John Preskill “Quantum Information and Spacetime”

it's a pleasure for me troduce professor John Prescott is the Richard P Fineman professor of theoretical physics and director of the Institute of quantum information and matter on whose external advising board I had the pleasure of serving John is a recovering particle theorists who has moved into the world of quantum information and and actually helped create the what you might call the second quantum revolution in which we understand now much more about the non-intuitive features of quantum mechanics and the role of information storage and computation that is presented by quantum physics and he's made many important contributions foundational contributions and our understanding of quantum error correction of fault-tolerant quantum computation of and I suspect we may hear about today something about information paradox in black holes for those of you in in Applied Physics who don't know what a black hole is it's a parametric amplifier whose idler is hidden behind the event horizon yeah so he's also made important contributions in topological physics and it's connection to both quantum field theory and quantum error correction so it's a great pleasure for me to welcome Jonathan for this second lecture well thanks a lot Steve I guess you'll all judge for yourselves whether I've really recovered from being a particle theorist after today's spot I mentioned toward the end last time that our progress and understanding quantum information is leading to new insights about the quantum structure of space-time and I want to amplify that idea in today's talk and then drill into it a little bit deeper in the third lecture tomorrow I started out my career doing particle physics and cosmology but about 20 years ago I started focusing most of my research effort on quantum information science quantum information science is a synthesis of three of the great themes of 20th century thought quantum theory computer science and information theory and incidentally just last Saturday we celebrated the hundredth anniversary of the birth of Claude Shannon and it's really quite striking to think about how Shannon's ideas today affect us on a daily basis and in quantum information science we're largely trying to take insights starting with Turing into the nature of computation and Shannon and to the properties of information bringing them into the quantum realm as I emphasized yesterday and in the next few slides I'll try to review some of the things we talked about yesterday for those of you who weren't there or have short memories I think the way we should think about this field is we are in the early stages of the exploration of a new frontier of physics well we might call the frontier of complexity or the entanglement frontier in physics we explore the short distance frontier of particle physics the long distance frontier of cosmology and those are very exciting and fundamental but so is this entanglement frontier we are just in the current age developing and perfecting the tools to build and control precisely very complex quantum systems producing very entangled states which are so complex that we can simulate them with our digital computers and we don't have the theoretical tools to understand very well their properties or to predict their behavior and that opens the opportunity for lots of new discoveries now on the quantum entanglement which is the focus of interest in quantum information science means the characteristic correlations among the parts of a quantum system which you can think about this way we can imagine a classical book which is a hundred pages long and then when we read one page of that book we learn one percent to the content of the book and after we've read all the pages one by one we know everything that's in the book but suppose it's a quantum book written in qubits instead of bits with the pages very highly entangled with one another that means that when we look at one of the pages we just see random gibberish we don't acquire any information that distinguishes one entangled book from another and that's because the information isn't recorded on the individual pages it resides almost entirely in the correlations among the pages that's quantum entanglement and what's so interesting is that these correlations are extremely complex so that if I wanted to express in terms of ordinary bits all of the correlations among just a few hundred qubits I would have to write down more bits than the number of atoms in the visible universe and that won't ever be possible even in principle and it opens the possibility that we are entering an age of quantum supremacy where quantum systems will be capable of performing tasks that wouldn't be possible at this were a classical world now we got excited a little over 20 years ago when Peter shor discovered that quantum computers if and when we succeed in building them will be able to break widely used cryptographic protocols but the broader lesson we learn from that is that there are mathematical problems that have the property of being hard for classical computers and easy for quantum computers can be done with resources that scale reasonably with this size of the input to the problem if we have a quantum computer and we're trying to understand better what are the problems in that class for which big quantum speedups are possible but from a physics perspective the most important thing about quantum computing is that we believe though we can't say for sure that a quantum computer will be able to simulate any phenomenon that occurs in nature but that doesn't mean that quantum computers don't have limitations there are some problems for which we don't expect quantum computers to be able to Chi achieve dramatic speed ups those include the hardest problems in the class and P and P complete problems and P means that once we find the solution we can easily verify the solution with a classical computer and for the np-complete problems we don't think that it's possible to do much better in worst case instances than do an exhaustive search for the solution and quantum computers can speed up exhaustive search but only a little bit quadratically not exponentially not very dramatic speed ups like we see for the factoring problem so one of the likely ways in which we'll use quantum computers is to simulate quantum phenom that are beyond the reach of digital computers so we'll be able to study properties of complex molecules exotic materials and also explore fundamental physics and new ways by simulating say strongly coupled quantum field theories or the quantum behavior of black holes or the state of the universe just after the Big Bang so we'd like to have quantum computers but it's a very challenging thing to realize large-scale quantum confusion computing in part because of the phenomenon of decoherence then we might imagine a cat which is both alive in the and dead at the same time in a coherent superposition of microscopically