Google has an online quantum computer simulator. It has ready made simulation scripts for Shor and Grover's algorithm. It allows you to enter your own scripts and save / recall them.

In a famous paper of 1936, the first work ever to introduce quantum logics,^{[28]} von Neumann first proved that quantum mechanics requires a propositional calculus substantially different from all classical logics and rigorously isolated a new algebraic structure for quantum logics. The concept of creating a propositional calculus for quantum logic was first outlined in a short section in von Neumann's 1932 work. But in 1936, the need for the new propositional calculus was demonstrated through several proofs. For example, photons cannot pass through two successive filters which are polarized perpendicularly (e.g., one horizontally and the other vertically), and therefore, a fortiori, it cannot pass if a third filter polarized diagonally is added to the other two, either before or after them in the succession. But if the third filter is added in between the other two, the photons will indeed pass through. And this experimental fact is translatable into logic as the non-commutativity of conjunction . It was also demonstrated that the laws of distribution of classical logic, and , are not valid for quantum theory. The reason for this is that a quantum disjunction, unlike the case for classical disjunction, can be true even when both of the disjuncts are false and this is, in turn, attributable to the fact that it is frequently the case, in quantum mechanics, that a pair of alternatives are semantically determinate, while each of its members are necessarily indeterminate. This latter property can be illustrated by a simple example. Suppose we are dealing with particles (such as electrons) of semi-integral spin (angular momentum) for which there are only two possible values: positive or negative. Then, a principle of indetermination establishes that the spin, relative to two different directions (e.g., x and y) results in a pair of incompatible quantities. Suppose that the state ɸ of a certain electron verifies the proposition "the spin of the electron in the x direction is positive." By the principle of indeterminacy, the value of the spin in the direction y will be completely indeterminate for ɸ. Hence, ɸ can verify neither the proposition "the spin in the direction of y is positive" nor the proposition "the spin in the direction of y is negative." Nevertheless, the disjunction of the propositions "the spin in the direction of y is positive or the spin in the direction of y is negative" must be true for ɸ. In the case of distribution, it is therefore possible to have a situation in which , while .

Von Neumann proposes to replace classical logics, with a logic constructed in orthomodular lattices, (isomorphic to the lattice of subspaces of the Hilbert space of a given physical system).^{[29]}

Near the beginning of the video a crucial point is made. Either of 2 quantum bits by itself does not store any data. It is the correlation between the two bits that hold the data.

Once you understand the simulation you can ask permission to use the real thing. I guess that that will not be necessary because the simulation does the same thing.

Shor's Factoring Algorithm is very important because it showed that a Quantum Computer will be more efficient than a normal computer when solving some important, practical problems (Factorization is important in the secure transmision of electronic data, like credit card numbers). This video shows briefly how Shor's algorithm can be simulated in Mathematica using the free Quantum add-on. Quantum is available at: http://homepage.cem.itesm.mx/lgomez/quantum/

1/4 intensity is due to electric field vector being diminished 2 times by square root of 2. Electric field is thus 1/2 and intensity will be the square of this at a value of 1/4.

Later in the talk Garrett uses a polarization rotator. This takes the output of one orthogonal polarizer and spins it 90 degrees so that it aligns with the second polarizer giving no relative loss to a single sheet of polarizer material. A single sheet of course has a loss of 1/2 when fed with unpolarized light. See image below.

David Mermin's "Stuff Left Behind" in terms of Von Neuman entropy.

Ron Garrett aka Erann Gat quantum video on quantum mechanics. It helps make quantum mechanics more clear by using very accessible experiments that use light as the test subject.

Where lambda are eigenvalues of the system. Very similar to Shannon entropy but I suppose with complex values.

Notes

The polarizer experiments that he shows are quite interesting.

This presentation builds on David Mermin presentation is like a second chapter to that

The 3 particle correlation shown near the end is the David Mermin "Stuff Left Behind" presentation

