 I'm going to tell you today about how some things discovered by physicists in the early 20th century have changed our view of the fundamental nature of information, which is at the heart of the information revolution that really got going in the end of the 20th century. Well, like other parts of mathematics, the theory of information and information processing originated as an abstraction from everyday life. If you're a student of Latin, you will know that calculation is a manipulation of pebbles and the digit is a finger or toe. And this abstraction was crystallized by people like Turing and Shannon and von Neumann into the theory of information and information processing. But unfortunately, these abstractions are too narrow. The quantum theory, which was developed in the early 20th century by physicists and chemists, now provides a more complete and natural, indeed, arena for developing the concepts, the abstract concepts of communication and computation. So what people who do information processing and storage and computing have used as information carriers like a signal on an ethernet cable or a hole in a punch card were what a physicist would call a classical systems. That is, their states are, in principle, reliably distinguishable and you can read the state of something without disturbing it. And almost so fundamental an idea that nobody even thought of saying it, but thoroughly describe two things, it suffices to describe each one separately. Well, those ideas actually were known to be already not right in the early 20th century in the context of physicists studying atoms and small particles, atoms, electrons and photons. And they found out that attempting to observe the state of a particle in general will disturb it and only obtain partial information about what the state was before you disturbed it. That's generally called the uncertainty principle. And they also found that two particles can exist in an entangled state in which they behave in ways that can't be explained by supposing that they were, that each particle had some state of its own, maybe a state that you don't know. So most of the 20th century passed by where quantum effects were understood because you needed quantum theory to design transistors and so on, but they were regarded from the point of view of information processing as a nuisance because they caused these tiny microscopic devices, they kept getting smaller and smaller, to become less reliable because of the effects of the uncertainty principle. And toward the end of the 20th century it was understood that quantum effects have positive consequences, that they make possible new kinds of information processing like quantum cryptography and dramatically speeding up some classically hard computations. And to understand how these things work you really need to understand entanglement, which is an idea that the average person on the street doesn't really understand. So I'm going to try to explain the difference between ordinary classical information or what we call information which everybody knows about because we're in the middle of an information society and quantum information by comparing quantum information to the information in a dream. Unlike the information in a book, the information in a dream is private in the sense that if you try to explain your dream to somebody or describe it, you forget what the dream was and only remember what you said about it. And it's inherently private, you can lie about your dream and nobody will catch you except maybe your spouse. But unlike the theory of dreams, despite the best efforts of Siegmund Freud, there is a well-developed mathematical theory of quantum information which I will not belabor you with most of the details of. So there are important differences but also there are important similarities to ordinary, what a physicist would call and what I will henceforth call classical information. We all know that information can be reduced to very simple primitives, just the digits 0 and 1. For example, you can encode a letter of the alphabet in five bits. And all processing of information can be reduced to basic logic operations like and and not acting on these bits. So you only have to handle the bits one or two at a time to do anything that you would need to do with classical information. Similarly, quantum information is reducible to what we call qubits, which are quantum systems that are capable of two distinguishable states. An example would be a polarized photon or a spin one-half atom. And similarly, any processing that you want to do to quantum information can be done by acting on these qubits one and two at a time. And just as classical bits, the fact that information processing, the software is independent of hardware so that you can, by making the hardware smaller and faster, you make information processors more and more powerful. They can do the same computations. Similarly, in principle, although we don't have not proceeded very far towards a quantum computer, the things you do with quantum information are independent of the particular physical embodiment. That's why I say they are fungible among the different quantum systems, just as classical bits are fungible among the different storage and transmission media that you use for them. So what is the central mathematical principle of quantum mechanics and of quantum information? It's called the superposition principle, and it says that between any two reliably distinguishable states of a physical system, there are other states that are not reliably distinguishable from either original state. Now, of course, we observe this phenomenon all the time. If we scribble a note, there may be a letter that might be an A or it might be an O, and if we scribble it badly enough or it gets smeared in the rain, it's not perfectly distinguishable. But this is a different kind of imperfect distinguishability. It says that with the most perfect equipment, you could not distinguish these two states. So quantum systems are systems in which there are distinguishable states, but not all states are distinguishable. And the way these states behave is that it states correspond to directions in space, and any two perpendicular directions correspond to distinguishable states, and any two directions that are not perpendicular correspond to states that are intrinsically imperfectly distinguishable. So any direction is a possible state, but states are only reliably distinguishable if their directions are perpendicular. And in the simplest case, which is two reliably distinguishable states, we're talking about directions in two-dimensional