distinguishable States real cats that we observe are always completely dead or completely alive and that's because we can't avoid interactions between the cat in the environment which in effect measure the cat and projected onto a state which is completely alive or completely dead and that phenomenon of decoherence that collapse due to interactions with the environment helps us to understand why although quantum physics holds sway at the microscopic scale classical physics is quite adequate for describing most of our ordinary experience if we want to resist decoherence as if we must if a large-scale quantum computer is going to work properly we have to find a way of preventing information that's being processed from leaking to the outside world the environment shouldn't be able to learn anything about the state of the quantum computer during the computation until the very end when we do a final readout so quantum computing is hard because on the one hand we'd like qubits to interact strongly with one another so we can perform the quantum gates that are needed for information processing but we don't want the qubits interact with the outside world with the environment except we do want to be able to control the qubits and ultimately measure them and it's difficult to find a physical setting in which we can satisfy all of these does it errata reasonably well but we've learned in principle how to make quantum computing scalable through the idea of quantum error correction and the essence of the idea is that if we want to protect information that is being stored or processed coherently from the outside world then we should encode it in a highly entangled state so that when the environment interacts with the parts of the system no information is revealed about the encoded state just like that hundred page book where we look at the pages one at a time we don't learn anything about the information content of the book the information that we're protecting should be processed or recorded in a highly entangled state where the environment interacting with the parts of the system locally can't get access to that information so it can remain protected so we can imagine preparing a state of a cat in an encoded form which is both dead and alive and maintaining it in effect perfectly isolated from the outside world for a very long time and also we've learned how to process information when it's in that encoded form now these ideas that we're mastering having to do with quantum information storage and information processing are really deep in general ideas about physics and so we should anticipate that those ideas will connect with other problems in physics perhaps in unanticipated ways and in fact we're seeing that happen increasingly often in recent years for example there's been a surge of interest among physicists who work on quantum field theory and quantum gravity in quantum entanglement and other quantum information concepts and in a way that's not so surprising because the quantum gravity community has been struggling for 40 years with a very deep puzzle which really has its origins in quantum entanglement the entanglement between the inside and the outside of a black hole which is responsible for Hawking radiation so let me give you a little tease - you anticipate where I'm heading in the talk let me talk about Einstein for a minute now we just celebrated recently the hundredth anniversary of the general relativity field equations and Einstein 20 years later after discovering the field equation I was still intellectually active he was 56 that doesn't sound as old to me now as it used to and Einstein and Rosen wrote an interesting paper in 1935 in which they laid out what they saw as the fundamental problem confronting physics that the incompatibility of our understanding of physics at the atomic scale and our understanding of gravitation on large scales and their point of view was that in order to reconcile the two we should figure out how to understand atomic phenomena from the point of view of the field equations most of us today would turn that around taking quantum phenomena as the really primary concept and try to understand the geometry described by the field equation starting from a quantum description now this paper was interesting for another reason that they introduced the concept that we now call an einstein-rosen bridge they constructed a few a solution to the field equations which has the geometry of a wormhole in space they took two Schwarzschild solutions and they glued them together at the horizon to sphere technically that's what they did and they didn't really understand what they were doing all that well they didn't understand the geometry of the Schwarzschild solution and what they didn't understand the dynamics of the solution that they had constructed but nevertheless it was a original and somewhat seminal idea it's you can ask my colleague Kip Thorne about that he's gotten a lot of mileage out of this idea of both scientifically and cinematically now it happens that just less than two months earlier there was another paper but I'll get to that in a minute because I wanted to point out that if there had been Twitter in 1935 Einstein might have been inclined to tweet and in fact if you look at his papers you can find many statements of 140 characters that would make suitable tweets so in this one for example you'll find every field must adhere to the fundamental principle that singularities of the field are to be excluded most of us would agree with that today and in the conclusion of the paper we hear that they've taken an important step towards reconciling atomic phenomena with the gravitational field equations well that might be debatable but coming to that other paper only about six weeks earlier there appeared a paper by a einstein-rosen and their friend Podolski it was the one I mentioned yesterday which introduced the puzzles that are eyes from quantum correlations from quantum entanglement and this paper too had some tweetable statements in it I'm being a little unfair now because we know that Podolski wrote this paper and then the reportedly Einstein wasn't too happy with the way it was written but maybe if Einstein were tweeting Podolski would have written as tweets too so one of which was to judge the success of a physical theory we should ask ourselves is the theory correct and is the description given by a complete and the conclusion of the paper was that those three authors felt that the phenomenon of entanglement indicated that something was missing however on an optimistic note they thought a more complete theory ought to be possible now these are two papers on very different subjects written just weeks apart one is about quantum entanglement the other is about unusual topology and the gravitational field equations and it's kind of remarkable that from our current perspective these papers really seem to be about the same thing that quantum entanglement and wormholes our complement in