Transcript

The Quantum Conspiracy: What Popularizers of QM Don't Want You to Know >> GLECKLER: Hi everyone. I'm Arthur Gleckler and I'm happy to introduce Dr. Ron Garret here, who's going to be speaking about quantum mechanics today. He's a former Googler from the very early days of the company, around 2000. He was the lead engineer on the first release of AdWords and the original author of the Google translation console. He also wrote the first billing system that Google used. Also, for many years, he worked at the NASA Jet Propulsion Lab in Pasadena, specializing in AI and robotics. I'm hoping to convince him to come back and talk about his experiences, debugging spacecraft 250 million miles from the earth. Here's Ron. >> GARRET: Thanks. So I'm told that my abstract caused a little bit of a kerfuffle so let me start out with a couple of disclaimers upfront to kind of manage expectations. The title of the talk was intended to be tongue-in-cheek. There is no actual conspiracy, at least, as far as I know but there is a fairly big disconnect between what you read about quantum mechanics in the popular press and what the actual underlying truth is, and that's what this talk is about. I am not a physicist. Do we have any actual physicists in the crowd? Oh, boy, okay. You can make sure you keep me honest. I'm a software engineer. I came upon this about actually 20 years ago when I read an article in Scientific American and I thought, "This can't possibly be right." And it took me 10 years to finally find a physicist at Caltech who could explain to me why in fact it wasn't right and at that point, everything just kind of clicked and quantum mechanics made a lot more sense to me than it did before. And that's what this talk is about. It's about–it's about a different way to think about QM that hasn't gotten very much attention and dispels this idea that quantum mechanics is sort of intractably weird. Somebody has said to me once that quantum mechanics obeys the law of conservation of weirdness. And to a certain extent, that is true. There is a certain amount of–quantum mechanics extracts a toll on your intuition and some of that will never go away. But I don't think that quantum mechanics needs to be fundamentally any more incomprehensible than, say, relativity which most technical people seem to have no trouble wrapping their brains around nowadays. So with that sort of expectation management out of the way, I want to start out by inviting you to think about the question, "What does it mean to 'measure' something?" So, imagine that we're sitting here doing some experiment. We have some system–let me grab a pointer–that we want to measure some property of it so we have some, you know, sensor here like a camera, and it gathers some data and we feed it to a computer, and that data shall pop up from the screen, and we look at that with our eyes, and we form some mental image in our head, and how do we know that this mental image that we form in our head actually corresponds to underlying physical reality? Well, one indication that we have of this is that we can do experiments more than once and observe that we get consistent results. So, for example, what color is this? >> Green. >> GARRET: Green, yes. So we can observe through our common everyday experience that the results of measurements are consistent across space and time. And, I mean, this is really a very reliable aspect of our universe, but it's actually a very deep mystery why this is so. And Einstein famously said that, "The most incomprehensible thing about the universe is that it is comprehensible." We can actually do experiments and get results that are consistent across space and time and we don't really know why that, any inherent reason, why that should be the case. Now, there is one very plausible sounding explanation of why this is the case, and that is that the results of these measurements are actually an accurate reflection of some underlying metaphysical reality. That reality, that there really is a universe out there and when we measure it, we're getting back actual information about that underlying physical reality. And that is the reason why these measurements are consistent because reality sees to it that that's the case. Well, it turns out that we can demonstrate that that's not true. I'm about to lead you down a rabbit hole but my purpose in leading you down this rabbit hole is to do it in such a way that you can find your way back out again. So I'm going to do it very carefully, step by step, and tell you in advance where we're going. I'm going to start out by reviewing the usual QM story. What you will read if you go to a popular account of quantum mechanics that you read, you know, pick up at Amazon or a bookstore or read about it in Wired or whatever. I'll then show you how that story can't possibly be true, because if that story were true, it would lead to a violation of relativity, in particular, it would lead to faster-than-light communication. And it doesn't do this in the usual way that most people think that it leads to faster-than-light communication, it does it in a more subtle way that really hasn't gotten a lot of attention. So you physicists in the room, bear with me. Then, I'm going to walk you through some of the actual underlying mathematics of quantum mechanics, in a way that is accessible to anyone who knows–can do basic algebra and knows what algorithm is. And finally, tell a new story based on our understanding of what the underlying mathematics actually says about what's really going on and hopefully we'll achieve enlightenment at the end of that. So, is there anybody here who has not heard of the two-slit experiment? All right, good. I will just blast through this very quickly. So, we have a–this is–you have a laser that shines through two-slits, and you get an interference pattern that shows that light is a wave and can interfere with itself like any other wave. And there are two strange things about this. If you look at the results of this experiment with very low intensity light, what you find is, and this isn't showing up very well, but this top image here shows just some dots scattered randomly. And then dots get denser and denser and denser until down here at the bottom you have a dense enough pattern of dots that you can start to see this interference pattern start to emerge. And this is an actual photograph of laser light going through a single slit and going through two slits. And you can see this interference pattern here, this is actual, an actual photograph of the same experiment, this particular one happened to be done with electrons but the underlying physics is the same. And the–the thing to notice here is that the total amount of light that you get in this pattern when there are two slits is brighter than the overall amount of light that you get with one slit which is what you would expect but that there are some places here where you have these dark bands that were bright up here when you only had one slit. And this is the interesting part that you want to kind of focus your attention on because what this means is that there's a spot here where light was shining and then you open up an extra path for light to get to the screen and that spot goes dark. And that is the manifestation of interference. But the strange thing about it is that this is not a continuous phenomenon, it's an accumulation of all these particles. Now, I can actually–I used to think that this was a fairly subtle experiment that you need a specialized equipment to conduct this experiment. It turns out it's not true, you can actually do this experiment yourself. These are some pencil leads that I've taped together with scotch tape and this is an ordinary laser pointer. And if I pass these, and there's just some very narrow gaps between these leads, so you can actually see this happen if I pass the leads in front of this pointer, you can see the light start to spread out. And if you're close enough, you can actually see the interference bands. I don't think you can see it in the back. But if you're interested, after the talk, come up, I'll give you a closer look at it. You can actually see the interference pattern. The point here is