space, like directions a compass needle could point. And this is nicely illustrated by the behavior of polarized photons. So I've got a perspective drawing here of a beam of light in which the photons are coming along. They're all horizontally polarized in this perspective view. And we can use the fact that photons can be horizontally polarized or vertically polarized as down here as a way of distinguishing them, because if you send photons into a calcite crystal or oriented this way, and they're horizontally polarized, they'll go straight through, whereas if they're vertically polarized, they'll get deviated while they're in the crystal and then come out on a shifted path. And this means if we wanted to encode bits in the photons, we could fit one bit in each photon and count them as they came out in these two different paths at the receiving end. Well, of course, the photons can be polarized also at any intermediate angle. So if a photon is coming towards you, it can be polarized at any angle relative to the vertical if it's sending towards you. And what happens if you send some of these theta polarized photons into the same equipment? Well, you might think that they, since they're intermediate between horizontal and vertical, they would be deviated by an intermediate amount. But that's not what happens. What happens is that some of them become horizontal and the other ones become vertical. And these two things happen with probabilities that depend on the angle theta. In other words, identically prepared systems, these theta polarized photons, behave differently. This is one of the amazing things about quantum mechanics. They behave differently and randomly and they forget their original polarization. We're going to come back to that a lot. So if I have a crystal like this and two detectors, I can use it, as I said before, to carry one bit of information per photon if I prepare these photons in these rectilinear states, that is horizontal and vertical, avoiding other directions. And if I wanted to, I could rotate the whole equipment 45 degrees and send through a chain of 45 and 135 degree photons, and again distinguishing them perfectly reliably. But there's no way to distinguish all four kinds, because the measurement that distinguishes vertical from horizontal randomizes the diagonal kinds of photons. The measurement that distinguishes the diagonal photons randomizes the rectilinear ones. And this fundamental limitation is what gives rise to the possibility of quantum money, quantum banknotes, and of quantum cryptography. But before I describe those things, I'm going to use this pedagogical analogy from my colleague William Wooters at Williams College. He says, how do you explain how a quantum system behaves when you measure it? He said, well, it's really very much like the old-fashioned kind of school where the students were not supposed to ask any questions and where they were only supposed to answer the questions the teacher asks. So the pupil is the quantum system, and the teacher is the measuring apparatus. So here we have a polarized photon coming along, and the teacher says, is your polarization vertical or horizontal? And the pupil says, I'm polarized about 55 degrees. I believe I asked you a question. Are you vertical or horizontal? Horizontal, sir. Have you ever had any other polarization? No, sir. I was always horizontal. So that's how quantum systems behave when you measure them. So this impossibility of determining the polarization of a single unknown photon carries with it equivalent impossibilities. You can't measure it exactly. You can't clone it, because if you could clone it into a lot of identical copies, then you could measure them along all different axes and find out in which case you could measure, determine the angle theta with arbitrary precision. Now there is a device that amplifies photons, that clones them in a sense, and that's called a laser. But lasers don't work very well unless they have an input signal that isn't too weak. And if you supply a laser with an input signal that is just a single photon, the laser generates enough noise so that even though the output is brighter, it's no more useful in distinguishing the polarization than the input was. In fact, if you send a single photon into one of these ideal lasers, two things happen equally often. You get two copies of the original, or you get the one original and one photon of random polarization. So how is this used for making money that's impossible to counterfeit? This was discovered by Stephen Wiesner around 1970 and only published in 1983. The idea is you take 20 polarized photons of these four kinds and put them in perfectly reflective boxes on the money, 20 of them. Now that's possible in principle but actual photons die off in a fraction of a second or even a fraction of a millisecond if you try to put them in reflective boxes. But going on with what's possible in principle, we then say the mint that prints this money knows what's stored there and when it's presented at a bank, the bank makes the correct kind of measurement on each of those photons and verifies that they're all as expected. That is, it makes a diagonal measurement on the first one, a rectilinear measurement on the second, and so on. And if they all pass inspection, the bank gives you your 100 Wiesners of Gold. Whereas if even one of them fails, then you get arrested. So whereas ordinary bank notes often contain a warning about how long you go to prison if you've counterfeit them, this one just has a saying in Latin that says known duplicabo or meaning I shall not be duplicated. Well this is pretty impractical because the photons don't last long enough but a related device is much more practical and that is using the photons to carry quantum information rather than to store it. And this at the bottom of the figure you see the quantum cryptography apparatus. This is clearly a piece of experimental equipment built by theorists. Gilles Brassard my collaborator at University of Montreal and our students including John Smolen here at IBM who largely helped me build this equipment. And it allows the users to generate a shared secret information by communicating over a public channel and a channel subject to eavesdropping by their adversary even though they share no secret information initially which is a useful cryptographic feat. And now this has been scaled up and done by real experimentalists so over distances of hundreds of kilometers. Well I would say the most remarkable manifestation of quantum information is however entanglement and I want to talk about where entanglement comes from how