ways of looking at the same phenomenon so that's where I'm heading let's talk about black holes a little bit we've been puzzled by black holes for a long time ever since Stephen Hawking discovered Hawking radiation we have not understood well what happens to information that gets lost behind a black hole horizon and not for lack of trying we still have a very incomplete understanding of the answer to this question and when there's a deep puzzle like this where principles of physics seem to collide in this case relativistic causality and the microscopic reversibility of quantum dynamics that means there's a great opportunity for big insights for big discoveries so let me remind you about black holes everybody loves black holes right I mean how can you not love an object made of nothing but pure warped space-time geometry and they're dangerous objects to the astronaut who foolishly ventures inside a black hole crosses at event horizon will not be able to return to the outside or to communicate to anyone who remains outside so just to recall the idea of a black hole we use the concept with a light cone you know if there's a flash of light at an event in space-time and we can imagine spherically outgoing light signal and if we look follow that expanding shell of light as a function of time it defines a cone in space-time that's the future light cone of that event in space-time and everything that can be influenced by that event P is in the future light cone of P and what we find by solving the gravitational field equations is that in the black hole geometry the light cones tip inward so that behind the horizon the future light cone lies entirely further deeper inside the black hole and that means it's impossible to escape and not only the anyone behind the horizon will be drawn closer and closer to the singularity where huge gravitational forces will tear you apart now sometimes we like to describe this geometry using a different kind of space-time diagram than the one I just drew a Penrose diagram I'll explain that because I'll show a few of them during the talk and the idea the Penrose diagram is to capture in a very clear way the causal structure of the space-time the geometry of light cones each one of the points in the diagram represents a two sphere and it's drawn so that the future light cone of a point is bounded by two 45 degree lines going to the left and to the right and so the picture makes clear that once you're behind the event horizon the future light cone can't avoid the singularity and we can formally map future and past infinity so they appear to be a finite distance away and we can fit them on the page so now black holes are radiate why is that classically nothing can escape from a black hole quantumly radiation leaks out and one way to think about that is in the vacuum we know there are quantum fluctuations which you can picture as virtual pairs of particles which appear and then Rhian aisle eight and it may sometimes happen that a virtual pair is created one member of the pair gets lost behind the horizon and the other escapes to infinity carrying away some positive energy so the black hole loses a little bit of its mass carried away by the radiation and the characteristic wavelength of the radiation is determined by the size of the black hole it has a thermal wavelength it's thermal radiation with a typical wavelength comparable to the black hole's radius there's another way of thinking about this which makes it a little clearer what it has to do with quantum entanglement because even in the vacuum of a quantum field theory that is a highly intense stay we can picture this most easily if we think about a one dimensional space and one time dimension as in this diagram here and in the vacuum there is a lot of entanglement between the right side of the space and left side and to make that operationally meaningful we can imagine an observer who is uniformly accelerated along this world line and that observer has a horizon light rays coming from the left side of the space can't catch up with the observer so everything that observer can measure is only determined by the right side of the space and correspondingly even though the vacuum is a pure state that observer sees a highly mix tape because of the entanglement and actually it's a thermal state why should it be thermal well one way of understanding that mathematically is that if I look at the world line of an accelerated observer the coordinates are periodic as a function of the accelerated observers proper time with a period given by 2 pi over the acceleration and because of the formal correspondence between the time evolution operator of quantum mechanics and the Boltzmann factor of statistical physics periodicity and imaginary time with period beta means temperature 1 over beta so in this case the temperature in natural units is just the acceleration divided by 2 pi and if you're wondering why when you step on the accelerator of your Maserati and take off your eyes are not flooded by thermal radiation you should put in the numbers remembering that h-bar is small and C is big so an acceleration of 1 ge corresponds to a temperature of about 10 to the minus 20 K the wavelength of the thermal radiation is comparable to the distance you would have to travel with this acceleration in order to reach a relativistic speed but now consider a black hole and let's imagine we're in a rocket quasi statically lowering ourselves closer and closer to the rock as we get closer to the horizon to prevent ourselves from falling in and we have to pump up the thrust of our rocket so there's more and more thrust as we get closer and closer to the rise and that means we have a larger and larger proper acceleration compared to inertial observers who fall into the black hole and correspondingly the state which looks like it's vacuum to that freely falling observer appears to be filled with thermal radiation quanta which are hotter and hotter because our proper acceleration is larger and larger as we quasi statically get closer and closer to the horizon so there's a thermal atmosphere near the black hole horizon most of those thermal quanto when they try to escape fall back into the black hole but a few of them that are directed nearly radially outward will be able to escape when they reach a distance far away they've been very substantially redshifted and wind up with the temperature which is corresponds to a thermal wavelength comparable to the black hole size now once we know the relationship between the temperature and the mass of a black hole we can just integrate the thermodynamic identity tds equals d e to find the entropy of a black hole and we get an interesting answer that the entropy is given by the area of the black hole horizon times 1/4 in natural units Planck units of area the Planck length is a very short length and correspondingly this is a very large entropy for a solar mass black hole about ten to the seventy eight about 20 orders of magnitude