that this is not a subtle phenomenon and it's not something that you need expensive equipment to reproduce. This is an everyday experience for modern humans, at least. Okay, so this is not yet intractably weird, because there are all kinds of explanations that we can postulate about how this might be happening. So, for example, photons and electrons might be real particles that have real locations and velocities like our intuitions about particles that might be pushed around by some kind of underlying wave. And–oh, I forgot to mention–whoops–the location where these particles accumulate is random, there's no known way to predict other than statistically where these particles are going to end up on the screen. So, the randomness might just be due to some underlying real physical property that we just don't know how to measure. But it turns out we can eliminate this possibility as well. And the way we do that is by asking–by trying to track the path of a particle and ask with–on its way to the screen, on its way to producing this interference pattern, which of these two slits did it go through? And we could do that. We can add detectors to the slits and we can measure which of the two slits a particle went through. But it turns out that when we do that the interference pattern goes away and the phenomenon that we were trying to get a better grip on has changed. And it turns out that this is an inherent feature of quantum mechanics, that any modification that we make to this experiment that allows us to determine, even in principle, which of these slits this particle went through destroys the interference. This is the famous wave particle duality, any modification that allows us to determine even in principle–yeah? >> You ask people in the VC to mute their mics so… >> GARRET: So, I've been asked to ask the people on the VC to mute their mics, did I get that right? Okay. Okay, anyway, so the conclusion from this observation is that something has to be–something has to be at both slits in order to produce interference. And the reason we know that is because we don't actually need both of these detectors, one of them is enough, because if we have one detector and it fails to register that we know the particle went through the other slit. Now, that particle going through the other slit, it never interacted with anything, the particle never interacted with anything, but because it allows us to know where the particle was even though we didn't actually measure it, that's enough to destroy the interference and so this particle that's over here must somehow have known that we were looking over here even though the particle itself wasn't there. So, something must have been there to be able to tell that we had a detector here, but we don't know what that is. Now, it turns out, this is again, this is a universal property of quantum mechanics. It holds for any kind of particle–in practice, that means photons and electrons because that's all there is in this universe unless you start getting into nuclear physics. And any kind of measurement, and any kind two-slit experiment, any experiment where you provide two different paths for the particle to potentially go down and bring it back together without knowing which way it actually went will produce interference and any modification that you make that lets you figure you out where it went will destroy that interference. Now, this is still not intractably weird because we can still tell a reasonable story about why this might happen. So, maybe measurement does something to this system. These are after all very small particles and very delicate systems and so maybe it's just physically impossible to make a measurement without disturbing the system in a way that is the cause of the destruction of this interference. Maybe the wave function collapses and becomes a particle somehow, this is the famous Copenhagen interpretation of quantum mechanics. But it turns out that we can rule out that possibility as well. And the way we rule out that possibility is by asking, "How and when does this collapse, this purported collapse, happen?" Collapse has a number of features that ought to make us very suspicious of it just at priority without even doing any experiments. It's a discontinuous and nonreversible phenomenon that once you know that a particle has gone through one slit or the other, you can't go roll back time and undo that. And if you look at the mathematics of quantum mechanics, which we'll get to later, there's nothing in the math that's discontinuous. And more than that, all the math is actually time reversible so we can make a–hypothesize that this collapse happens, but this is fundamentally at odds with the mathematics of what quantum mechanics–of how quantum mechanic says that our universe works. So we can actually do better than that. We can actually do an experiment to show that collapse, if it happens, is a much subtler phenomenon than it would've first appear and this is the famous–this is the quantum mystery number two, the famous quantum eraser. Now, I–this is a two-slit experiment that I have now reduced to something more abstract. So we have some particle source, this can be photons or electrons. We have some abstract way of splitting up particles so that it has two different paths to go down and some abstract way of recombining that particle so that both of those paths end up in the same place so that we get interference. And here notionally, we have one detector at one of these dark fringes and another one at a bright fringe so that if we introduce some kind of an abstract measurement on one of these branches then the interference pattern fusses out, that we now have the path, the amount of light, in each detector. So now we don't have fringes anymore, we have the spread-out pattern that we saw and the single slit version of the experiment. Like I said, there are lots of different ways that you can do this–actually, let me go on to the next one. So it turns out that you can erase this, that there are physical ways that if this measurement, certain kinds of proto measurements, that you can do here, you can then go back and erase after the fact and restore the interference. And here's a concrete example of that. If we depolarize light and–I'm actually going to show you this in just a second, so bear with me if you don't understand what I'm about to say, you use polarized light and you do the measurement by rotating that light 90 degrees and then erase it by filtering it 45 degrees, that is an actual concrete example of a quantum eraser. And I can't show you the interference part of it but I can show you the erasure part. So what I have here is some Polaroid film, this is the same stuff that you find in polarized sunglasses. And, I first want to convince you, has anyone not played with this stuff before? Okay, so again, real quick. If the–if the axis of the film are aligned then you can see through it. Can everybody see through this? And if I rotate it at 90 degrees then you can't see through it anymore and that effect is independent of the absolute orientation of the film so the light that's going through to your eyes starts at unpolarized over here, it gets–passes through this film and becomes polarized, let's say in this direction, and I can demonstrate then that it has become polarized in that direction by filtering it out using a filter at 90 degrees. And there's also this cool adjunct to the experiment that you can do by adding a filter at 45 degrees. If you put it in front or behind, nothing happens which is pretty much what you–what you'd expect. But if you slide this in between, then suddenly you can see through it again. Pretty cool, huh? And the reason for that is because if you start out with polarized light and you filter it at 45 degrees then some of it gets through and it's now polarized in this direction. And now I can do the same operation again, which is now the relative orientation of these two are 45 degrees so some of it gets through again. But that's not what I want to show you, that's not the cool part. Cool part is this stuff. This is what they didn't show you, what they didn't show you in high school. This film is actually a polarization rotator.

