Scott–Potter set theory

Transcription

>> LLOYD: So it's great to be here. Perimeter Institute had me give a popular lecture last night, but, of course, the reason I'm really here is to find out what's going on here because this is the epicentre of quantum information processing and quantum computing in the world. And so I am really here to learn about stuff. I also need to apologize to Olaf Dreyer. Last night, when I said that the birds were quick to escape Hamburg in the wintertime, I would like to point out that they're also just as quick to come back in the spring, not merely for the good beer. Alright! I'd like to talk to you about the Quantum Mechanics of Closed Timelike Curves. Of course, this sounds pretty wacky and it is wacky because it is the quantum mechanics of time travel, essentially. In quantum mechanics we are accustomed to the situation where our intuitions don't work out to be true, and when you add in time travel, your intuitions doubly don't work out to be true. Now, this may sound kind of wacky fictiony, and in fact if you look at the history of time travel there's a long literature about time travel. Interestingly, for most of it, if you go into the distant past, it's about time travel to the future, which actually isn't that hard because we're doingthat right now. In the Mahabharata there's a great scene where a king goes to visit Brahma in his beautiful palace and he's there for a few days partying like mad. When he comes back countless eons have passed and his entire civilization has decayed into dust. There's a beautiful Japanese story called Urashima Taro where a fisherman, Urashima Taro, frees a sea turtle from a net and then a few days later a very beautiful turtle princess comes and invites him to the house of the Sea King. He takes her invitation, and they're married. It's a great time and he lives there in great splendour for three years, but he misses his parents and so he begs her to let him go back. So she says "Ok, you can go back and I'm going to give you this box covered with beautiful seashells, but you must not open the box." You can already see where this is going. So Urashima Taro comes back, he lands on the beach of his island. Everything looks strange. He goes to his village, and his village is no longer there. He finally finds this monument to his parents and to himself saying that he died 300 years ago. In despair, he says "Well gosh! Maybe if I open the box I'll be able to see them again" so he opens the box and this mist comes out and envelopes him. He feels himself rapidly turning into a very old man, and then he dies. There's another, an Irish legend about Finn McCool, the famous Irish hero, who goes to visit the King of the Fairies and he stays there for a few days partying like mad, which if you've read the legends of Finn McCool you know he could really party like mad, and he comes back. When he goes away, the King of the Fairies gives him a magic horse, which he rides back on, and he tells him 'Whatever you do, don't get off the horse.' So what does he do? Of course he gets off the horse! Just like Urashima Taro, whenever they give you a magic talisman and tell you not do something, you always do it. So he gets off the horse and as soon as his foot touches the ground, then he turns into an old man and dies. Right, we detect a pattern here in these stories. So the first story is about travel into the past. Time travel into the past shows up in the 1700s, but they don't really get the notion of time travel. There's also Mark Twain's famous story, A Connecticut Yankee In King Arthur's Court, in which this Connecticut engineer is hit on the head on the job site and he ends up in King Arthur's court, where he decides to modernize things and hence wreaks total havoc, destroys the civilization, et cetera, which is kind of a metaphor for modern times too. But it's not until H. G. Wells' famous story, The Time Machine, that we see the kind of picture of time travel that we're familiar with from movies and books today. In that famous story, there's a machine which the time traveller enters and it allows him or her to go backwards in time to a specific date and then come back to the future, or back to the present. Now, once these stories got started about real time travel, people really began to think about the real contradictions inherent in time travel. You've got to be careful in thinking about time travel because there are a number of paradoxes, which I'll describe. I'll describe how our theory and, I should say our experiment as well because with Aephraim Steinberg we did an experiment I'll describe to you, effectively sending a photon a few billionths of a second backwards in time. What's the first thing you would think of trying to do if you have a photon interacting with its past self? What would you have it try to do? Would you have it buy its former self a beer, which would be the nice thing to do? No! Of course, we have it try to kill itself! We have it try to go backwards in time and kill its former self and then we see what happens. I'll give you a hint. You know those movies where they say at the end 'No animals were harmed in the filming of this movie'? Well, to say that no photons were harmed in the course of this experiment would be an exaggeration. So in time travel stories, there are basically two fundamental paradoxes about time travel. One is the so-called Grandfather Paradox, where the time traveller goes back in time and, either inadvertently or on purpose, kills her grandfather before he meets her grandmother so she doesn't exist so she can't go backwards in time so what the hell is that about? How does that work? There are essentially two resolutions of this paradox. One is that when she does this, she can kill her grandfather and in doing so she enters an alternate world. So there's a famous--Is it a Ray Bradbury?--story called The Sound of Distant Thunder, where the time traveller goes back to the Jurassic Period, determined not to change anything, but he inadvertently steps on a butterfly. And because of the famous Butterfly Effect, when he returns to the present everything is weird. It's sort of like it was, but the politicians are different and the language is spelled in a different fashion, et cetera. So that's one version in which you enter into an alternative universe when you come back to the present, and the other is that you can't do anything that's inconsistent with the past. If you look at movies about this, for instance the famous movie, Back to the Future, not to mention the slightly less famous movie, Hot Tub Time Machine, which I haven't seen but I have had it described to me, is of the type where you go back into the past, you change the past, and you enter into an alternative future. The other type, which is exemplified by the movie and the book, Harry Potter and the Prisoner of Azkaban. Who here has seen this movie? Ok, great! Alright, I'm glad some people have. Don't you guys ever get out? I know we're a little far from town here, but come on! In that movie, Harry and his wizarding buddies are trying and doing all this stuff, and all kinds of weird crap is happening and they can't figure out what the heck is going on. But in the end, they figure out that what's happening is that they've been interacting with themselves coming backwards in time, but everything is self-consistent. So things are weird and strange, but they're self-consistent. For the Grandfather Paradox, this would correspond to a situation in which the time traveller is unable to kill her grandfather no matter what she does. I'm sorry, but I'm going to use you for this example since you're sitting in the front row. So she points the gun at her grandfather and --Blam!--pulls the trigger and-- Whoop!