to understand it and what it can be used for. It arises naturally during interaction between quantum systems because of the superposition principle that you already know about. Now remember that I said that any processing on quantum data can be done by one or two qubit operations. So this means if I have a bunch of qubits passing through these wires or quantum wires or optical fibers then anything I want to do to the state of that bundle of qubits can be done by acting on them one and two at a time. Well one at a time means just rotating a photon's polarization by some angle. Two at a time would mean using one photon say the one in this wire to control what happens to the one in that wire. And the only two qubit interaction we need is the one that corresponds to what in classical information processing is called the exclusive war. In other words a conditional bit flip where the control bit if it's a zero nothing happens to the target bit and if it's a one the target bit gets flipped between one to zero over zero to one. So let's do that here and we'll use a vertical photon to represent a one and a horizontal photon to represent a zero and we'll keep the horizontal photon to be orange and the vertical photon to be green just so we can tell them apart. This is standard notation for quantum states introduced by Dirac this sort of half angle bracket meaning that this one and zero are not classical one and zero there are two reliably distinguishable quantum states. So what happens here is if the control qubit is a one the target qubit gets rotated 90 degrees. If the control qubit is a zero the target qubit doesn't get rotated. So what would this do? So this is a quantum version of this exclusive war. We call the control dot by quantum people. What does a superposition of inputs do? It's a quantum computing element so it has to obey the superposition principle. Well the superposition of inputs is the direction intermediate between horizontal and vertical. For example a 45 degree diagonal direction which can be represented by this vector in a two dimensional space. And what it does is not as what you might think of rotating this target qubit by half the angle and making 245 degrees no what it does is a superposition of these two situations. It has to do that to follow the superposition principle and that means it produces a superposition of both photons being horizontal and both photons being vertical. And that's called an entangled state. And the fancy color and lettering suggests it's a different kind of state which can't be even described in the language that we use for classical information. This entangled state is a state that is, if you have to say what it is it's a state which is perfectly definite even though the two photons don't have polarizations of their own. It's a state of sameness of their polarizations. So this means that it's not the same as both photons being 45 degrees. Well let's see why it isn't. Why this state is not the same as that state. And we just have to do a little four dimensional geometry here. So I'm trying to show that this direction in four dimensional space is a different direction from this direction. Now to see that the entangled state on the left is different from the one on the right we have to do a little algebra and it makes it easier instead of to draw these arrows to call h for a horizontal photon and v for a vertical photon. So this diagonal photon is h plus v divided by square root of 2 and the 135 degree diagonal that is the perpendicular diagonal direction is h minus v over square root of 2 and you can see those directions are perpendicular. It's like this is h, this is v, this is one diagonal and this is the other diagonal. So the two diagonals are perpendicular and the vertical and horizontal are perpendicular. Well we know that a single photon lives in a two dimensional space and if we have two separate photons a green one and an orange one we can think of four distinguishable directions because whenever I say two letters the first one is green. I'm going to pronounce it with a green tone of voice. H, H, H, V... Wait am I getting it? H, H, H, V, V, H and V, V are all distinguishable because for example the first two can be distinguished by measuring the orange photon. So we know that the states of two photons live in a four dimensional space and we can work out the state of this entangled system on the left it just says what it is. It's H, H plus V, V divided by square root of 2. We got that. Well what about this one? Well this one is H plus V and so is this one H plus V and if we expand that out it's H, H plus H, V plus V, H plus V, V all divided by 2 and that's a different direction in four dimensional space. So this says that this state can be described by giving a polarization to each photon and this one cannot be described the best you can say is it's a state of sameness of the polarization. Well here's an example of William Wooter's idea of the students behaving randomly. An entangled pair of students, I'm going to call them Remus and Romulus are very bad students. They don't answer any sensible answer to any question. They always answer randomly. But they always give the same answer even when you question them separately. So a teacher could ask Remus what color is grass and growing grass and he says pink. Another teacher asks Romulus and you get the same answer. Now if you weren't happy with that metaphor there's another one from my own past. I was in San Francisco in 1967 known as the Summer of Love and there it was very easy to meet people who thought they were perfectly attuned to one another even though they had no opinions about anything. And the hippies at that time thought that if you had enough LSD then everybody could be in perfect harmony with everybody else. Now they were not known to be very good at mathematics and now that we have a mathematical theory of entanglement we know that entanglement is monogamous which is something else the hippies weren't very good at. And the more entangled two systems are with each other the less entangled they can be with anything else. Well I'll say now how these entangled particles behave in the laboratory and how to explain it in everyday language if you don't want to deal with four-dimensional geometry. Well so as I said the two photons are created at the same time they come out of some apparatus and if you measure either one of them along any axis it gives a random result. For example it turns out to be vertical here and the other one turns out also to be vertical and even physicists will say it causes the other one to become vertical but that is a