If I'd stick this in here, I can actually take this light that's polarizing this direction, I can rotate it to 90 degrees that it's polarizing the same direction as this film. And the thing to notice here is that the apparent brightness here in the center is the same as it is up here. There's before, if we did the high school version of the experiment–whoops–you've gotten quite a bit of loss. So, I really can take–I can take light and I can polarize it and I can take that polarized light and I can rotate it by 90 degrees and so I can create two different paths so I can tell which way the light went through. I can tell whether the light is going through here or whether the light is going through here. Actually, let me back up a step. My claim is, without this filter, it's a little hard. My claim is that the light that's coming out of here is different than the light that's coming out of here, so I can tell which way it went. And the way that I can demonstrate that to you is with this measurement apparatus that lets me filter out this light and tell which way it went. So this is a measurement. This should collapse the wave function according to the Copenhagen interpretation. But I can undo this, and the way I undo it is by filtering at 45 degrees. And now, I have to ask you for a little bit of suspension of disbelief because this is not high precision optical equipment and my angles aren't aligned just right. And if you look very closely, you will actually be able to tell the difference between these two paths, but the difference is now much less than it was before. See that? Oh, I'm sorry. I didn't realize there are people over there. Okay, I'll just show you all this at close range afterwards. So there's a measurement. There's an erasure of that measurement. The light is going this way so the–in time, the erasure has to happen after the measurement. And if I actually had a laser to shine through this, I can demonstrate to you that the interference would go away and would come back. So again, these are not subtle effects and they're not effects that you necessarily need high precision equipment to reproduce, it's an everyday experiment. This is $30 worth of polarizing film. So this leaves us with the philosophical conundrum that is embodied by Schrodinger's cat. If there's no collapse, then if we set up a radioactive source that triggers some kind of a mechanism that will break a bottle of poison that will kill a cat that's in a sealed box, can this–what happens? Quantum mechanics says that this cat isn't a quantum superposition of being alive and dead which is intuitively absurd but as far as we can tell, that's really what happens. So if that's not intractably weird enough for you, this is the third quantum mystery entanglement which is usually described as sort of an ancillary phenomenon to all these other mysteries. And, oh, yes. It's sort of–oh, by the way, has everybody seen this picture? Has anybody not seen this picture before? Okay. This is what the production of quantum entangled photons really looks like. You take a–this is not–this is not an actual photograph, this is a drawing. But this is an actual photograph of the output of one of these gadgets. There's an ultraviolet laser that shines through a crystal of some material called beta barium borate, details don't matter, and this crystal has this interesting property that it will absorb photons of ultraviolet light and reemit them in–and that it'll kick an electron up to an excited state, and then that electron will drop back down to its ground state and it will do it in two steps. And so it will kick up in one step, down in two, and in the process of coming back down, it will emit two photons instead of one so what comes out of this system is visible light. And the photons always come out in pairs and they come out in matched sets because of fundamental conservation laws. We have law–conservation of energy and momentum and so–and electron–a photon that, say, comes out over here is a red, photon will be matched by one that comes out over here as a blue photon, and the same thing over here. And in the middle, you get this band where you get photons that come out at the same wavelength and it just matched in position so a photon over here will be matched by one over here. And they'll also be matched in polarization as it turns out. So this is the way it's depicted conceptually. You have these ultraviolet lasers. It's called a down convert–this crystal is called a down converter, and you send these photons off to opposite sides of the universe and what–and then you measure some property out of them. Let's say, we split them according to polarization or position them in, it doesn't matter, any quantum state variables, as long as you can filter them that way and what you find is that they're perfectly anti-correlated because of the conservations laws. So if you get a photon up here, on the right side of the experiment at the upper detector, that will always be matched by a photon over here on the left side of the experiment at the down detector. An unfortunate artifact of the English language that words "left" and "lower" both start with a letter L so I'm going to switch back and forth between them. This is what the–what Einstein famously called, "Spooky action at a distance." If you take this phenomenon, this isn't controversial, this is an experimental–an undeniable experimental result, this really does happen, combine that with a fact if there is no collapse with the wave function and the inescapable conclusion seems to be that as it was put in Wired as recently as last June, that when an aspect of one photon's quantum state is measured, the other photon changes in response even when the two photons are separated by large distances. And this would seem to be impossible because it would seem to violate relativity because it's an instantaneous effect and we know that we can't communicate information faster than light, because if we could do that then we could communicate information back within time and that would cause all kinds of problems with causality and just be a horrible mess. So these instantaneous effects are supposed to be impossible but there is one thing that comes to save us and that is this quantum randomness. We don't actually have any control over whether the photon on one side ends up at the upper detector or the lower detector and so we can't actually send information up here to there, we know that if we see the photon up on the upper detector over here, then our counterpart across the universe must have seen it at the lower detector over there but we haven't transmitted any information from A to B. And you can actually prove this mathematically that it's impossible to transmit information using this phenomenon. But it turns out that the proof of the impossibility has a loophole. So that is the end of step one. I'm now going to on the step two and show why the story that I've just told you can't possibly be true. So let's summarize and take stock. A split/combine experiment produces interference. Any which-way measurement destroys that interference, there's some which-way that the proto measurements that we can go back and erase after the fact and restore the interference and measurements on entangled particles are perfectly anti-correlated. So the quantum conspiracy is that all of these things cannot possibly be true and here's why. So this is a thought experiment. This experiment has not actually been done that, again, with tongue slightly in cheek, I've done the Einstein-Podolsky-Rosen-Garret paradox, and it's two-slit experiments that are fed by quantum entangled photons produced by one of these down converter setups in the middle, and I want you to consider the question of if we measure on the left, do we destroy the interference on the right? So, if the answer is yes, then we have faster-than-light communication because this interference is a macroscopic effect. It's really easy to see if you have interference or not. You just look at it with your eyes. You don't need any kind of delicate detectors or anything so you just measure over here and take measurement away, measure, take the measurement away, and over there on the other side of the universe, this interference pattern will come and go and