--at the last minute a quantum fluctuation whisks the bullet out of the way. The prevailing theory of closed timelike curves is due to David Deutsch. The quantum mechanics of closed timelike curves is due to David Deutsch. This is a theory of the first kind where you can enter into an alternative universe. The one I'm going to tell you about: projective closed timelike curves, is of the second kind where you cannot actually go back and kill your grandfather. Ok? Alright. Are there any questions at this point? Any more time travel stories you want me to hear about? I've been collecting time travel movies. Any good time travel movies you want me to go and watch? >> AUDIENCE MEMBER: I just have a question. How would you know you've travelled back in time if you can't interact? >> LLOYD: If you can't what? >> AUDIENCE MEMBER: If you can't interact? >> LLOYD: No, you can interact! You can interact. You cannot cause something to happen, which you know not to be the case. Right? Obviously, this is someone who has not seen Prisoner of Azkaban. [laughter] >> LLOYD: What's your problem? So you can interact with the past and make all kinds of weird things happen, and in the future you will remember that those weird things happened because you can certainly interact with things but you can't go and actually change the past. So you can't go back to that horrible blind date that you had when you were fifteen, you can't go back and undo that. I'm sorry. [laughter] >> LLOYD: Yes? >> AUDIENCE MEMBER: Primer. >> LLOYD: Primer? >> AUDIENCE MEMBER: Primer. That's the time travel movie. >> LLOYD: Ah! Yeah, I've heard that this is a really awesome movie. Who here has seen Primer? Yeah, I've heard this is really awesome. This is definitely on my list of things. And there's this one, Eight Monkeys? Or Ten Monkeys? >> AUDIENCE MEMBER: Twelve. >> LLOYD: Twelve Monkeys! Twelve Monkeys. Twelve Monkeys, I'm told, with one slight mishap is of the second kind where the time traveller goes back in time and tries to change some horrible thing happening and it doesn't happen. I've heard it's an awesome movie, too. Yeah? >> AUDIENCE MEMBER: Have you seen The Time Traveller's Wife? >> LLOYD: Oh! The book, The Time Traveller's Wife. You know I started reading that and, maybe I'm too close to the subject, I had to stop after fifty pages. Which category is that? >> AUDIENCE MEMBER: The movie is like Azkaban. It's very good, but it asks... The Grandfather Paradox is 'Can you go back in time and kill your grandfather?', this asks 'Can you go back in time and modify your most intimate relationships?' >> LLOYD: Right. Yeah. >> AUDIENCE MEMBER: Well, I thought it was a very good movie. >> LLOYD: Excellent. So is this possible to modify your most...? >> AUDIENCE MEMBER: You gotta go see it. >> LLOYD: Ok, I gotta go see the movie. Ok, alright. [laughter] >> LLOYD: Yeah? >> LEUNG: So I haven't seen this one but Charlie told me about one in Futurama. >> LLOYD: Futurama. See, the problem is that there are so many of these things that I can't see them all. So which type does this fall under? >> LEUNG: Basically, it is about the Grandfather Paradox again. He managed to kill his grandfather. >> LLOYD: He did manage. Ok. There you go. Awesome. >> AUDIENCE MEMBER: He became his own grandfather. >> LLOYD: Ah! Ok, this brings up another one. I always say that if I went back in time and met my grandfather, I really was very fond of my grandfather, I would certainly buy him a beer. And only if we had too many beers and got in a fight would I accidentally kill him. There's another famous story about the Grandfather Paradox in which the time traveller goes back in time, goes to a club, meets a beautiful woman, sleeps with her, they do not practice safe sex, which I do not recommend, then she gets pregnant and in turn she gives birth to his mother. So he is his own grandfather. That's another Grandfather Paradox. Thank you for bringing that up. This is the second major paradox about time travel. Let me explain it in less... Since I talked about quantum hanky panky last night, I'm uncomfortable being seen as the resident quantum pornographer. [laughter] >> LLOYD: Hey, if you look at technologies, one of the main things that drives their introduction is pornography. So if we could come up with quantum pornography that might be good. Pictures of naked electron--Well, never mind. The second major paradox is what's called the Unproved Theorem Paradox. In this paradox the time traveller reads a cool proof of a theorem in a book, and she goes back in time and she shows the proof to a mathematician. The mathematician says "Wow! What a cool proof! I'm going to include it in my book." The book, of course, is the same book in which she obtained the proof in the first place. This is actually quite disturbing because you have a carefully constructed, beautiful proof that came from nowhere. It was never produced. Actually, this other version of the grandfather story is, in fact, the Unproved Theorem Paradox in disguise because if you go back and become your own grandfather, then a quarter of your DNA came from nowhere. It was never subjected to natural selection or anything like that so, in fact, the problem there is this Unproved Theorem Paradox. I'm trying to do a lot of research on the philosophy and things of time travel. I haven't been able to find time travel paradoxes which are not variations on these two themes. So any theory on time travel in quantum mechanics or elsewhere has got to come up with a resolution of these paradoxes, and so I will tell you what our resolutions of those paradoxes are. But first, let me give a little bit of history. We probably wouldn't be discussing this at all if Kurt Gödel, in 1948, hadn't... Gödel, when he was a famous logician and when he got to the Institute of Advanced Study Einstein was there, and Gödel and Einstein were buddies so Gödel learned general relativity. Gödel because he was fond of... paradoxes he decided that he would say "Wow, maybe it's possible to have a space-time that has closed timelike curves in it. So you have this funny space-time manifold and it could be possible that you have a path that goes backwards and you end up in the past." The way this normally looks is the famous coffee cup picture, so time goes up here. This is in a 1+1 dimensional space-time. Down here, space is just some big circle, and here is the handle of the coffee cup. Time is flowing here, but down here when you go around this handle, time goes like that and you can end up interacting with yourself in the past. Ok? Gödel space-times are really weird-looking. They're these massive clouds of swirling dust. It seems to be important to have rotation in these space-times to have closed timelike curves, but they do indeed have closed timelike curves. This is what's called a timelike wormhole. Lest you think that this is some weird wacky thing... By the way, Google Images has some great pictures of Gödel space-times. I've been told that Gödel, when he told Einstein about this, it was sort of a birthday present for Einstein, and Einstein hated it. [laughter] Einstein was not Mr. Paradox Guy. He didn't like it. He didn't like quantum mechanics, he didn't like paradoxes. Everything was supposed to be cut and dried. But not so Gödel. It's also a fact, when you take any space-time and it's rotating sufficiently and there's a sufficient amount of mass, you'll get a closed timelike curve. The interior of a Kerr black hole, a rotating black hole, inevitably has closed timelike curves in it. Back me up, Olaf or some other gravitational person here. This is actually a fact. Unless you think "Oh! Who cares what happens inside a Kerr black hole?" If our space-time itself, if our universe, is actually over-dense and so it's collapsing, then it is effectively a black hole. And if there's even a tiny bit of net rotation of the whole universe, then there would be closed timelike curves within our universe. General relativity allows closed timelike curves and, you could say in some circumstances, it even encourages them to happen. How do you deal with this? How, in particular, do you deal with things like the Second Law of Thermodynamics going around here? What's the quantum structure of the quantum states? Can you even have a quantum Hilbert space picture of what's going on inside this closed timelike curve? The answer to that actually is that you can have a quantum Hilbert space picture but you cannot assign a