very bad way of thinking as I will tell you later. So how would we explain this? Perhaps the easiest explanation that comes to mind is to say well the apparatus isn't actually producing the same state each time it's producing a pair of photons of the same polarization but from one shot to another every time you press the trigger on it it'll give a different polarization so sometimes it'll send out two vertical ones sometimes two at angle theta sometimes two at horizontal and so on. It's very easy to imagine that kind of random two-bullet shooting gun but this doesn't work as an explanation because if you set up the equipment to measure the vertical versus horizontal polarization well sometimes this source would emit two diagonal photons and if each of them had a diagonal state and interacted with the apparatus without any communication with the other one then they would each behave randomly and that means they would sometimes come out with opposite polarizations. In fact when you do the experiment they always behave the same. Now this is a toy model of well a toy version of a more complete argument that says any property at all that you would try to attribute to these two photons does not explain the strength of the correlation of their polarizations not merely to say did they have a polarization but did they have any property that you could attribute to them separately. This was the famous result that John Bell discovered in 1964. Well how do you explain it? Well the way that people do but they know that it's a bad way of talking is instantaneous action at a distance and the reason that's a bad way of talking is so we create this pair of Einstein-Pedolsky-Rosen particles that's another name for entangled particles and one of them happens to pass through a vertical polarizing filter and that means that it was vertical and the other one is vertical too it sends a message that says oh you've got to become vertical and so when we measure it it turns out to be vertical. Well of course that violates special relativity that messages can't travel faster than the speed of light besides even if you could send messages faster than the speed of light this photon amount of bounced off a lot of mirrors and be in some place how would you even know where to send the message to. So that's not a good explanation. Quantum mechanics gives the right explanation and you can go back to the algebra that I gave on a few slides ago and get exactly the predictions of the oppositeness I mean the sameness along any axis but if you have to explain it to somebody at a dinner party and you say well let's start thinking in four-dimensional geometry we've got H's and V's and so on my experience is this doesn't work very well so you have to come up with something else to say about it and so this is not really very rigorous but it's a little better than saying that it sends an instantaneous message to the other particle telling it what to do. So what can say it sends a random uncontrollable message backward in time that is when this photon gets measured it decides at that instant to be vertical and then it decides it always was vertical this dotted line is the message backward in time and of course it's twin Romulus over here of course since it had just decided to be vertical it's twin of course has to be vertical too and whenever you get around to measuring Romulus he will turn out also to be vertical well this sounds like even worse thinking than this hippie entanglement because if you could send a message backward in time you could tell your broker what stocks to buy or sell yesterday and of course I mean even if you're not rich and don't own any stock you could certainly avoid some mistakes that you've made in life by just giving yourself good advice well the answer to this argument is that the word message is really not right if you can't control a message it doesn't work as a message you can't help, you can't give yourself useful advice or your broker buy a message that you can't control and so entanglement behaves like having a pair of magic coins that no matter how far apart you take them and toss them you'll always get the same answer but you can't control either one now this is one of two logical situations in which a message backward in time is harmless the other one is the Cassandra myth where the message gets propagated it gets chosen by the future event but then when you send it into the past nobody believes it well how does this entanglement, what can we use it for? well one of my favorite things for using it for is what is called quantum teleportation which is a way around the problem of getting complete information out of one quantum system and putting it into another and it looks like that would be impossible because there's no way of measuring the state of a single photon and getting its polarization exactly so how can you get that information out of one photon and put it into another one that has never been near the first one well in fact if you tried to do that you would measure it and you would get some information and you'd use that to produce a copy here but it wouldn't be a perfect copy, the polarization might be wrong by 10 or 15 degrees because you wouldn't have learned what this polarization was but you would have ended up spoiling this photon so here's how we get around that using entanglement and now we have three photons we have an unknown photon over here whose state we want to transplant to a different photon and then we have an entangled pair of photons here that were never anywhere near this one they're just entangled with each other and what we do is we do a measurement of A and B and we don't ask them what's the properties of A we ask them what's their relation so we measure the relation between A and B and then we take the result of that measurement and we report it to the location of particle C and then we use that to rotate particle C into what turns out if you do the math into an exact replica of the state of particle A before you destroyed it so we don't clone the information because we have to destroy A before we can produce the copy and we don't send it faster than the speed of light because this message goes only as fast as the speed of light and if you try to measure this particle C before you've applied the corrective treatment it behaves completely randomly so despite the name it's not a way of transportation it's just a useful primitive in quantum information processing which goes on among other things in the operation of a quantum computer well here's my human analog of quantum teleportation suppose we have somebody let's say call