alphas and two betas. Each. Then use these constraints. Just do a little algebra. You can show that this is a PMF. In other words, these 16 products of four numbers add up to one. Not only that, if you plug them into the table for the non contextual model type and solve simplify, you'll come up with this table right here. So very simple proof that every factorizable model instances, also non contextual, therefore must satisfy all the CHS inequalities. Finally, the trivial. This is just for completeness. It's completely true. Everything is 1/4. Obviously this is a legal model. Not very interesting. All uniform, no correlations, all zero. S functions, all zero. So what? Well, it comes at the bottom of a tree. I'm going to show you. Here's all the models all together, just to show you what they look like. Divided up by row, showing by color. If you put it into a four by four table, the colors differentiate the rows. And finally the constraints that make all these into conditional probability vectors. Now, here's the tree I'm talking about. This is interesting because you'll notice I've got red and green boxes. Except for quantum mechanics, which is a blue box, blue outline. All the red boxes, all the model instances satisfy the CHS inequalities. All the green boxes not only violate the CHS inequalities, but some of the model instances in the green boxes even violate a serial bound. So the only. There's nobody else in this whole list, this whole tree that can replicate quantum mechanics. Either the S functions are too big or the S functions are too small. So you always have something that won't work. Anyway. This is something I've never seen before. It took me, I don't know, a couple years to figure out all the different cases. And these are all proper inclusions, by the way, the arrows indicate set inclusion. These are all proper. I had to come up with all the counterexamples. Thank God for Mathematica, I'll tell you that. Anyway, here's another view. QM and non contextuality have a non empty intersection. There are model instances, they're the same. They're both. They're QM and the contrast non contextual. There are QM model instances that are not non contextual. All the factorizable model instances RCI post OD model instances are non contextual. The only model instance that's QM and also factorizable is a trivial one. That's the only one. Okay, so moving on. The three theorems of John Bell claim there's three theorems and there's other people who written papers like this, but I kind of stole some ideas from all of them. But anyway, the three terms of John Bell I say hidden variables model cannot replicate at least some of the statistical predictions of QM if it's based on either RCI plus OD, that's his first paper in 64. Factorizable factorization condition, that's one of his other papers in 76. And non contextuality, which is his paper in 1966. Actually he wrote it before he wrote the first paper but there was an editorial snafu and it didn't get published till 66. But anyway, that's the hierarchy that I'm talking about. Now here's the plots of correlations that kind of confirm. Here's Bell's 64 paper 7666. This is RCI plus OD. These are plots of correlations. I've never seen these QM correlations. You also see this in the literature. These polytopes for the Bell. These are exactly. These blue solids are exactly the Bell CHS HS inequality. The facets of those inequalities, if you plot them in space 3D, I mean you have to fix 1, C1 is fixed minus 1 in the first column, minus 1/2 in the second 0, 1, 2 and 1. The other three correlations are allowed to go free based on a constant for that column. And I've never seen this factorizable column plots either. That's new to me anyway. Thank God for Mathematica again. But anyway, you can see the progression of RCI plus OD is a subset of factorizable subset of non contextuality. Now these are just correlations, so that doesn't prove the theorems but that confirms them. Visual confirmation, which is kind of nice. What you can prove is that the convex hull this is a basic theorem. This is my version of the hidden variables theorem that you usually see. The convex hull of all the 16 RCI plus ID modal instances equals the set of all NC model instances. In other words, if you take the columns in the table of the RC, the 16 RCI plus outcome deterministic model instances, you combine them with any PMF. The Rho ks are non negative numbers which sum to 1. So this is a convex combination. You get a vector of conditional vector of numbers which is exactly the list of the non contextual conditional probabilities. Magic, in other words, non contextual and factorizability. This is just a schematic picture. It's meant to be kind of like a complex polygon and this is the interior of it. The RC/IOD RCI+OD on the outside is