quantum state, just to telegraph some of the punches. The next hint of how you might deal with quantum mechanics of closed timelike curves comes from John Wheeler. This is not anything he published. Amusingly, this is reported by Feynman in his Nobel Prize acceptance speech, which is online so you can get it. And you go look at it, and he starts off with this dry stuff but then he starts talking about his experience and there's this place where he says-- I'm going to try to paraphrase it in a Feynman-esque kind of way-- he says "Well, Johnny Wheeler called me up from the Institute and he said 'I know why all electrons have the same mass.'" And Feynman said "Really?" And Wheeler then said "Yes, and I also know why they have the same mass as the positron." And Feynman said, "Why? Why?" And Wheeler said, "Well..." By the way, time is always go up in this picture because anything to do with general relativity has time going up, so get used to it. Wheeler says, "Well, look. Electrons and positrons are always created in pairs." e+, e-. "And they're always destroyed in pairs." e-, e+. "So we can think that what's going on is that there's just one electron that's going forward and backward in time. When it's going forward in time, it's an electron. When it gets destroyed, it turns around and becomes a positron. So the reason that they all have the same mass is because there's only one of them!" "And the reason why they have opposite charge but identical charge is that when you do charge-time reversal you take plus-charge to minus-charge by a famous theorem of quantum field theory: the CPT theorem." So then Feynman says, "This was totally crazy, but I did steal from this the notion that positrons were electrons going backwards in time." So in fact, at the heart of contemporary quantum field theory was the notion that positrons are electrons going backwards in time, there's this crazy idea of Wheeler's that says "Look, there's only one electron and one positron." And, of course, if you go and look at Feynman diagrams you realise why this isn't really true because you can have other Feymman diagrams like this and they're connected by photons and so there's probably not only one electron in the universe. Though it's kind of a nice idea if there's only one electron. Our theory, which is with Lorenzo Maccone, Yutaka Shikano, Raul Garcia-Patron, and then also we have Aephraim Steinberg's experimental group at University of Toronto who did the experiment. What I'm going to tell you, you can think of as essentially the mathematization of this Wheeler notion of positrons being electrons going backwards in time. It's going to rely strictly on entanglement and, indeed, when you create electrons and positrons in pairs out of the vacuum, their spins are entangled singlet states. And when you destroy them, the only place they can be destroyed is in entangled singlet states. This is going to be key for how this theory of time travel works. Let me continue with the theory a little bit because I think it's important to know about. I have this Master's degree in History and Philosophy of Science from Cambridge and so I think that the history of ideas is actually quite important to understanding what the next idea is going to come from. Around 1988, Kip Thorne and Ulvi Yurtsever, and later in the early 1990s, Jim Hartle and David Politzer looked at path integral approaches. Path integrals, of course, what you do is you take a bunch of classical trajectories, you assign them an action, and you sum e^(iS) over all classical trajectories. What they did is they say, 'We only take classical trajectories, classical paths, which are self-consistent.' In classical mechanics, you don't have this many-worlds stuff that David Deutsch advocated, but we're only going to take classical paths that are self-consistent so we sum over self-consistent classical paths. This is a perfectly reasonable idea and you can formulate it. The problem is that path integrals are very hard to evaluate so they never got very far with this approach, but they were able to look at it. Then David Deutsch, in 1990, came up with this... Maybe I should actually start... yeah, this is the right order. So in 1990...no, no, no, this is not the right order. Yeah, it's the right order. Nah! It's not the right order. [laughs] I want to especially mention that Charlie Bennett and Ben Schumacher, Charlie Bennett in particular starting with the invention of teleportation, which was... When was teleportation? It was around... Do you remember, Debbie? '94? >> LEUNG: '92. >> LLOYD: '92. Yeah, so starting with teleportation, Charlie Bennett talked about... With teleportation you have--I'll be kind of graphically suggestive about this-- you have an entangled singlet. And then you make a measurement right here. And then you send classical information over here. You perform some operation dependent on this classical information. This is a Bell measurement. And then you do something right here, and if you have a state psi, you end up getting state psi here. Charlie Bennett always talked about teleportation as if this part-- Where did the information go?-- as if the quantum information went here and this is the quantum information going backwards in time and forwards again. By the way, this idea is at the essence of what I'm going to tell you about. Unfortunately, Bennett and Schumacher have never published anything on this. They've talked about it for years. They've never really developed, so far as I can tell, any explicit theory about this so you could also say what we're doing is developing a theory out of this metaphorical description that Charlie Bennett has been using for decades now. Ok, so now let's get down to it. This is all history and now it's going to be math and stuff like that so are there any more historical questions, comments, or things like that? Were you bored by hearing the history of this? Maybe you were. That's ok. Ok. I'm sorry? >> AUDIENCE MEMBER: Which paradox does arbitrage come in? >> LLOYD: Arbitrage? >> AUDIENCE MEMBER: Yeah, why didn't you send that photon half a billionth of a second after the stock market was updated and tell itself to... >> LLOYD: So that's consistent with both. This is like a many-worlds version. This is 'enter another world.' And this is 'same self-consistent world.' So if you send information about what the price of the Swiss Franc is going to be back in time and then invest in the Swiss Franc, as long as the amount of your investment is sufficiently small that the history of the Swiss Franc is the same then it's ok. But if you try to buy all the Swiss Francs, then it will cause things to go haywire. Then it would fall in this many-worlds version. You can imagine making money off of time travel in either of these worlds. In this way, you can make a lot more money, in this world, in the Deutsch version, the many-worlds version. Ok. Ok, so enough of this fun fooling around with history, fooling around with the past, which is of course what time travel is about. Let me now actually tell you how these things work. The first thing I'd like to tell you is about David Deutsch's theory. So I'm going to review. Who here is familiar with Deutsch's theory of closed timelike curves? Yeah, some people are. So let me tell you how Deutsch-- I'll switch colours. Blue for serious stuff. Let me describe to you what this Deutsch paper in 1990 does. By the way, even though we think our theory of time travel is better for reasons I'll tell you, this is a beautiful and elegant theory because it's very hard to formulate quantum mechanics in these contexts. These path integral approaches are a good start, but it's tough to deal with these paradoxes. Let me tell you what Deutsch suggested and how he dealt with this paradox here. So Deutsch's-- Oh, I guess I won't use this one. I'll throw that one on the ground. I'll use this one. Ok. So the way that Deutsch's theory works is like this. You have normal, what he called, chronology-respecting degrees of freedom, and let's call this the state rho-A. And then you have the closed timeline curve, and this could