her Alice who has witnessed a complicated crime in Chicago and the FBI wants to know what the story was but they know that her memory is in a kind of fragile dream-like form and they have to ask her just the right questions in just the right order and some of these questions have sensitive information in it that they don't want to disclose to the Chicago police so for sure the Chicago police are going to ask her wrong questions it'll just confuse her so they tell her they'd like to come to Washington but she says that she doesn't like to travel and if they subpoena her she'll probably get uncooperative so they decide to send one of their own guys down there but that isn't very good because these guys all have opinions and they don't trust each other to interrogate her alone interview her alone I should say interrogate has a more sinister meaning so well then Remus volunteers she says I don't know anything about this case so I'm not going to influence her unlike any view besides I like to travel just to ask my brother so Remus goes to Chicago to meet Alice and they explain to them they're not supposed to talk about the crime or anything they're just supposed to have a speed date and decide whether they like each other well pretty soon they decide they can't stand each other and the police tell Alice she can go home and then they get on the phone to Washington and say well Alice and Remus don't get on and they have actually maneuvered themselves into a state of perfect oppositeness and that means you can go to Romulus and ask him all the same questions you would have asked Alice except you have to turn the answer around and whenever he says yes Alice would have said no so that's the human analog of quantum teleportation for what it's worth I think after Wooters told me his analogy and then I kind of overdid it he may be sorry well the principle I mentioned earlier that is that if two particles are perfectly entangled with each other they can't be entangled with anyone else and indeed the kind of classical correlation that is of two things each being random but having the same random state because they're like they're two coins that actually are both heads or both tails not because they're in a state of oppositeness of mysterious oppositeness ordinary classical correlation typically comes about from attempts to clone entanglement now of course cloning you can't do it because entanglement is monogamous so here's what happens suppose Alice and Bob maneuver themselves into an entangled state a state of perfect sameness of polarization and then Bob decides he wants to become entangled with somebody else so I call her Judy down here and so he does the same maneuver they did up here but the only the effect of that then is that the entanglement with Alice is spoiled and it's merely classical correlation so Bob is correlated with Alice along some axes but not along others and also he's correlated with Judy along so this is just ordinary classical correlation like we're all used to and it doesn't display the hallmarks of entanglement so let's speak about the origin of quantum randomness how entanglement explains the origin of quantum randomness I should put back here going back to here all of these actions are reversible if I stopped here and just undid this interaction I'd get back to this state and then undid this interaction I'd get back to that state so let's look at this in the case of polarized photons so we have these polarized photons come in here and what I said before is that some of them go into this beam and become horizontal and some of them go into this beam and become vertical but what I really should have said is that they do not yet behave probabilistically what every one of them goes into a superposition of being horizontal in the upper beam and vertical in the lower beam in fact they all go into the same superposed state but when this state gets to these measuring apparatus these detectors then it has to decide whether it's going to be horizontal and in the top beam and vertical in the bottom beam so if we avoid the measurement and just let those two photons these photons go into the two separate beams now these are photons that haven't interacted with anything yet and therefore we can switch the horizontal photon to a vertical photon by rotating it 90 degrees and similarly the vertical into horizontal and we can say put an optical element called a halfway plate that does that that takes horizontals and makes them vertical and vertical makes them horizontal and then put them back through the same crystal the same size crystal of the same material and they will recombine and be back to their original polarization so what has happened here is that I've produced an entangled state and then I've de-entangled it and I go back to everything the way it was originally and what this means is that the public embarrassment of the pupil in having to say what his polarization is in front of the whole class is what makes him forget the original polarization in principle if you took the teacher and all the other students and any mouse that was listening and made them all forget what they'd heard the student could get his original polarization back so now I've argued that classical ordinary information and information processing is a special case of quantum information processing and we should really develop the whole theory on the quantum foundation and that means we've got the obligation to explain what we mean by a classical bit well that's easy we just say a classical bit is a qubit with one of two standard distinguishable values for example horizontal and vertical and a classical wire is a wire that carries qubits that carries these zeros and ones faithfully but randomizes superpositions of them now why would a wire randomize a superposition it's because the ordinary wires that we have in most of our computers the signal passing along the wire I'm drawing this as a thick classical wire is really equivalent to a quantum signal that interacts with an environment down here I'm representing the environment here by another wire and it interacts by this gate that I just showed you about this controlled knot gate and what that means is if the signal is a zero or a one the environment gets a copy of it and if it's anything in between the environment becomes entangled with it but if you lose track of the environment it's helpful if it escapes out the window or gets lost in 10 to the 23rd other photons that are in the room then the remaining one this is the student whose classmates have gone out to recess and you can't get them to forget what they heard the