larger than the entropy of the Sun for example and so there's a tension between two things we know about black holes on the one hand they are extremely simple objects made of nothing but geometry which we can completely characterize by just a few parameters like mass angular momentum in charge but quantum mechanically they appear to be extremely complex objects to have many many microstates if indeed that's the right way to interpret the entropy of a black hole as it is for other quantum systems okay so now let's come to the question of what happens when a black hole forms and evaporates we can imagine preparing a pure quantum state allowing it to undergo gravitational collapse of black hole forms and the black hole starts to evaporate it evaporates very slowly but if we wait long enough the black hole disappears completely and then we can ask what happened to the information that fell through the black hole horizon where is it now well in the case of other objects that thermalize in the mid thermal radiation and cool off we anticipate that information which was thermalize is not destroyed but gets transformed into a form carried away by the radiation and which is very very scrambled which is possible to read in principle but very hard to read and practice what a black hole is different from other thermal objects because of the event horizon and then makes it harder to understand how the information could be emitted in some highly scrambled form one way of looking at it as I described yesterday is because the geometry of the black hole is very highly deform it's possible to draw a single slice through the geometry which I've drawn in green here which is a space-like slice so you can think of it as a slice of constant time but it can be chosen so that it crosses both the collapsing body from which the black hole formed and nearly all of the outgoing Hawking radiation that was emitted while the black hole was evaporating so if there's quantum information encoded in the initial state and it is released in the Hawking radiation that means that information is actually at two different places at the same time that would be cloning of a quantum state copying a quantum state and as I emphasized yesterday the principles of quantum mechanics don't allow us to copy unknown quantum states so now we're really stuck because either we have to admit that the information is permanently lost and give up on the traffic reversibility of quantum dynamics or we have to accept that cloning occurs and either way we have to recast the foundations of quantum theory one way here's here's a way of describing it with the Penrose diagram where that green slice is a little bit easier to interpret the point is that I can choose this space like slice so that it stays away from any regions of high curvature it stays far away from the singularity so we would expect to be able to describe physics there using semi classical reasoning and yet the quantum information that fell in with the collapsing body and the quantum information carried by the outgoing Hawking radiation both cross that single slice well one rather desperate way out of this puzzle is the idea of black hole complementarity which asserts that contrary to what you would naively expect we ought not to think of the interior and exterior of the black hole as two different subsystems of a single system more properly and a theory of quantum gravity would have to explain why it's true we should think of the interior and the exterior as two complementary ways of looking at the same system one description is appropriate for the observer who falls through the horizon the others our description is appropriate for the observer who stays outside it's rather hard to reconcile the two descriptions but they are just two complementary ways of describing the same system so of each carries the same quantum information that doesn't mean that information has been cloned it just means we have two ways to interpret the same quantum information well that's the right idea we should be able to subject it to some consistency tests we should be able to ask whether it is operationally impossible within domain of semi classical physics that we think we understand to verify that quantum information has been cloned so we can think about that this way as opposed Alice and Bob want to do such an experiment to verify the cloning so Alice carries some quantum information into the black hole and Bob waits outside collecting the Hawking radiation after Alice falls through trying to decode Alice's information from the Hawking radiation and once he successfully decoded it he then jumps into the black hole as well hoping to unite with Alice and verify that her information is also intact behind the horizon but if bob has to wait too long it'll be very hard to do the verification that Alice still has the information in fact if he waits a time longer than our log R where R is the radius of the black hole expressed in the natural Planck units then it'll really be too late because by the time Bob falls in he would have to receive a signal that was emitted by Alice less than a Planck time after she crossed the horizon in order for Bob to receive that message before he hits the singularity so if the information came out in a time less than R log R then Bob would be able to verify that Alice's information is both inside and outside the horizon but if it takes a time our log R or longer for the information to come out then we don't seem to have any operational meaning for the cloning it can't be verified and at least not in any way we understand so how long does it take for the information to come out well there's sort of a standard conventional view of how that works which is this then if a black hole forms and then it starts to evaporate at first the quantum radiation the Hawking radiation really is featureless it doesn't reveal anything about what fell into the black hole but after a while the black hole has radiated away half of its initial entropy and now the radiation system is a larger system has more degrees of freedom than the remaining black hole and at that point the information starts to be revealed so at first the Hawking radiation is really just featureless thermal radiation and you can think of that as being true because that radiation is still highly entangled with the microstates of the black hole but once enough radiation has come out there aren't enough microstates for the radiation to be completely mixed now it's the black hole which is nearly completely mixed and each qubit merging in the Hawking radiation becomes entangled with the previously emitted qubit so that at the end of the evaporation process the state is again a pure state and the information has been revealed but informations doesn't start to come out until most of the mass has been radiated away or about half of it that's been radiated away and that takes a really long time because the lifetime of black hole goes like radius cubed and flanked