be many degrees of freedom. I'm just going to do it as if it's two qubits but it could be many different degrees of freedom. Let's call this rho-B. And you have some interaction between these things. Then the question is: How do you make sense of what happens up here? Now interestingly, Deutsch does not tell you what happens over here. He does not assign this a degree of freedom, this thing going backwards in time. Which is sort of funny if you think of the coffee cup picture because there's something happening in the handle of the coffee cup, but Deutsch does not give you a state for this which is already a hint that something is a little fishy. Deutsch's self-consistency condition basically says that-- Here we have rho-prime of A and here we have rho-prime-- Sorry, I should put it right here so at this point right here you have rho of AB. No, let's actually... Yes. So when the thing comes out...let's call it rho-prime of AB. When it goes in, it's in the state rho-A tensor rho-B. And then what he asks is that the state that reduced density matrix for the system that comes out of this closed timelike curve is the state rho-B. Right? You can see why this gives you a self-consistency condition because it says that the state that enters the closed timelike curve in the future is the same as the state that emerges from the closed timelike curve in the past. So this condition is the trace over A, U rho-A tensor rho-B U-dagger, is equal to rho-B. Ok. This is Deutsch's self-consistency condition. Alright? So you see this makes sense, right? He wants the states to be self-consistent and he wants this state to be the same as this state there. Moreover, this is because this is a superoperator. This is the same as saying if I have some superoperator which is this interacted with rho-A via U, and then just look at B. We are asking that the state rho-B be an eigenstate with eigenvalue 1 of this corresponding superoperator, this process. And because superoperators always have an eigenstate with eigenvalue 1, such states always exist. Ok? So it's a nice self-consistency condition. It looks good. Ok. Now, let's look at actually what happens then if we do something where... No, let me just leave it at that. Let me just mention some things that are a little disturbing about this before I go on. I'll tell you our theory, and then I will compare the two of them. There is something a little disturbing here. Deutsch is assuming that when the time traveller exits from the curve, and the rest of the universe is out there, that they are in this tensor product state, which means that the time traveller, when she exits from the curve, is completely uncorrelated with the stuff outside of the curve. That is, she emerges in a universe where none of her memories are any good. They don't correspond to the universe she sees. That's a little disturbing already because that's certainly not this Urashima Taro kind of time travel or H. G. Wells kind of time travel. It's like you end up in this weird place that has nothing whatsoever to do with what you remember. And the reason for this, of course, is that... I would describe this in quantum information terms as I would personally prefer a closed timelike curve to behave like a quantum channel and quantum channels preserve correlations with the surroundings. But by demanding that only the reduced density matrix be the same as it enters the curve in the future and emerges in the past, in this case you are actually erasing all memories of outside. This is not behaving like a quantum channel, so this is a bit disturbing. I should say that after we wrote our paper, we corresponded with Deutsch about this and he said that he found aspects of his theory unsatisfactory and I believe that this might be one of those aspects that he found unsatisfactory. Yeah? >> AUDIENCE MEMBER: In this result, is the superoperator linear? >> LLOYD: Yeah, sure. Any superoperator can be written as an unitary interaction with an environment when you start out in the tensor product state. This is whosey-whatsit's theorem. I don't know whose theorem it is. The transformation that B undergoes when I take U, it interacts with A, and I take the trace over A. This is certainly a legitimate quantum mechanical transformation. It corresponds to a superoperator; it is linear. So this is a superoperator and all superoperators have an eigenstate with eigenvalue 1. There is a non-linearity in Deutsch's theory. This transformation is linear. There is a non-linearity because what Deutsch is saying is that the only rho that we can allow are things that satisfy this. So you can't put anything in here, you can only put rhos that satisfy this. Not anything can go through this closed timelike curve. There's a non-linearity in the sense that you're selecting out of the set of possible states just these rhos-sub-B from which it can happen. There may be multiple rho-sub-B for which this is the case. That is, the eigenvalue may be degenerate. In that case, for reasons that I'll describe in just a second, Deutsch says "Take the one with maximum entropy." Actually, I'll describe it in a second, so the reason for that is that if you don't take the rho-sub-B that has maximum entropy then you immediately run into this Unproved Theorem Paradox because the Unproved Theorem Paradox is perfectly self-consistent. You can have the Unproved Theorem go through this, everything is fine. If you think of this as some classical transformation and then...that's bad. Deutsch really doesn't like that. If you read his paper, which is a really excellent and interesting paper, he spends a lot of time talking about this Unproved Theorem Paradox and how much he's upset by it. So he says "Ok, you take the maximum entropy: 1." Alright. So. Let me now contrast this with-- any more questions about this? I'm going to stop talking about this Deutsch thing right now. There's more disturbing things as we'll see in just a second about this. Actually, I'll mention one more disturbing thing which is that Scott Aaronson, John Watrous, and Todd Brun and a bunch of other people have shown that this is absurdly computationally powerful. So this non-linearity of selecting out the particular states allows you to solve anything. Deutsch CTCs allow you to solve anything in PSPACE. Plus computation: both classical and quantum computation. It says that PSPACE is equal to polynomial time so you can solve any problem in polynomial space in polynomial time. For you computational complexity people out there, and I know you're there because I see you, you know that anything that is this case is really bad, and if you say that you can do this then computer scientists will immediately disbelieve that this is possible, all physical evidence aside. That's actually kind of a bad thing about this. I should say that the way I got involved in this is there is a paper from IBM in which, the summer before this last one, where they objected to this Aaronson result. Is anybody co-author on this paper here in this room right now? [laughs] Because I want you to step up and defend it if you are. Anyway, so this caused a big argument. This got us, we found this argument to be annoying so we decided to work on these closed timelike curves ourselves and when Charlie Bennett came...I'm sorry? >> LEUNG: What annoys you about the argument? >> LLOYD: What annoys me about the argument? Because I don't believe it. That's why. [laughter] >> LEUNG: The opposing idea or you don't believe the argument, you said? >> LLOYD: So the Aaronson paper is correct, I went through it very carefully. The IBM argument says that in the presence of closed timelike curves you cannot prepare your computer in a particular problem state in order to solve that problem. Now, I don't know if this argument is correct or not, but the two papers start from different assumptions and I actually quite distrust the argument from the IBM paper. Anyway, we brought Charlie Bennett to MIT and we put him and Scott Aaronson in a steel cage and we had them duke it out. It was inconclusive in the end. I would prefer not to discuss this paper because I