vertexes, non contextual and factorizable on the inside. Of course, this is just a schematic. It's really a nine dimensional subspace, a polytope and nine dimensional subspace of 16 dimensional Euclidean space. Then you can go and take the convex combinations of all OD256 outcome deterministic model and says guess what? You get all the generic ones including qm, which would suggest you can make a hidden variables model for quantum mechanics. You can. It's outcome deterministic, but it's not local. It won't be local unless you get inside this intersection here of non contextuality and quantum mechanics. Then you can do it with a local one. Okay, and that's what I show here. Here's a non contextual one, angles 135° 0° 90° 45. That's this model right here. That's a QM model. It's also a non contextual model. You set Rho K equals 1, 4 for 2, 8, 9, 15 and 0. Otherwise you can also write it. There's an equal sign as a convex combination of these four RCI plus OD model instances, all with weight 1/4 will equal this model over here. So there's a hidden variables model for this QM model instance, but it's not. I'm sorry, yes, it's a local outcome deterministic model, but that's okay because this just happens to be one of the QM models which is also local. Also a local one, non contextual, I should say. You can turn this into a classical hidden variables model. I detail it a little here, but I'm probably running out of time, so let me go to the next one. Here's one which is not contextual. This is the famous one that violates the CHS inequalities. It has s function equal to 2 times square root of 2, which is bigger than 2. So it cannot be non contextual because all non contextual model satisfy the CHS inequalities. And this is where Mathematica. Just unbelievable found me. These 13 model instances which are outcome deterministic, they're all zeros and ones, they go across. By the way, this is one model, this is the second model, this is the third, and so on. And these are the weights that go with them. They're all some cosine function R squared of 67.5 degrees. In some combination or another, I never would have found those Weights with pen and paper, obviously. Anyway, so this would seem to suggest that there's a deterministic model for quantum mechanics. But what's wrong with those OD models? None of them satisfy remote context independence. They're non local. So this is a perfectly legitimate outcome deterministic hidden variables model for this QM model instance. But it's not local. So what does this all mean? This is my last slide. A fable. Suppose Alice and Bob ask their friend Eve to try to fool them. And I think they're receiving data from an EPRB experiment. EPRB experiment. So Eve sends out four numbers. A1, A2, B1, B2. She sends A1 and A2 to Alice. If Alice chooses her first measurement, Alice gets the number A1. If Alice chooses her second measurement, she gets the number A2. Similarly for Bob. So what does Alice, I mean, what does Eve do? She's got to have some method of sending out. So let's suppose she has a probability distribution. Now this is not a conditional probability vector. This is a PMF. Now row one through row 16 is a PMF. They add up to one. So she's got, for each of the 16 possible numbers she sends out, she's got some probability. She gives a 1 and a 2 to Alice, B1 and B2 to Bob. They choose their measurement, they get their output. Now what happens when you compute the marginals? Let's compute the compatible marginals of that distribution. What are those numbers in there? Those are the non contextual ones. Those are exactly the non contextual conditional. If you view these as conditional problem double conditionals, those are the non contextual conditional probabilities. Therefore, Eve will only be able to fool Alice and Bob if it's a non contextual model instance, like that first one I showed. If it's the second one, she won't be able to fool them. So what's the conclusion? The conclusion is my final conclusion. In a QM world in which the behavior can be contextual, it is not always possible to fake the results in a way that ignores future measurements. That's deep stuff anyway. That's exactly sounds a lot like what Bohr was saying, trying to say to Al, al, baby, you got to wait for the measurement to know what's actually going to happen. And this, this is kind of what the theory says. And those experiments, like the ones I showed Zeilinger and so on at the beginning, who did those experiments? So far I have concluded, and the 2018 when supposedly closed all the loopholes and said, yeah, nature violates the inequalities Anyway, this is just my way of looking at that situation. I think it's more intuitive than the traditional hidden variables approach, but it comes to the same conclusion, obviously. Anyway, that's it. Thank you.