units and that's a really long time for a solar mass black hole it's about 10 to 67 years however we can re-examine this question in the case where Alice carries her information into a black hole that is already old we've reached the point where the black hole is past the halfway point and the Hawking radiation that's coming out now is entangled with the radiation that was emitted previously and we can ask what happens then and in that case we can argue that the information really does come out really fast so we don't have to wait very long don't have to receive many Hawking quanta of radiation after Alice jumps in for Bob to be able to decode from the Hawking radiation Alice's quantum state and this next slide is a little technical but I wanted to show it to you because it's very representative of the way quantum information theorists think about things so I want to describe this situation I have a black hole which is very highly entangled with radiation that's been immediate permitted previously and Bob has carefully collected all that radiation and he has it encoded in his laboratory and then Alice is going to throw in a few qubits into the black hole and then there will be some unitary map which will describe how Alice's qubits get mixed up with the other qubits and coated in the black hole and then a little bit of radiation leaks out which Bob also collects and the question is can Bob acting on the system that was initially entangled with the black hole plus the additional emitted radiation decode Alice's qubits so the way we can think about that is imagine that Alice's qubits are entangled with some other system I called it the reference system and so then what Bob is trying to do is to extract from the system that he has access to in his lab the subsystem that is entangled with the reference system because those are Alice's qubits how can e do it or not well we can ask the question this way does the reference system decouple from the black hole after this radiation is emitted where by D couple I mean does the reference system become uncorrelated with the black hole and its quantum state be described as a product state with no correlations if it does decouple then since the overall state is pure and then was entangled with something initially the final state is pure and n since its uncorrelated with B Prime has to be entangled with ER what's in Bob's lab and that means Bob and principal can decode Alice's state that makes sense now the question is what kind of mixing transformation will have the property and how much radiation has to be emitted for Bob to be able to do the decoding and as a first approximation we can say well just suppose that mixing transformation is just a random unitary matrix chosen randomly with respect to the invariant the unitary group and then we can do the calculation and say that with high probability the state will be close to a decoupled state once enough radiation has come out and how much is enough well if Alice Through in K cubits Bob only has to collect hay plus some constant number of qubits in order for the state to be exponentially closed in that constant to a product state so you know if Alice threw in a thousand cubits and a thousand plus 100 cubits came back the decoupling will work incredibly well and Alice Alice's qubits can be decoded by Bob very accurately so it comes out fast if the mixing is very thorough and then we have to ask how long does it really take to do the mixing and so what we suggested when Hayden and I analyzed this is that the dynamics of the black hole will be very rapidly mixing it's highly chaotic and roughly speaking we can understand how it goes by imagining a random quantum circuit the quantum circuit consists of pairs of qubits in the black hole which interact according to some unitary transformations and in every time step we pair up the qubits and perform some random unitary and the timescale that fixes the time in between steps is determined by the black holes inverse temperature essentially its size and then the information can spread very rapidly in a number of time steps which is just logarithmic in the qubits it can spread among all the qubits so the mixing will occur at a time which goes like the inverse temperature times the log of the entropy that's the hard log R because the entropy goes like R squared and beta goes like R so it seems that the information in this situation really should come out in time R log R and we just barely barely satisfy the criterion that the cloning isn't verifiable and so the this picture gave rise to a conjecture for which we have additional evidence now that black holes are very fast thermalize errs that they can mix up information in time which goes like inverse temperature times the log of the entropy and then nothing else can mix faster than that so how fast is that it takes a time of about 10 to the 67 years for a solar mass black hole to get halfway through the evaporation process but for things to mix and for the qubits to come out so that Bob can decode that only takes a time hour log R which is a millisecond or so it's a lot faster than you might have naively expected so this and other evidence seem to indicate that the black hole complementarity idea was on the right track and meanwhile we had other evidence which in really is is more compelling that information is not destroyed by black holes coming from the holographic duality between Vulcan boundary the a DSC at t-duality well we learn from one wall the Sina was how to put a black hole in a bottle where the walls of the bottle or what we call CF T or conformal field theory and the inside of the bottle is a negatively curved space a TS anti-de sitter space and the amazing thing about this duality is that everything that happens in the bulk inside the bottle has an equivalent description of quantum degrees of freedom evolving only on the boundary so I can consider the formation and evaporation of a black hole in the bulk and it can be equivalently described in terms of dynamics on the boundary where there's no gravitation and there's no black hole and no place for information to hide so at least in this setting in anti-de sitter space where we understand quantum gravity best there doesn't seem to be any way for information to be destroyed when a black hole forms and evaporates now this still leaves a lot of unexplained it doesn't give us a very clear picture of how the information is escaping from behind the black hole horizon and it doesn't make clear how the experience of an observer who falls inside the black hole is encoded in the boundary so there are open questions but it's a very strong indication that black holes can form an evaporate without destroying information but as I mentioned toward the end of my talk last time this idea of black hole complementarity was challenged a few years ago by the group called ants black hole complementarity tries to reconcile three ideas each of which seems a reasonable on its own that a black