you can send Morse code instantaneously. That's obviously impossible, so the answer must be no. But if the answer is no, then we know the position of one particle but we have interference regardless and that contradicts the fundamental principle of quantum mechanics, which is that we can't know the position of the particle and still have it interfere. Now, this is not yet an iron-clad argument. There is one other possibility that I have not mentioned here. Can anybody think of what it is? Any of the physicists in the crowd? Oh, good. So, this one last possibility, and that is that if there was no interference to begin with. It might be that entanglement sort of counts as one of these subtle proto-measurements that destroys the interference so we didn't have any interference to begin with. But it turns out that doesn't get us out of this faster-than-light conundrum because, fine, if that's the case, then we can still produce faster-than-light communications by putting in a quantum eraser and destroy the entanglement. Entanglement is a very–is a very delicate property. Physicists work very hard to produce it and maintain it. It's very easy to destroy. So, just destroy the entanglement and produce interference where there was none before and, again, we have a faster-than-light signal-ly mechanism. Now, that is a very compelling argument that the story that I have told you, the usual quantum story is wrong. That argument is, in fact, correct. The story that I have told up until now is, in fact, wrong, and that's why. >> You said this experiment hasn't been done, why not? >> GARRET: Because all physicists know what the outcome will be. And I'm about–I'm about to tell you what the outcome will be. And by the time I finish telling you, you will be convinced enough that the outcome is what I tell you that it will be that you won't need–feel the need to do the experiment either. I promise you–hmm? >> GLECKLER: Why don't you repeat the question. >> GARRET: Oh, I'm sorry. The question was, why hasn't this experiment been done. Yeah? >> So the entanglement is about polarization, in which way it goes through a splitter, is that about exactly the same polarization, we know they're the same? >> GARRET: That's right. So, I assuming here, like I said, there are lots of different ways that you can do the split-combine experiments. You can use some thing called a Mach-Zehnder interferometer. You can use something called a Stern-Gerlach device. You can, you know, lots of different ways. When I talk about polarization, the way that the splitting is done is with the device called the polarizing beam splitter, which is exactly like one of these, except instead of just absorbing half the photons and letting half the–half of them go through, it reflects them. So it look like a mirror, a half-silvered mirror, and what comes out this direction is photons that are polarized one direction and what gets reflected at the other direction are photons that are polarized in the opposite direction. Okay, so we are now in the depths of the rabbit hole and now, it's time to find our way back out. We're going to do some math. Don't panic, it will not be as bad as you think. This is the mathematics of quantum mechanics right here. This is the famous Schrodinger wave equation. It's–this is the free variable here is this thing called psi. Psi is the quantum wave function and it obeys the dynamics of this partial differential equation, which those of you who are proficient in partial differential equations, will recognize as a wave equation like any other equation that describes waves, and that's why these particles seem to propagate like wave because this is the map that describes how they propagate. The point here is that these, the dynamics of this thing, are continuous and time-reversible. All wave equations have continuous time-reversible dynamics. And the second part of quantum mechanics is that you take this quantum wave function, which is a function of position and time and is a complex number, you take the norm, the magnitude of that complex number and square it, and that gives you the probability of measuring a particle at this position X at a time T. That's really all you need to know about the mathematics of quantum mechanics. There are some things to note about this. There is this distinction between the underlying amplitudes, which are complex numbers, and the probabilities, which are of where we find these particles, which is the only thing that we can measure which are real numbers. And the reason that particles can interfere is because complex numbers with magnitudes greater than zero can add to zero. You can have a complex number that points off in this direction, another complex number that point off in that direction. They always have magnitudes greater than zero but they can add and destructively interfere. The dynamics are continuous, time-symmetric, fully deterministic and hence, reversible, and so there's no place where you–there's no place you can find anything that resembles collapse in this math. And no randomness either, by the way. Going from amplitudes to probabilities by taking this wave function and squaring it has no physical justification or whatsoever. It's purely a hack. But it's a hack that works really, really well. So here's what the actual math looks like for the two-slit experiment. This is, and if you go to actual papers on quantum physics you won't find this in the popular press, this is what you will find in physics journals. This is–or physics texts–this is the state, the amplitude of the photon being at the upper detector. This is the amplitude of photon being at the lower detector and we have to divide by the square root of two in order to make the total probability come out to be one. So to figure out the probability, we take this number, which is a complex number, and take the modules and square it. And when we do that, it's almost exactly the same as just squaring A plus B in 8th grade algebra class, you get A squared plus B squared plus AB plus BA. You have to take these complex conjugates because they're complex numbers, and the square root of negative one pops in there to do some weird things, those kind of details don't matter. The point is this is a complex number, psi U is a complex number. Its magnitude is a real number and when you square it it's a positive real number. So here we have a positive real number, here we have another positive real number, the sum of those two has to be a positive real number. But over here we have two different complex numbers and two different complex numbers that were multiplied together so this sum can be negative. So this is–this is where the interference comes from. This is the mathematical manifestation of interference in quantum mechanics. That's what it looks like in terms of Greek symbols. So, what happens when we add detectors? Well, when we add detectors, the amplitude starts to look like this. We have the amplitude for the photon to be at the upper detector times the amplitude for the detector to be in the state where it shows that the particle is at the upper detector and the same thing at the lower detector. So this is just the mathematical description of that. And when you–when you–when you take the amplitude to that and square it, here's what you get, same thing as before, psi U squared plus psi L squared. And this, which looks an awful lot like an interference term, right, which is weird because I just got though telling you that if we have a detector that we know–tells us which way the particle went, that destroys the interference. Well, there's this subtle difference here between what we have now-what we have before, and that's this weird notation here, which is called the rock–bracket notation. You don't need to concern yourself with it. Just take my word for it when I tell you that this quantity here is the amplitude for the detector to spontaneously switch between indicating that the particle is at the upper slit and the lower slit. In other words, it's a measure of the reliability of the detector. That just comes out when you do the math. And if the detector is working properly, that value–oops–this–this value is the amplitude for it to spontaneously switch between UNL and spontaneously switch between L and