don't think it's very illuminating. Sorry, shouldn't have brought it up. When we looked at this I said to myself, "I actually know of another way of doing closed timelike curves" because about eight or nine years ago I worked on this problem of how information escapes from black holes. By the way, this talk has every wacky possible thing you could imagine. It's got closed timelike curves, we're going to have teleportation, that's the least wacky of things, and we're going to have how information escapes from black holes. So let me review how this model works. I said "There's something very funny about this" because there's another Aaronson paper, which says that quantum computing plus post-selection is equivalent to solving the computational class of probabilistic polynomial time, affectionately known as PP. When I told my children that there was a computational complexity class known as PP, they felt that this was really hysterical. [laughter] >> LLOYD: So because of my work on escaping from black holes, I happen to know that you can make a theory of closed timelike curves based on quantum mechanical post-selections, which I'll now present to you, which I thought all along was equivalent to Deutsch's theory. I should note that Charlie Bennett and Ben Schumacher, while they've been talking about this going backwards in time in a method that's very close to what I'm going to describe, they were also not aware that their theory was different from Deutsch's. Now, if you look at these two things together, if you can get post-selection and quantum computing, you can get closed timelike curves, which I'll show you in just a second, then you've proved that PP equals PSPACE, which would be really news to lots of people. You would have collapsed part of the polynomial hierarchy, which would be pretty amazing. So the first way we started on this is that we said "Hey! Look, we can prove that PP equals PSPACE!" If you've ever worked on these things you know that after a weekend of working on it, you realize that you were wrong but it was fun for about a weekend. The resolution was that we realised that, in fact, the closed timelike curves via post-selection are not equivalent to Deutsch's closed timelike curves. So let me tell you how this works. Let's look at first escape from a black hole. How does escape from a black hole work? Ok. Here again is time going up. Here is space. Here is my picture of a black hole. This is the event horizon of the black hole. This is the singularity, where everything gets smushed into nothingness. Note that the singularity of a black hole is spacelike. It's not actually pointlike, it's spacelike which is kind of interesting. Note also the horizon is lightlike. Light goes at forty-five degree angles here. You can see why the horizon is lightlike because if you're right at the horizon of a black hole and you send off a beam of light that's just trying to escape from it at an angle, then this beam of light will just keep rotating around the black hole. If you want to take a path that hugs the horizon, it's a lightlike path so the horizon is a lightlike surface. Here is the picture of what happens. We have this poor innocuous state falls into the black hole. It can be your favourite evil professor or something falling into the black hole. And it's going to get smushed at the horizon, but wait! Wait! There's more to this picture! Because outside of the horizon, Hawking radiation is being created and the way that Hawking radiation works is that-- Let's do it like this-- you have pairs of particles are created from the vacuum that are created in singlet states because the only thing you can create if you create something out of nothing is in a singlet because all conserved quantum numbers have to be zero. Alright? They're created as singlets. Part of this vacuum fluctuation has negative energy and it falls into the black hole, thereby reducing its mass, and the other part has positive energy and it escapes to infinity, thereby carrying part of the mass of the black hole off to infinity and that's how black holes evaporate. Ok. Now, there's a lot of debate about what happens about information in black holes, but there's one mechanism proposed by Gary Horowitz and Juan Maldacena, two heavyweight string theorist types. The mechanism is the following... I'm going to describe it in the way that makes it sound most plausible even though nobody knows if this mechanism takes place or not, but just go with me for a second. Maybe it's the case that the only way, just in the same way that the only way to create something from nothing is for it to be in a singlet state, maybe the only way you can have it go away to nothing, like at the singularity, is for it to be destroyed as a singlet state. Suppose that every state that falls in the black hole gets destroyed as a singlet or projected onto a singlet, that would be more explicit, suppose that the singularity projects incoming stuff onto a singlet together with half of a Hawking radiation pair. Alright? I mean, who knows, right? The great thing about being a theorist is that we have no idea what happens at the singularity so let's just say it's whatever we want it to be! Experimentalists are not allowed this kind of leeway. Now, you can say "Well, what is the state out here?" Aha! We recognize that this is just like teleportation. Ok, here's a singlet state right here. Here's the state psi. But here, we measure and get the singlet. We measure and get the singlet and what that means, if you are familiar with teleportation which I know lots of people are, in the singlet state is one where Alice sends to Bob the information "Whoa! Don't do anything! You already have the state psi." And so I'll draw this like this because now we have projection-- this is creation of a singlet-- this is projection onto a singlet, and so here's this nice picture. This is the kind of thing that Charlie Bennett was fond of drawing. Oh look! The information goes back here, up here, and out here, right here. Alright? Is everybody ok with this? In this case, if you project onto a singlet, which is a non-linear operation. It's like saying we make a Bell state measurement and we toss out all three quarters of it and we renormalize the problem. Get this. So the renormalization of the probability to 1 is non-linear and so this is non-linear quantum mechanics, which is dangerous. That's how you can solve these hard problems using it. But at any rate, the state escapes from the black hole. Ok? Are people happy with this? Yeah. >> AUDIENCE MEMBER: But wouldn't we need to know the Bell state measurement? >> LLOYD: Right, so in ordinary teleportation-- You know, ordinary teleportation. Everybody can do that. People have been doing this for decades. You know, anybody can do that and you make a Bell state measurement. Actually, making all the Bell state measurements is hard. You make the Bell state measurement, you get two bits of information. Alice sends those two bits to Bob, and then Bob does something as a function of those bits. So the idea here is that a non-linear process takes place where instead of having an ordinary quantum measurement, it-- for whatever reason, ike you're at the singularity of a black hole-- it projects you onto the singlet part. So it only gives you the singlet part. All the other stuff gets tossed away and now has probability 0. >> AUDIENCE MEMBER: So only the ones that make it out were projected onto the singlet. >> LLOYD: Right, only the ones that make it out were projected on the singlet. This is totally illegal in ordinary quantum mechanics so that's a very good question. It's illegal, so this is something non-linear and bad. And you can see it's bad because already you have things that are propagating faster than the speed of light, you clearly have violated the No Cloning Theorem because, of course, this could be back down here. So you're definitely doing bad things. Yeah. >> AUDIENCE MEMBER: Well, you've partly addressed my question, but what's unanswered here is what determines the physics of when it goes back into the future from the outside? >> LLOYD: Interestingly, the back into the future part... The Back To The Future part: note that one of the main actors in that is named Christopher Lloyd, no relation I believe. [laughs] Only distant relation. Interestingly, this part down here is really the uncontroversial part. That's just entanglement. That's just an entangled singlet state. The controversial part, of course, is this projection part. >> AUDIENCE MEMBER: Well, sure, but it could go at any one of those times including earlier in the past. >> LLOYD: Yeah, well, of course. Let me segue, because I realize that in indulging myself and telling you about history and talking about time travel movies and stuff like that I'm going to go short on time. As everybody knows--by the way, I try this out on people I meet on the street who have nothing to do with physics and they know this-- so as everybody knows if you can go faster than the speed of light, you can go backwards in time. Everybody knows this. That's just the way it is. Here is the model for post-selected, or projective, closed timelike curves. We create a singlet down here. We project-- This is 'create singlet.' We project onto a singlet up here and renormalize probabilities. Renormalize probs to 1. This is all really easy to do and calculate because it's just as if you were taking a Bell state measurement and you say "What's the conditional probability of everything else, given I got a singlet?" So all probabilities in this theory are calculated using the conditional probabilities that you got a singlet here. Here's this other thing. Then you have some transformation, and then something else comes out at the end. Ok? So probabilities of events... ...Equals conditional probabilities. It's a very well-defined theory. The one thing you have to say is "What is the probability of this as 0?" and the answer then is nothing. It can't happen. Because conditional probabilities are not defined if the probability outcome is 0. This just doesn't happen, which is good because now you can see already that we are going to always have things that are self-consistent because in this picture of closed timelike curves what happens is a valid conditional probability for a sequence of events in quantum mechanics. It might be hugely improbable if you don't do the projection on a singlet, but it's still possible. So you can't get things that are exactly impossible like, for instance, killing your grandfather. Ok. Once you have this... I want again to give Charlie Bennett and Ben Schumacher credit even though I've actually rather had an annoying time dealing with them over this because they have talked about things like this for a long time. I know that Raymond told me-- I didn't know about this paper-- that you guys did an experiment here with NMR to look at this notion of "Oh look! Things are going backwards in time. Let's look at what happens with this." And I want to give them credit for talking about this for years but I'm not going to give them credit for, and I wish to express my annoyance at them for, not writing this up as a paper so we can actually see what they mean because they never wrote it up. It only exists in the form of four transparencies on a talk of Charlie Bennett. So they have a theory, which is like this, but it's not clear what they meant by it. In fact, when we talked to them, they actually use quite a different language and I'm not sure if we agree on stuff. At any rate, Charlie told me that they were unaware that their theory was different from Deutsch's and this theory is definitely different from Deutsch's as I'll now show you. I'll show you by giving you a picture of a quantum circuit for our Grandfather Paradox experiment. I'll show that Deutsch's theory and our theory give different results for what happens. I can do this over here. Let's do a Grandfather Paradox. The simplest version of a Grandfather Paradox is something like this. Here's our closed timelike curve. Ok. Zero equals dead, one equals alive. We're just going to switch this over to having our photon killing itself. If you do a sigma-x, you flip this around the x-axis right here, then what happens is zero equals dead, one is equal to alive. And you see if you're alive here, you're dead here. You get turned into dead. If you're dead here, you get turned into alive here. So this is the Grandfather Paradox. The very simplest thing that you can imagine about it. And moreover, in our experiment we're going to ask this qubit to declare whether it's dead or alive. So here we're going to measure, up here. So here we measure it, we couple it to two other qubits via a controlled-NOT, and then we're going to ask it to declare if it's dead or alive. By the way, I think if you look at the Grandfather Paradox in Charlie Bennett's notes, again trying to decipher what they meant by this, it's a different paradox and this might be why they didn't figure out that their result was different from Deutsch's. We also asked Charlie Bennett to be a co-author on our paper after we'd done the experiment and discovered that they'd been doing this before. After this delayed the submission of the paper by four months while he decided, in the end, that he hadn't contributed enough to it. This is the kind of thing that happens in science. What they hey. You gotta negotiate with people, people get upset if you don't give them credit, et cetera. So I'm trying very hard to give them credit and express my annoyance. [laughs] What happens here? Well, in Deutsch this is totally ok. Remember, Deutsch always works, it always gives you something. What is the state that work here? What is the state where if I pop it in here, it comes around here, and it's still the same state? >> AUDIENCE MEMBER: The Hadamard state. >> LLOYD: I'm sorry? >> AUDIENCE MEMBER: The Hadamard state: equal superposition of zero and one. >> LLOYD: Right, that's right. That will work. However, Deutsch asks us to take the maximum entropy state. >> AUDIENCE MEMBER: So a mixture then. >> LLOYD: Right. So, here, for Deutsch, basically--I'll call this rho-B again-- rho-B is equal to 0.5(|0><0|+|1><1|). Actually, you see right now since this was very alert we could also have a sigma-x state. That would also be fine because it's an eigenstate of sigma-x, right. This shows you why Deutsch has to give you the maximum entropy state because there are several states. The minus one would also work. There are several states that have this criteria and this state right here is like the never-created proof of the theorem: it's a special state, sigma-x up, and went around and everything is totally fine. It's self-consistent, but it's some special state and nobody ever picked out why it should be sigma-x, why it should be spin-x up. So Deutsch says, "Ok, we gotta use this state to make it fully mixed so we don't run into this Unproved Theorem Paradox." But now you see something truly strange and alarming here, which is that even though the density matrix for the state, the time traveller as she enters into the closed timelike curve in the future, is the same density matrix in the past somehow during the transition around this curve the actual state has been mixed up so zeros become one and ones become zero. This is a bad closed timelike curve to enter alive because you're dead when you exit. Of course, it's a good one if you're dead because you exit alive so we could have both destruction of life and we could have creation of life from nothing from this. So that's pretty weird. Deutsch's criterion which looks perfectly fine to begin with, now you see when you apply it to this Grandfather Paradox, you say, "Hey! Hold it now, I thought you said the state was the same." And the answer is that the density matrix is the same, but your particular part of the density matrix may have gotten completely screwed up, which is bad if you were alive when you entered the curve. Ok. >> AUDIENCE MEMBER: Sir? >> LLOYD: Yes? >> AUDIENCE MEMBER: What happens if you think the density matrix is lack of knowledge? It's actually a pure state somewhere... >> LLOYD: Right, of course, as you can tell when you have this kind of system