remaining one behaves randomly and this means that a classical channel is a quantum channel with an eavesdropper and a classical computer is a quantum computer with eavesdroppers on all its wires so among other things the quantum theory of information explains the close connection between cryptography the art of defeating eavesdroppers and privacy and computation and entanglement so if entanglement is ubiquitous and almost every interaction between two systems produces entanglement why wasn't it discovered until the 20th century? the reason is because of monogamy most systems in nature other than tiny ones like atoms or photons especially photons interact so strongly with their environment that they become entangled with it almost immediately and that means that if you lose track of any of these things that have become entangled the remaining ones behave as just as if they're classically correlated in other words we have a world that appears to be full of randomness and correlations among things that are individually random which can all be explained by they all have some particular state and we just don't know what it is and yet that whole view arises as a side effect of this subtle thing that we didn't know about until the 20th century and we didn't realize that it had to do with information processing until the last 30 years of the 20th century well of course the main reason people are so excited about quantum information is a practical reason that is if you could build a quantum computer it would greatly speed up some hard problems like the most famous one is factoring large numbers now here's a problem here's an example of a large number it's if you are very smart you can realize that this number is the result of multiplying these two now in fact you don't need a quantum computer to do that to multiply these two numbers you could do it on a quiet weekend this 3 times 7 is 21 that's where that 1 comes from it carries a 2 and then so on if not too many people are bothering you you could actually do it and you could prove that this times this equals that but what's hard to do on a pencil and paper or even on a pretty powerful classical computer is to take this number and figure out that these are its two unique factors however this job is easy relatively easy for a quantum computer not a whole lot harder than multiplying and the reason is well I won't say the reason exactly yet but it works because during the processing even though the question and the answer are classical information the fast algorithm for doing this involves entangled intermediate states so we have to build a computer in which the intermediate data is protected from eavesdropping until the computation is done of course we're for most of my life we've been facing the end of Moore's law but it's really happening that computers can't keep increasing exponentially in their power and cheapness because they're getting to be already near atomic dimensions so can quantum computers give Moore's law a new lease on life and how soon will we have them well I'm going to be somewhat discouraging about that because there is a whole theory being developed of the classes of problems that quantum computers would probably help for or are known to help for and ones where they wouldn't so it's much more complicated there's some problems which we have every reason to believe are hard even for a quantum computer and then some problems that are easy like multiplication for a classical computer and certainly a quantum computer and then some number of these intermediate problems which appear to be hard for a classical computer but easy for a quantum computer of course in order to build a quantum computer you have to keep the eavesdroppers out of it and that looks like an impossible job but it isn't impossible because you don't have to isolate it completely from its environment if it can be isolated about a little more than 99% from its environment quantum error correction techniques which are heavily being researched in this laboratory now will do the rest so the quantum computation doesn't have to be perfect an example of a quantum error correcting code is something that will take a state of one qubit and encode it into an entangled state of five qubits such that even one of these five qubits can be damaged and then undoing this operation sucks all the errors out and throws them away into these ancillary qubits and the original one comes out unscathed now extending that kind of idea for a whole computation involves continually feeding in clean qubits into the processor sucking the errors out and then doing your processing and then doing it over and over again and it's able to correct even errors that occur during the error correction itself so this is a field of great interest and activity to design efficient quantum fault tolerant computations so in conclusion I would say that quantum information provides a coherent basis for the theory of communication computing and interaction between systems in which classical behavior is just a special case and a classical channel is just a quantum channel with an eavesdropper and a classical computer is a quantum computer that's handicapped by having eavesdroppers on all its wires so the right question isn't why do quantum computers speed up some computations and not others it's why does the lack of privacy slow down some computation of course lack of privacy eavesdropping is bad for privacy but actually slows down computations there's some things which if somebody is looking over your shoulder are really much harder to do and so I would finally say that this ought to be part of liberal arts curriculum just like the roundness of the earth even non-science majors should learn a little bit about quantum information and entanglement because it is so fundamental to everything about the world that we inhabit although it was only realized in this last century now I have a few extra topics one of them is the famous Einstein Bohr debate and how Einstein I would say suffered from a tragic misconception there is the kind of questions people often ask people who are working in quantum computing which is well really what is a qubit how much information is contained in a qubit compared to a classical bit isn't a qubit just the same thing as an analog bit that something can have a continuous value between 0 and 1 instead of just having to be a digital value and the other is how do these quantum speed ups work where do they come from well let's look at the first one so this weird behavior of subatomic particles was discovered in the early 20th century by physicists and Niels