U, those are both zero so this term goes away. That's the math of how measurement destroys probability. And the interesting thing about this is that measurement is a continuum. It's not a dichotomy. The math tells us that we can measure just a little bit or we can measure mostly but not quite. And we have vary–it's a varying levels of interference that we get depending on whether the measurement that we're making is reliable or not. That's what the math says. So what about entanglement? Well, this is, if you go to a physics paper that talks about entangled particles, this is what you will see as the mathematical description of a pair of entangled particles. What this means is that you have an amplitude for the particle on the left to be in the up state and the particle on the right to be in the down state superimposed with an amplitude for the particle on the left to be in the down state and the particle on the right to be in the up state; again, divided by the square root of two. Now, this looks a lot like the–or the unmeasured two-slit description. But there's some notational sleight of hand going on here because this is shorthand for this and it's an unfamiliar notation. This vertical bar, followed by the bracket, this is a term. So you got–this is a quantum wave function here. This is another quantum wave function here. This is the wave function for the upper particle. This is a wave function for the lower particle. Another way to write that is–you don't have to use arrows, that's just a notational convenience, so I could call this the left upper particle and the right downward particle, and the left downward particle on the right upper particle. And this is just another way of writing this Psi, this quantum wave function. So this and this are the same thing in different notations. And this should now look familiar. This is exactly the same as–oops–as this, the two-slit experiment with the detector, module of a few labels. And so, that is now the answer to the first part of the EPRG Paradox. In fact, entanglement does count as a proto-measurement that destroys interference. But it's actually much deeper that that. According to the math, entanglement and measurement or the exact same phenomenon, the math is exactly the same, and that is why entanglement destroys interference because entanglement is measurement. And I have a lot more to say about that later in the talk. So okay, so there's no interference but now what about this last–the idea of creating the interference using a quantum eraser. So let's take another look at our–I'm running little short on time so I'm just going to blast through this. This is what the state equation looks like for the quantum eraser after the so-called measurement but before erasure. So you have an upper photon that's horizontally polarized because we've–let's assume we start with a vertically polarized light going in here and we measure by rotating 90 degrees. So we have now the upper photon rotated from vertical to horizontal and the lower photon still vertical and this you will now–you should recognize as a measured and therefore non-interfering state. And it turns out that if you filter now at 45 degrees this is the state function, the quantum wave function, that you end up with. You now have a photon that's either in the upper or lower slit and that's either horizontally or vertically polarized, and this kind of makes sense because if you think of these as vectors and you have a horizontal polarization plus a vertical polarization, that's a 45-degree polarization, which is exactly what you would expect to see if we're filtering it 45 degrees, right? But remember the square root of two term here that I told you was there in order to make the total probability to come out to be one? Now, that's a two root though, and if you run the math on this you find out that the total probability is not one, it's one-half. So either we've made a mistake or half our photons have gone missing. Well, in fact, half our photons have gone missing which is also shouldn't be too surprising because we filtered–we put this 45-degree–we put this filter in place. This filter is filtering out half the photons that go through it. If it–if they come in at 45 degrees then half of them come out polarized this way and the other half get blocked. So it turns out that the other half, the half that didn't get through, have a different wave function that has a negative sign over here, which again, makes intuitive sense because the filter lets these–the filter is at 45 degrees so it lets this axis through and this axis, which is the H minus V axis it blocks. And these photons interfere with themselves and these photons also interfere with themselves. So the photons that passed through the filter display interference fringes and the photons that don't pass also display interference but it turns out that they're anti-fringes. They're exactly like the bright spots in the interference fringes for the photons that got filtered out, exactly lined up with the dark spots of the fringes for the photons that were let through. And they sum together to produce what we perceived when we look at it as non-interference. So this quantum eraser doesn't actually erase anything and it doesn't produce interference, it just filters out interference that was actually already there all along. And it turns out that we can actually do this in the EPR experiment too. And the way that we–but in order to do it, we have to transmit classical information from one side of the other in order to do the filtering. The way it works is you make a record of all the photons that you collected over here and keep it in order so you've got this record of first photon was at the upper detector, the second photon was at the down detector, and so on and so forth, and over here you keep track of which photons ended up where on your screen, and then you take this record and you transmit it over here by some classical slower-than-light channel. And you look at all the up photons and sure enough there's an interference pattern, and you look at all the down photons and sure enough there's an interference pattern. But the only way to see that is to take classical information and move it from here to here. And that is the last nail in the coffin. I was very disappointed when I learned this 10 years ago because I was really counting on winning a Nobel Prize and taking over the world but, oh, well, this is the next best thing. So the take-home message up to this point is measurement and entanglement are the same phenomenon, and what you will find in many, many accounts and even some professional accounts, is that they're completely different. That measurement is this common everyday thing that we can sort of intuitively grasp and entanglement is the quintessential quantum mystery and in fact they are really the exact same thing. Now, having come to that realization we can now tell a different story about quantum mechanics that to my software engineer's mind is much more intuitively pleasing than any of the other competing alternatives. So Copenhagen is the most popular but as we've seen it, it's scientifically untenable. There just is no collapse. Their–the next most popular interpretation is the so-called "many worlds" interpretation where it says that anytime that a particle can go multiple ways, the entire universe splits and the math actually supports that, but I personally find that that takes a heavier toll on my intuition than I'm really willing to concede. There's another thing that nobody's ever heard of called the "transactional" interpretation by fellow named Cramer at the University of Washington. Actually, if you're really interested in this stuff, I encourage you to take a look at because it is kind of interesting. It postulates that the backwards in time solution for Maxwell's equation are physically real and if you make that assumption then you can explain a lot of stuff, but I don't have time to get into that. What I want to talk about here is the quantum information theory which I have dubbed the "zero-worlds" interpretation of quantum mechanics. It's an extension of classical information theory with complex numbers and if you run through that math you get some very interesting results. So here is a lightning