all your intuitions about quantum mechanics and what you've been taught, you've got to be careful about. If you think of it as a lack of knowledge, say, from an outside observer, like the person out here monitoring the situation while we're here, we'll say "Well, I don't know what it was. Hold it, let me go look over here." They'll get statistics up here. What they'll find is, basically, in Deutsch's case if this one is zero the next on is one, if this one is one the next one is zero. So what it says is "Well, I don't know if she was alive or dead when she went in, but, by gum, whatever she was if she was alive she turned out dead and if she was dead she turned out alive." So I think it's still ok with that. You didn't know it beforehand, and then you found out. Yeah? >> AUDIENCE MEMBER: I think if you're using Deutsch's recipe for this problem, then you shouldn't use CNOTs as a part of your unitary. >> LLOYD: Yeah, it is. I didn't derive to you that this is the maximum entropy state from this, but if you include it with the CNOTs, take the trace over the CNOT gates, you'll find that the fully mixed state satisfies Deutsch's self-consistency criterion and because it's pretty clearly the maximum entropy state-- Unless you want to disagree with that, too --then this is the Deutsch prediction. >> AUDIENCE MEMBER: But zero plus one doesn't hold. >> LLOYD: Oh! If you have the CNOTs, yes. Sorry about that. If I have this, then it's ok. You're right. I'm sorry, you're right. You're right. I'm sorry, I didn't mean to. I was unfairly scoffing at you. You're exactly right. If you put this in here, then this state doesn't work if you're actually doing these measurement interactions. Completely correct. Ok. What happened with P-CTCs? Well, what happens here is that this can never happen. If I think of it without the CNOTs to make your life easy, you see what happens is that the singlet right here gets changed into a triplet which has zero overlap with the singlet so that the projection onto the singlet is zero. In the raw Grandfather Paradox, it's an example where it doesn't happen. On the other hand, what happens-- Let's suppose that you actually have something which is just e^(i theta sigma-x), so you're performing a partial rotation around the x-axis, and e^(-i theta sigma-x) is equal to cos(theta)*I-i sin(theta sigma-x). What happens then is that this projection to the triplet knocks out this, so in fact the qubit never gets flipped. It only behaves like the identity, you just take the identity part, no matter how small it is, amplify it back up to one, and so what P-CTCs say is that you get zero, zero, one, one. If the time traveller entered the curve alive, she exits the curve alive. If she enters the curve dead, she exits the curve dead. And that's because these projective closed timelike curves behave like idealised quantum channels. They preserve not merely the state of the system when it is a closed timelike curve, they also preserve any legitimate correlations with variables out here. So if you remember being alive when you enter the curve, or somebody, maybe if your mother, remembers that you were alive when you enter the curve she will see you emerge alive in the past. Ok? This already shows you that Deutsch's closed timelike curves are different from these projective closed timelike curves and the main difference is this quantum channel version. I'll also tell you I'm out of time, but let me just tell you what happens with the second Grandfather Paradox, the Unproved Theorem Paradox. Let me see if I can get this right This is always tricky. Here's the closed timelike curve. What happens is the time traveller reads a theorem in the future... This is a qubit right here. Here's the theorem. And then, in the past, she writes the theorem onto this qubit. She tells a mathematician what the theorem is, and then she goes her merry way. This is the Unproved Theorem Paradox. She needs to be able to read the qubit in the future and then write it back in the past, and I claim that this is the quantum version of this Unproved Theorem Paradox. Sorry that I'm rushing through it a little bit. 'Read theorem'. And this is 'tell theorem'. To read the theorem she has to be in the state zero, so she knows what the theorem is, and then she tells it to the mathematician. This is the mathematician. This is the time traveller. Now you can ask, 'What happens here? What is the state right here?' Now, remember when Deutsch does this circuit what happens is he identifies this right here with this right here, but now any theorem will do. Could be zero, that's fine, one will do as well. So he has to take the maximum entropy state in order to get rid of this paradox. Basically, Deutsch says, 'Ok, look, it's gotta be the maximum entropy state up here so you get a mixture of |0><0| with |0><0| and |1><1| with |1><1|.' You see that Deutsch, by introducing this extra entropy in the problem by having this maximum entropy state, he has randomized the theorem so it's not some special proof or anything like that. However, you also see another problem or feature, let's say-- it's not a bug, it's a feature-- of Deutsch's theory which is that you're introducing entropy. You're taking pure states to mixed states, whereas a moment's thought about this process will tell you that if this state is a pure state and you project out part of a pure state, then, by gum, this state is a pure state. P-CTCs take pure states to pure states, Deutsch's takes pure states to mixtures. This is Deutsch. And what about P-CTCs? Well, people who have been doing quantum information for more than six or seven years might recognize this as an entanglement-swapping circuit. We create entanglement over here and we swap the entanglement over here, so what we get right here is a pure state which is |0><0|+|1><1|. Now you see a neat feature, which is that we never even worried about this Unproved Theorem Paradox until we formulated the theory, and then we looked at how the Unproved Theorem Paradox plays out once you do the theory and you find "Look! The theorem is in a complete mixture! How did it happen?" It's because entanglement stepped in to save the day and it says "Look! We have this pure state. Nothing ever told the theorem to be one thing or another. We know that it's a pure state, so it's gotta be an entangled pure state with equal amplitudes for zero and one." So the theorem is just a pile of garbage. Let me just close by saying, I need to thank Aephraim Steinberg and his colleagues. We have a paper that describes this with the experiment. We did the experiment because you can do the experiment because it's just like teleportation where you toss out three quarters of the results. So anybody--well, not just anybody-- but lots of people can do teleportation experiments so we can actually make this happen. We can't deterministically send information back into the past and mess with it, but we can do an experiment, which in a post-selected fashion, is completely equivalent. Here's the place where I differ with Charlie and Ben. They call this a simulation of time travel. I would point out that, in some sense, this is a bit more. In fact, we're being very honest by saying... If you think of the initial teleportation experiments, they were all post-selected too, and if you actually calculate the fidelity of those experiments without post-selection, the fidelity is 0.111, or something like that. When people like Zeilinger and DiMartini reported fidelities of 80%, 0.8, in their teleportation experiments, that doesn't mean that you could give them a qubit and have them teleport that qubit with fidelity 0.8. It means that in a post-selected fashion, when they post-select for the experiment succeeding, then they would teleport your qubit with fidelity of 0.8. Here, I'm telling you right now that we're going to post-select it, and in a post-selected fashion this is completely equivalent to sending things back in time. How does this work? You create the singlet, you create this other state, you create this time evolution. We actually have four qubits in this experiment. And