Bohr became the main spokesman of the new theory and he said that physicists have to learn to accept it not everyone agrees with the way he described it but the two new phenomena were this indeterminacy the fact that individual particles even when they're completely controlled in how they're prepared behave differently they behave randomly an entanglement which I just talked about a lot there was two particles that no matter how far apart behaves in ways that can't be explained they are individually random but too strongly correlated to have been acting independently so Einstein was really impressed by both of these things and didn't like either of them he called the first one the indeterminacy God playing dice and he says I don't believe that God plays dice an entanglement he said called it a spooky action at a distance or in German it's Spukhafte Fernwerkung which the idea was if two things are too far apart to have any plausible influence on each other it almost looks like some paranormal thing is going on there shouldn't be a way for what one of them does to influence the other and he spent the rest of his life trying to find a more naturalistic explanation of the these quantum phenomena in which every effect would have a nearby cause so he has two problems here he's got an effect without a cause that's random behavior and an effect which if you try to find a cause for it the cause isn't nearby and this was just unacceptable meanwhile the rest of the physics world went on and started using these phenomena and the mathematics that explained them and yet they couldn't agree with the right language to describe what was happening so one of the famous slogans I'm not sure who it came from was people argue about what's really going on in quantum systems they don't disagree about what will happen when you do an experiment but they disagree about how to describe what's going to happen and the serious minded quantum physicist says just shut up and calculate don't tell me what you think is happening and he might say it was echoing what Bohr said to Einstein when Einstein said that God doesn't play dice and Bohr says stop telling God what to do well now it's pretty clear that this most celebrated scientific mind of the 20th century that the one scientist whose name is a household word was not flexible enough to take this new fact in and his mistake was in viewing entanglement as some kind of influence of one particle on another and the paper that he wrote with Podolski and Rosen describing the predicament that this phenomenon of entanglement produced in quantum mechanics and there must be some better theory than quantum mechanics because this is unacceptable this was called the Einstein-Podolski-Rosen paper and it came out in 1935 and one of the important notions in that paper was what they called an element of reality if you can determine some property of a system without touching the system without touching anything nearby the system to predict perfectly what it would do there must be some this is what Einstein-Podolski and Rosen said there must be some element of reality in the system that you haven't touched that was already there before you worked on the other one the logical jumping to a conclusion that they did was to have the idea that these elements of reality a thing that is about not about what you do to a system but something that was always there before you touched it that this element of reality had to be localized so in other words the right way to think about it is to say that it's not true that if the whole isn't a perfectly definite state that each part must be in a perfectly definite state an entangled state is a different kind of state of the whole which is perfectly definite but requires the parts to behave randomly and making any measurement on one of the two particles gives you a random result that allows you to perfectly predict what the other particle would do if you made the same measurement on it and that's pretty much the way everybody thinks of it now even though they still can't agree with what language to describe it with now another person around the same time as Einstein he had a really better understanding of entanglement than Einstein did and he called this effect steering that is that if you do you measure one system you find out exactly what the other system is sort of remotely steering another system finding out exactly what it would do under certain conditions but steering is a really bad name for as anybody who's driven a car would know because what we're talking about is a case where you turn the steering wheel of your car and it has completely unpredictable effect on the steering wheel of your car but it has the same effect on the other guy's car of course if he turns the steering wheel the same thing so if you had cars like that you wouldn't realize there was anything strange about them except that they were terribly dangerous until you compared your crash reports afterwards and you realized this eerie correlation was present a mistakenly believing that entanglement could be used for long range communication Nick Herbert published a paper in a study tried to patent this imagined application of it the refutation of these wrong ideas in the early 1980s by Deeks and Wooters and Zurich is part of what led to the birth of modern quantum information theory but this wrong idea like perpetual motion is so appealing that it is perpetually being rediscovered and as I said earlier the proper understanding of entanglement could indicate but also how if you generate an entangled state and lose part of it the remaining parts behave randomly so the intense correlation the monogamy, the inability to make multiple copies of the same correlation and the random behavior of the parts are all things that fit together mathematically and you can't have one without having the others well people often ask how much information is in n qubits compared to n classical bits or n analog variables and this is somewhat ill-posed question because it neglects entanglement and also there's two kinds of information in the state how much information is required to specify it and how much information can you get out of it so let's look at the separate answers to these questions the information required to specify a digital state of n bits is n bits and the information you can get out of it is n bits if you have n real numbers numbers between 0 and 1 it takes an infinite amount of information to specify the number but with any particular hardware you can only get limited precision on the answer so that's an example of whether there's more information in