introduction to classical information theory. It's the study of this quantity called the Shannon entropy of a system A, which can be in any one of a number of classical states, and it's defined as the sum of the probability that the system is in sum state A times the log of that probability and then you take a negative sign. And intuitively, it's a measure of the amount of randomness that's in the system A. So just to simplify things for the purpose of this talk, if the system has an equal probability of being in one of N states, then the entropy is just the log of N. So when N is one and the system is definitely in one state then the entropy is zero. And if it can be in one of two states with equal probability and we take this log base two, we measure information content in bits, then it has one bit of randomness in it. You can define all kinds of other derived quantities like the joint entropy of multiple systems and the conditional entropy, and this quantity here which is called the information entropy which is a measure of how much information a system A contains about a system B, and the interesting thing to note about it is it's the sum of some of these other quantities that had been defined up here. And the information entropy ranges between zero and one, where zero means that this system has no–system A has no information about system B, they're completely uncorrelated, and one means that they're perfectly correlated. So for example, because they're sums we can describe these quantities as Venn diagrams. So this circle here on the left is system A and this circle on the right is the system B, and the total entropy is contained inside these circles, the total entropy for each system, and the information entropy is here in the intersection, and that leaves the conditional entropy out here because the information entropy is the system's individual entropy minus the conditional entropy, just simple addition. This is the important part. If we flip two coins so that they're completely independent of each other, this is what the numbers end up looking like. The conditional entropy, the coin A has one bit of randomness and coin B has one bit of randomness so the total entropy in the system is two bits of randomness. The system as a whole of these two coins can be in one of four states, the log of four is two, and there's no information that one coin contains about the other. By way of contrast, if we have a coin with a sensor, just looking at that coin telling us whether it's landed heads or tails then–and the sensor is working properly, then we have one bit of information entropy because the sensor gives us perfect information about the coin and vice versa by the way. The coin gives us perfect information about the sensor, there's no directionality here, and the total entropy in the system is one bit so it's only going to be in one of two states, heads and sensor says heads, or tails and sensor says tails. If we extend, do the same math again except using complex numbers instead of real numbers then you end up with something called the Von Neumann entropy which is called S and this hairy-looking equation over here, which I don't have time to go into, but the intuition is kind of the same as it was before when we talked about how interference was produced. Because we're now dealing with complex numbers rather than real numbers, the information entropy is no longer restricted to the range zero and one. And, in fact, entropies are no longer restricted to be positive real numbers, they can be negative. And this turns out to be, if you do the math, the entropy diagram for a pair of entangled particles. You get a negative bit of entropy over here. And the information entropy, the amount of information that one particle quantum information now, that one particle contains about the other is two bits. So you can think about two entangled particles, the math is telling us that these particles are now, somehow, better than perfectly correlated. They have become super correlated and the total entropy of this system, the sum of all these numbers, is zero. There's no randomness. That's not yet the cool part. What happens when we take a measurement? Well, when we take measurement, we have a particle that becomes entangled with a macroscopic system of particles. So what happens if we have three mutually entangled particles? You end up with a Venn diagram that looks like this and if we assume a two-state system then the actual numbers come out looking like this. You've got one bit of information entropy between A and C, one bit between A and B, and you've got this negative, this weird negative entropies over here and it's all kind of mind-boggling. But let's imagine that this particle down here is the one that we're measuring and this is our–these two particles here are our measurement apparatus. And let's look just at the measurement apparatus and ignore the fact that we're actually measuring a particle here. So we're going to take this particle C, we're just going to throw it out for a minute. It turns out that ignoring C is exactly what is represented by this trace operator here that ends up showing up in the math. Look what happens. This one bit of information entropy–we lose this boundary so this one, the negative one, cancel out and become a zero, same thing over here, and what we have is if we ignore this is exactly the same system from an information theoretical point of view as a coin with a sensor, we have two classical particles that are perfectly correlated with each other in a classical sense. I should remind you of the experiment that we did at the beginning of the talk where everybody agreed that that's something was green. And we can get that from quantum mechanics, we get these two systems that are in classical correlation. But we did that not by actual–not by having an objective physical reality that we reflect but actually by ignoring the thing that we're measuring, or that we think we're measuring. As it turns out that this extends to any macroscopic ecosystem. If you add an arbitrary number of particles the entropy diagram ends up looking exactly the same. So, this is now the mathematical description of a quantum measurement. You have the system that you're measuring. It's particle Q. It interacts and gets entangled with A particle, which is in the parlance of the theory called an ancilla which is why they label it A, and that ancilla then gets entangled with a macroscopic measurement apparatus in system of 10 to the 23 particles, and the entropy diagram ends up looking exactly the same where this entire system has the same quantum information, theoretical information content, as the third particle in a three-particle entangled system. So, that is now a description of what measurement looks like purely in terms of quantum mechanics. And the interesting thing about that is it describes all of the microscopic phenomenon that we see that we naively observe about classical measurements but it's purely in terms of quantum mechanics, which means that it's reversible. So, their–so, somehow, we ought to be able to undo an actual physical classical measurement but in practice we can't seem to. And the reason for that is because in order to do that–in theory, it's possible, but in practice we would have to undo all of the entanglements in this macroscopic system. So, for me, to now go back in–back in time, you know, we aren't really going back in time, but for me to erase all of your memories of having seen this green thing on the screen and agreed that it was green at the beginning of the screen, I would have to undo this enormous web of entanglement that has since proliferated at the speed of light. I have to bring all those particles back together and recombine them. And in principle, that's possible and in practice, obviously, it's not, which is why classical measurement seems to be irreversible despite the fact that the physics of the universe say that everything is reversible. So, this has some philosophical implications. I call it the zero universe interpretation of quantum mechanics. If you really–if you buy this as a description of what the physics of the universe is really like, then it tells you unambiguously that it is not the case; that the reason that measurements are