hole and scramble information without destroying it that an observer who falls through the horizon doesn't see anything unusual at the moment of horizon crossing but will later on be destroyed by gravitational forces at the singularity and that an observer who stays outside a black hole doesn't see anything unusual just the usual laws of local quantum physics and they argued that we can't really reconcile all three points of view and that they advocated that the most conservative way out is to give up on the second assumption to say that an observer crossing the horizon does not pass through unscathed but gets destroyed in a seething fire wall right at the moment of horizon crossing and they said this crazy thing because of monogamy of entanglement I explained it yesterday but I'll remind you that in classical systems correlations our polyamorous can be shared in many ways so when Adam and Betty read the same newspaper and become correlated nothing prevents Charlie from reading the paper as well and then he's just strongly correlated with Betty and with Adam as they are with one another quantum correlations are harder to share so that if Betty and Adam are very highly entangled Betty uses up all our ability to entangle the and can't be correlated with charlie and if on the other hand Betty is correlated with Charlie in a fully entangled State then she can't be correlated with annum and the monogamy is frustrating Betty wants to be highly entangled with Adam and Charlie there's no way to do it she can only entangle more with Charlie by becoming less entangled with them and as I mentioned this is an important idea in cryptography and in the study of quantum matter and in black holes where it's the basis of the firewall idea that amps proposed because as they noted if it's really true that an observer who crosses the event horizon notices nothing unusual then Hawking radiation which is being emitted now should be highly entangled with degrees of freedom behind the horizon that's because the freely falling observer just sees vacuum no particles and therefore in the vacuum there has to be a lot of entanglement between just inside the horizon and just outside which is the reason there's Hawking radiation and on the other hand if information is revealed from the black hole then as I've already told you in the case of an old black hole the radiation that's coming out now is highly entangled with radiation that was emitted earlier and if by the time it pulls away from the black hole it's highly entangled with SystemC then it must already be entangled with C at the time it's close to the horizon and so that means the system B wants to be highly entangled with a so that freely falling observers think the horizon is smooth but it also wants to be highly entangled with C because that's necessary for information to come out of the black hole and it can't have it both ways because monogamy of entanglement so if information is really going to be a minute from the black hole we have to break the entanglement between a and B and that makes the horizon a very energetic place that's the firewall so it was surprising to me about this amps experiment is that this violation of the monogamy of entanglement seems to have an operational meaning it seems to be something that's verifiable I can consider a single infalling observer who is still far from the singularity and yet can receive information verifying that B and C are entangled and that a and B are entangled so this violation of a fundamental principle of quantum information seems to be something you could check in an experiment within the domain of validity of semi classical reasoning so it's really quite humbling that after any decades of work we can't answer such a simple question in a way that takes quantum physics into account as what's inside a black hole so what are the possible answers well one is that there's an unlimited amount of stuff inside a black hole that's the information loss scenario I can imagine keeping a black hole mass fixed by continuing to throw more and more stuff in as the radiation is emitted and then there's no limit to the amount of information that can get lost behind the horizon which never comes back in the Hawking radiation so there's an unlimited amount of stuff potentially behind the horizon well that would be rather disturbing from several points of view for one thing it would mean that we couldn't interpret the black hole entropy the area over for as counting microstates of the black hole as counting the amount of information that could be behind the horizon because the latter would actually be unlimited independent of the area and also as I've mentioned in the case of a DSC of t-duality on the boundary description there doesn't seem to be any place for the information to behind it oh the firewall answer is there's nothing at all behind the horizon or at least that's one of way of interpreting what a firewall is then not long after the black hole forms the singularity joins the horizon so an observer who encounters the singularity the horizon at that point is immediately reaching the end of the space-time and destroyed at the singularity but that's pretty crazy because it's not at all what we find when we study gravitation classically by solving the gravitational field equations would say the horizon should be smooth so this would be a quantum effect which would really be profound turning a smooth geometry into an end of space-time in a region where there doesn't seem to be any reason for a singularity to occur classically because the curvature is low until right at the singularity another possibility is that complementarity will still hold but it's a little more wildly non-local than we originally imagined and the lesson seems to be that we have to in some way relax our notion of locality in space-time so by that I mean as if the puzzle is that system B seems to be entangled both with the interior of the black hole and also with radiation emitted a long time ago which is now very far away there's no violation of monogamy if we can think of a and R as being complementary descriptions of the same system if we can think of the interior black hole as Rayleigh being another way of looking at that radiation which is very far away but that's pretty crazy because this radiation might be light-years away by now and if we take it seriously it means that by tickling the radiation we could have some effect which could be seen by a freely falling observer who falls through the horizon that would be very non-local physics so those are the possibilities that most immediately come to mind there's information loss there are firewalls there is complement complementarity that is profoundly non-local or we're really confused and way off track so how are we going to make progress how we can understand what's really going on well I think what we should do which we have other reasons to do anyway let's try to understand more deeply this holographic correspondence