then here, it's as if this measurement is like the photon entering the time machine. Photon enters the time machine. If the red light goes on, which means you got a singlet state, then everything in the experiment including all the measurements you did in the past yield exactly the same results as if the photon had gone back in time and tried to mess with itself. Ok? In a post-selected sense, this is time travel. Of course, not real time travel in the same sense that teleportation is not real teleportation. That's ok, I give them credit for it. Here, we should be more careful about talking about post-selection. What were the results of the experiment? Well, it's actually very nice. We find, indeed, that the photon never manages to kill itself in the past. The tool they use, a quantum dot source or a single photon quantum dot source and entangled pair source, is called a photon gun. That's it's technical name, which is useful for us because I can describe it in the following way. What this means is that when you take the photon gun and you point it off in that direction-- let's do it off in that direction in case there's a mirror there-- then the red light goes on a quarter of the time and you never manage to kill yourself in the past. Now you take the photon gun and start pointing it closer and closer... Let's say I, the photon, take the photon gun-- I don't want to take advantage of you any longer, even though we managed to save you the last time around. I take the photon gun and point it closer and closer to my head in attempted photon self-suic-- Well, it's not really suicide, it's more like... I don't know what killing yourself in the past is more like, but I attempt to kill myself in the past and what happens is I still fail. When the red light goes on, I still fail, but now the red light-- This is the probability of successful post-selection. The probability of the red light going on gets lower and lower, until finally-- so let's call this angle the angle phi. We start off at pi, pi/2, and we're going to end up at 0. Finally, when you get down to 0, the probability of successful post-selection goes to 0 because, with 100% probability, I'm going to kill myself when going to the past, that's not ok. It violates logic and so it doesn't happen. Alright. Let me summarize. Oh sorry, Raymond has got a hockey--mine is sharper! [laughs] I just picked this up because... [laughter] >> LLOYD: Look, Raymond, as I told you tomorrow, I was going to go over. [laughter] >> LLOYD: Let me summarize. Closed timelike curves are a perfectly legitimate part of general relativity. Therefore, it's important to figure out what happens in them quantum mechanically. Now, they may not be allowed in our universe or not, we don't know. Stephen Hawking says no. He has this chronology protection postulate, which says you can't have closed timelike curves. On the other hand, since he doesn't give any reason for why this is so-- It's like many of his other statements: 'It's just so and, by God, I believe it to be true!' We don't know if this is possible. It's certainly possible in the ordinary laws of physics. It's good then to figure out how quantum mechanics would work on this. Deutsch proposed this theory. We propose a separate theory and we believe that Ben Schumacher and Charlie Bennett, had they actually managed to write the paper and work out the theory, would have arrived at the same theory. It is different from Deutsch's theory. It has this nice feature that it can still be described in Hilbert space, but of course because you have post-selection you cannot uniquely assign a state to the system as it's moving along right here. We were also able to show that-- this was pretty tough because if you go read these Politzer and Hartle papers, they're about path integrals over Grassmannian variables and it's been a long time since I did a path integral over Grassmannian variables-- but you can show that they're equivalent to Politzer path integral method. Politzer only does it for a single qubit going backward in time, but they're equivalent for that in that case so we think that they're equivalent generically to these path integral methods. And we performed an experiment and, by gum, we're going to probably do this experiment soon as well and if you guys would like to do this experiment we'll be happy to collaborate with you to figure out the right way to do it. So, thank you very much. [applause] >> LAFLAMME: We're a bit late, but if somebody has a profound question... >> LLOYD: Raymond has a group meeting right now. Maybe you should go to your group meeting. [laughs] I'm happy to answer questions now and also later. Yes. >> LEUNG: It's more a comment back to your original objection to... >> LAFLAMME: Speak louder, so... >> LEUNG: Right, so just to comment back on the paper, the Deutsch response against Aaronson and Watrous... >> LLOYD: The IBM paper, yeah. >> LEUNG: Yes, I should say it while the crowd is still here. >> LLOYD: Yeah. >> LEUNG: I think the difference here is just that we object to the model as trying to compute on a fixed input and we think that a computation algorithm should work on an arbitrary input that is decided on the spot. >> LLOYD: Right, in the paper... Were you a co-author on the paper? >> LEUNG: Yes. >> LLOYD: So, in that paper... Actually, I should say I'm still confused about the paper so my understanding of it, correct me if I'm wrong, Aaronson and Watrous just said "Hey, suppose we have an input and we can put an input into this." We have access to these closed timelike curves and we're allowed to put in any input we want over here. And then we say "Ok, what kinds of problems can we solve?" The answer is that they can both solve problems in PSPACE. In your guys' paper, and now is your chance to correct me if I'm wrong, says "Hold it! You have to look at how you prepare this input, and that if you have these non-linear systems you're no longer allowed to think of things-- A mixture is necessarily... When you apply something to a mixture, it's no longer the same as applying it to the individual components of the mixture and then seeing what happens to the individual components." So you say "Well, you have to say how do you prepare this input and if it's entangled with some other state or something then you can't necessarily prepare that input." >> LEUNG: Basically, that changes what you mean by the input and, therefore, the fixed point changes as well. >> LLOYD: Yeah. >> LEUNG: The fixed point doesn't do anything for each of the individual components. >> LLOYD: I agree with that, but having witnessed the Scott Aaronson/Charlie Bennett steel cage fight I have to say that my impression was-- as I said, it was kind of a draw--the different assumptions are Scott says "We can prepare this input and we need to have this part of the system, and we want to look at what happens," whereas you guys say "We look at the whole universe and we ask what it means to prepare an input then and then if you just have a mixture of inputs coming in here then you have to redo the calculation again." I agree with that because the calculations in your paper are correct about that, but what I don't agree with, and I don't agree with this. I will now come clean and say I don't actually agree with it rather than say I don't know whether it's right. I don't agree that that's the correct way to talk about whether you can prepare an input or not. I don't think that Scott's way is wrong. I don't think that your way is wrong either. They're both different ways and each is equally self-consistent. So I, in fact, don't think that your paper really refutes Scott's result. Simply, you choose to have a different definition of what it means to choose an input. >> LAFLAMME: [speaking over LEUNG] Maybe we can have a rematch of the Bennett/Aaronson... [laughter] >> LLOYD: I've got the pointy hockey stick here. [laughter] >> LAFLAMME: I could go downstairs and try to find a cage. [laughter] >> LLOYD: Yeah, we can do it in the clean room. Oh, no, that's a bad idea. [laughs] >> LAFLAMME: Let's thank Seth again! >> LLOYD: Thanks. [applause]