consistent across space and time is because there's a real underlying metaphysical reality out there. It tells you in fact, the exact opposite, that what we really are is, as David Mermin puts it, "correlations without correlata." We are not made of atoms we are actually made of bits. We are our thoughts and these thoughts actually reside, if you will forgive stretching a metaphor to the breaking point, we are a simulation running on a quantum computer. I'm going to skip that. So, the take-home message is, going back to this Einstein quote that, "The most incomprehensible thing about the universe is that it's comprehensible." Here's an explanation in terms of physical theory of why the universe is comprehensible. Quantum mechanics actually predicts a comprehensible universe but at the cost of forcing you to believe that what you perceive as physical reality is not actually real it's actually an illusion. And the motto here is that "spooky action at a distance" is no more and no less mysterious than the "spooky action across time" that lets us perceive the universe as consistent from one moment to the next. They're both produced by the exact, same physical phenomenon, namely entanglement. And I'll leave you with this quote from an ancient Japanese Zen master and open the floor for questions. Uh-oh, the physicists are walking out. Yeah? >> So you have the three particles when you remove from the shell what the main two look like and they were building to the point in the sensor. Why not leave the–what happens when you leave a third particle away? And if you're taking that as another thought experiments and say, "Well, this is equivalent too." >> GARRET: Yeah. Well, this is–so these are all just math, right? >> Right. >> GARRET: So, these are all mathematical manipulations, the results of which we interpret in order to tell stories about what our world is like. And if you leave it in, then what you have is a description of the unadulterated underlying physical reality, which is quantum. And the reason that's hard to wrap your brain around is because your brain is classical, everything that you are is classical. You're made of classical bits that ones–or you're a Turing machine, you're not a quantum computer, but you're made out of a quantum computer. And that's why there's this fundamental disconnect that will always take a toll in our intuitions that will never go away because they're really fundamentally different. The difference between real numbers and complex numbers, the underlying reality is complex, but the thing that is processing the information that lets you think about these things is real. It's made of real numbers. That's really the underlying–the pithiest way I know of summarizing this so it just depends on what point of view you want to take. >> I'd like to make a short pitch for the multiple universes and hear your response. The multiple universes, one way to look at it is the natural equation, it allows for–because it's linear, you can have things occurring that live in the same equation and sometimes, say, with the dead and a live cat, you can see that you can actually sort of separate them, each goes its own way, they don't interact much, so a very good approximation is to consider them one at a time. However, when real quantum effects go then you cannot do it this way. So, there are–there are some things that you can separate and some not. So, now, you're saying that you have an interpretation of a one universe so… >> GARRET: No, not one. Zero. >> Zero, all right. >> GARRET: Very important distinction. One universe, one classical universe really is untenable. >> Well… >> GARRET: You can–you can–that's Copenhagen. >> Yeah. >> GARRET: You can–you can–yes, the–so the question was, what do I think about multiple universes? Multiple universes are just as tenable, according the math, as zero universes. The only thing that's not tenable is one classical universe, that's–that's the only thing that the math tells you unambiguously does not exist. And it's a matter of taste. You know, I personally–one of the things that the–that the math tells you if you think about it in terms of multiple universes, is that once you get beyond a certain level of separation these universes are forever inaccessible to us, even, you know, to any reasonable degree of approximation. And then there's this philosophical question of is the thing actually real and the analogy that–the best analogy that I've heard is imagine a photon that leaves the back of the sun and goes away from us and so it's forever outside our light cone, is that photon real? And my reply to that is there's a difference between the photon that leaves the back of the sun and travels away from us because there's always the possibility that somewhere out there's a mirror that's going to reflect that photon back to us and we won't know that until that mirror actually reflects it and it comes back and we can see. But in the case of multiple universes, we know unambiguously the math tells us so that once another universe splits off, it's never coming back. There's no way to bring it back. >> After a concurrent time, it will. >> GARRET: Excuse me? >> After a concurrent time, it will. >> GARRET: Okay, in any amount of time that we have any practical reason to care about. So, yes, if you're thinking cosmologically, multiple universes is a–is a tenable interpretation. If you're thinking in terms–if–if you want a story that you can help you to understand how the universe works in terms of your everyday life and what your fundamental nature is, I personally find that I gravitate more towards the information theoretic point of view and believing that–that I'm–I–the universe that I exist in is a very good high-quality simulation. But that's a matter of taste. >> So do multiple universes conserve mass energy? >> GARRET: That I don't know. I have to defer that to the physicists. That's a very good question. I don't–I actually don't know the answer to that. >> Could you amplify a bit more for what does a zero universe mean? >> GARRET: This–oh, by the way, the question back here was do multiple universes conserve mass energy. It's a very–a very good question because–and I don't know the answer. I should–I should ask–I need to ask a physicist who… >> I've asked this question of "many worlds" proponent and the answer I got is, yes, that you're not actually doubling the amount of mass, you're dividing the amount of mass into two effectively but… >> GARRET: Oh, okay. So somebody in the audience is saying that when universes split, the total amount of mass in the universe gets evenly divided between the two universes and I'm guessing that math works out in such a way that if you reduce the amount of mass in a classical universe uniformly by one half, that everything ends up working out the same as it did before. But I'm going to have to think about that. >> So this is just even one universe does not conserve mass energy with anything. Universes as whole… >> GARRET: Okay. >> And yet, in one–in the universe we're in, we don't seem to see this mass dwindling away. >> But it's, you know, because we're doing the same… >> GARRET: Yeah. So in that, we are now beyond the limits of my knowledge of this stuff. Yeah? >> For those of us who are intrigued and teased, where can we learn more? >> GARRET: So, I have paper about this. It's on the web. I highly recommend David Mermin's book, Boojums: All the Way Through, where he doesn't actually talk about this but he talks about the Bell inequality in a very accessible way, which is also–I didn't have time to talk about that but it's very worthwhile knowing about. Or send–send me an email. I can actually put you in touch with the guys whose research this talk is based on. They're down at Caltech, I think. Yeah? >> So is this an entirely local theory that all interactions are interacting locally propagating at the speed of light, or is it? >> GARRET: Well, so this get–so, the question was is this a local, purely local theory. It's quantum mechanics. And whether quantum mechanics is purely local is the subject of debate. And it depends–it depends on what you mean by purely local. Classically, it's not, and quantum mechanically, it is. You know, I–this sheds no extra light on that question. Okay, I guess, that's it.