in the case where we know the most about it that is anti de sitter space hopefully this which is our best handle on non perturbative quantum gravity well when fully fleshed-out help us to understand the black hole interior now this holographic duality is really remarkable it's kind of amazing I've tried to indicate it here in a drawing that's easier to interpret in which the spatial slices are two-dimensional the boundary has one spatial dimension and in order to indicate the negative curvature in the bulk I've drawn this two-dimensional disc as a pointer a disk all of the colored regions here geometrically had the same size but they appear to be smaller and smaller as you approach the boundary that encodes the negative curvature and the way this duality is supposed to work is that there is an exact correspondence between weakly coupled gravity in the bulk that means classical general relativity plus small quantum Corrections and a very strongly coupled quantum field theory on the boundary and there's a very complex dictionary that map's the observables that could be measured locally by the bulk observers to vary non-local observables on the boundary the bulk has an extra radial dimension emergent from the strongly coupled theory roughly speaking you can think of that as a distance scale of renormalization group scale the theory on the boundary is a conformal field theory so it has a scale as well as a spacetime position and advancing further and further into the bulk corresponds to looking at the boundary Theory had longer and longer length scales flowing into the infrared from that description it's not at all obvious why the bulk physics should appear to be local even and scales that are small compared to the curvature scale at the ball and that's something that's still not very completely understood but what does seem to be emerging from our recent insights is that the geometry itself is emergent that it is really a manifestation of quantum entanglement on the boundary so what are the hints pointing in that direction well one is the holographic entanglement entropy which has been known for about ten years we can ask the following question suppose we take the boundary and we split it into two parts but some connected region a on the boundary and the complementary region and we asked how entangled are those two with one another how entangle is a with its complement and that can be quantified by entropy which is a measure of how much information is missing to an observer who looks only at system a the information is missing because it resides in entanglement with the complementary region and the answer that ryu and taki and Hoggy game in the context of the holographic correspondence is that if i want to know that entropy then in this picture of a two dimensional bulk I should draw in the minimal surface in the vault which connects together the points of region a and measure its length that minimal surface because of the hyperbolic geometry and the vault will dive deep inside the bulk and then returned a because that's really the shortest path through the bulk geometry and the length of that path in units defined by the gravitational constant the same units we would use to relate the entropy of a black hole to its area that's the entropy of region a the amount of entanglement between a and its complement and in higher dimensions in three spatial dimensions I would consider a surface of minimal area and it really would be area divided by four G that gives the entropy if somebody asked me yesterday about measuring entropy and I said it's hard but this correspondence relate entropy to something which is more directly observable bulk observers can measure the geometry and thereby determine the entropy now we can also ask suppose that we have a that we have two different boundaries we have two different strongly coupled field theories and we entangle them with one another so now there's a quantum correlation between two different boundary theories each of which has a holographic dual description in terms of bulk geometry and what happens in that case and the answer is that those two boundaries become connected in the bulk by or home and if there's no entanglement between the two boundaries then there's no wormhole connection and the to bulk geometries are disconnected from one another so it was small to say nansuk skin' who had the quite ingenious idea of referring to this connection as ER equals EPR meaning Einstein and Rosen who discovered the bridge equals Einstein Podolsky and Rosen who first discussed quantum entanglement at least in this setting entanglement and wormhole connectedness are really two different ways of looking at the same thing so if we regard that as a general principle we can think about it this way suppose Alice and Bob are in different galaxies and they'd like to communicate with one another they would like to be united well in fact the wormhole that Einstein and Rosen described is not a traversable wormhole you can't get from one end to the other if Alice stump jumps into her end I shall never be able to emerge at Bob's end because the wormhole just keeps stretching and getting longer and longer and you can never get to the other side now that corresponds to a property of entanglement which I mentioned yesterday which is that we can't use entanglement for instantaneous communication Alice and Bob can't use many entangled pairs of qubits to immediately send a message from Alice's galaxy Bob's galaxy well we can do this suppose Alice and Bob are in different galaxies and they've had the foresight to prepare many entangled pairs where Alice has half of each pair and Bob has half of each pair and now Alice can gravitationally collapse all of her qubits and Bob can gravitationally collapse all of his and now they have two black holes which are entangled with one another according to ER equals EPR those two black holes are connected by a wormhole and so now Alice and Bob if they dare can both jump into their black holes and be united inside the wormhole where they'll be able to have a I hope fulfilling romantic relationship for a while but ultimately they're doomed to arrive at the singularity and be torn apart so it's a romantic but sad story now let me give you a preview of what I want to talk about tomorrow which is something I've been working on for the last year so with some brilliant postdocs pistachio Sheeta and Harlow so I've told you about that this amazing idea the holographic correspondence it's it's the most it's the best handle we currently have on understanding quantum gravity non-perturbative Lee and I've told you a little bit about quantum error correction which is the basis for our belief and hope that we will be able to operate large-scale quantum computers and what I would like to argue in tomorrow's lecture is that these two ideas are actually very closely related that the holographic correspondence is a type of quantum error correcting code that when I think of a local