the system then you can get out a quantum system with n particles has exponentially many complex numbers I haven't mentioned the fact that these numbers can be complex but there's an exponential amount of information in it and yet you can only get n bits out so you can get out less than if it were an analog variable and yet the amount of information required to describe the state is much more well there's another difference between digital, analog and quantum information that is why we are so excited about quantum computers and that is that there is good error correction for digital information there isn't good error correction for analog information if I have a slight error in a voltage that's 0.543 volts instead of 0.544 how do I know that it wasn't 0.544 to begin with rather than it was 0.543 and it had 100th of a volt added to it so there isn't good analog error correction but there is good quantum error correction and that means there's the hope of building reliable quantum computers so another way by the time I missed lunch and I'm getting pretty hungry by now if you think of a computer as a information processor and the stomach is a food processor the thing that's different between a classical computer and a quantum computer or the thing that's similar is you give it a classical input of n bits but the classical computer its intermediate state always has a particular one of these digital states so there's an intermediate state of the computer a quantum computer because of superposition and entanglement the intermediate state can be a superposition over exponentially many of these distinct states of its qubits where each of these numbers is an independent variable two to the nth two to the nth numbers waits on these elements of the superposition which can even be complex numbers it just makes it twice as bad and so we say a quantum computer is like a big stomach which has a lot of room for maneuvering to process the information which is just actually rotation in large dimensional space whereas a classical computer is limited and therefore it can do some kinds of problems better that's just a very moving argument I can speak of the particular most famous quantum algorithm which is Schor's algorithm for factoring now the first part of Schor's algorithm boils it down to a problem of period finding finding the period of a periodic function and it works we have we have in the computer we have two registers we call them the x register and the y register and we start out with them both in the zero state and the first thing the quantum computer does is taking a rather small number of steps it generates a uniform superposition over all the values of the x register so instead of both the x register being zero and the y register being zero the x register is a uniform superposition of zero with the y register and each individual value of x then the next thing we do is to reversibly compute this function this periodic function we compute it in superposition so we fill the computer up with a graph of this periodic function where it repeats a very large number of times and then we do something that I haven't shown you why it's easy but it is easy taking only a few quantum operations to make a Fourier transform of the x register so instead of having a periodic function we have something that has peaked which is very sharply peaked here at multiples of the inverse frequency and so then we just measure the x register and we get a random one of these peaks because it never finds itself in the space in between and if these peaks are sharp enough that's enough to determine the period of the periodic function and in the case of short algorithm that means you can factor the number now this is something actually very familiar to physicists it's the problem of multi-slit diffraction so or multi-slit interference as we know in the two-slit experiment if you send a single, if you send a light beam in here and you have this midpoint of the two slits lined up exactly with the axis of this horizontal axis what we'll get is a maximum probability of the photon landing here zero probability here goes up to a maximum down to a zero and so on in a sinusoidal way, whoops in a sinusoidal pattern and so this will allow me to measure the slit spacing by measuring the spacing of the interference pattern and what I sometimes do when I'm in a lecture room with a white wall is take a laser a laser pointer which has a very definite wavelength of light and shine it through my shirt onto the wall and you can see stripes on the wall whose spacing is inversely proportional to the distance between the threads in my shirt but anyway if we have two slits we get this kind of pattern and if you have enough photons we can determine the slit spacing but suppose somebody says alright I'm only going to give you one photon how far apart are the slits and then we have a problem because with this sinusoidal variation this is not guaranteed to be at a maximum it might be anywhere here except at one of these absolute minima so we can only get a little information about the slit spacing from the impact point of one photon and so let's say we will say well okay you're not going to give me more photons how about giving me more slits and of course your adversary will say take all the slits you want so I say okay I'll take a million slits here like this and we still only get one photon but now the interference pattern is extremely sharply peaked more sharply peaked the more slits there are so even one photon will give you a good estimate of the slit spacing and that's exactly what's happening in Schur's algorithm and you would say well why don't you just build a large diffraction grating and use that to factor large numbers the reason is that the number of slits is exponential in the size of the quantum computer register so in other words to factor to factor a hundred bit number you would need a diffraction grating with two to the hundred slits and even if they were very close together this would be several light years many light years in diameter and of course it wouldn't something that big you can't use it for a fast computation as well as being hard to build so this is essentially the quantum because of the nature of quantum information some problems that look like they require an exponentially large amount of classical resources to do this multi-slit interference can be folded up and made exponentially smaller and put into a quantum computer that has only a few hundred qubits or if we have good error correction and build it the way we know how to build it now a few billion qubits maybe would be needed