operator in the bulk deep inside the ball which can be described on the boundary as a very nominal operator that corresponds to this feature of quantum codes that if I want to protect some encoded qubit using the code I should encode it in a very highly entangled form so the bulk degree of freedom which can be observed locally corresponds to some highly entangled encoding on the boundary which is very robust against error so the bulk geometry actually deep inside the bulk remains intact even if we introduce errors on the boundary there's a redundancy in the encoding which makes the geometry very robust and part of the reason I think that's exciting is that it's another indication that the right way to think about geometry in quantum gravity is it's a feature of highly entangled States and that means that quantum geometry should be something that we can simulate and study in laboratory experiments experiments with the right kind of highly entangled States will manifest a kind of holographic duality which will allow us to understand more deeply through experiments features of quantum gravity like black hole evaporation last year one mall the Seine and I gave a presentation to an audience that we thought would be mostly biologists and we are trying to explain this connection between quantum information and quantum gravity so it's Juan's idea to explain it this way quantum gravity is like life and we really want to understand life and the boundary theory is chemistry and everything about life is really fundamentally explained by chemistry if we just knew how and the quantum information theorists are chemists they understand things about the structure of entanglement which is relevant to the physics of the boundary and the quantum gravity theorist thinks all that's very nice but what they really care about is life and so what we're trying to find is a kind of molecular biology we want to understand life on the basis of the underlying chemistry and our fruit fly is the black hole information it is the toy problem which whiff we understand it in a non-death won't be able to go even further to explain life the universe and everything now if you listen to string theorists these days you might hear a conversation like this one string theorist says to the other well this is very exciting entanglement is something to say about the emergence of space-time but then the other string theory says yeah I hope so but it's hard to work on and it's true it is hard to work on these ideas because there are very few scientists who can confidently wield the tools of both quantum information and quantum gravity and I think that's in the process of changing that there's a new generation of young scientists who have been internalizing the ideas and concepts and tools of both quantum information and quantum gravity and that's going to accelerate progress or as one of my collaborators recently put it now is the time for quantum information scientists to jump into black holes and if they do I think great progress will result so tomorrow more about holographic quantum codes so the question is how do i distinguish bulk observers and boundary observers and can these observers communicate well oh yeah they can a bulk observer can go to the boundary but the way I distinguish is you can think of the boundary theory as being a kind of boundary condition for the bulk theory and so everything that happens in the bulk and this has that this works has a lot to do with the special causal structure of anti-de sitter space everything that happens in the bulk can be understood in terms of something that you know happens on the boundary and the dictionary that I spoke of you know relates it's most it's a more interesting dictionary let me put it that way when I consider bulk observers who are deep inside the bulk when then the corresponding boundary observable could be something very non-local as the bulk observer gets closer and closer to the boundary it becomes a more boring dictionary where we wind up with you know boundary observable localized where the bulk observer arrives at the boundary space is made of entanglement spaces space-time is a four-dimensional so the question is if we have a picture in which geometry emerges it from entanglement can we explain why space-time is four-dimensional well I wish we could but no I don't think so because I don't think four dimensions is the only possibility that makes sense that there are quantum theories of gravity and other dimensionalities all of which can be realized in some type of holographic description I mean it might not be you know in general wealth you know on we it is our misfortune to live not in anti-de sitter space but to sitter space at the cosmological constant which is positive instead of negative and it is anti de sitter space for which this holographic correspondence has been best understood I actually think holography is a much more general thing and that we can understand geometry in anti-de sitter space or asymptotically flat cases in terms of some alternative strongly coupled highly entangled non gravitational theory but we have a much clearer idea of how that works in other cases but this picture does not indicate anything special about three plus one dimensions sorry in some sense and so I guess which that would then be in the classical general relativity equations yeah um quantum mechanics being linear well as far as I mean you're you you know a lot more about quantum KS than I do don't you so one way of thinking one thing I'm way of thinking about the randomness I just mean by that that if we consider of evolving the system forward in time and then acting with a local operator and then evolving backward in time we'll get a much much different state than the initial state so their sensitivity to use small perturbations and it's a very extreme sensitivity it arises because the quantum information spreads out among the degrees of freedom very rapidly now part of your question was what does that correspond to in classical general relativity and really the the rapid spreading of information is quite closely related to the exponential redshift as you get closer and closer to the horizon and you can think of that the inverse temperature is entering to determine the length scale is you know being that that's that's what's relevant for determining how the ratio scales when you get close to the horizon how long it takes to yeah you can calculate it I mean and it's well you can calculate it to a good approximation if you start out with a big black hole because you can follow the evaporation of a black hole and just using semi classical reasoning until very late in the process once it gets close to the Planck scale in the final stages of the evaporation then it becomes much harder to calculate exactly what's going on but that's a that's a tiny fraction of the total lifetime so in fact it's something that we can calculate and that's was the basis of my claim that for a solar mass black hole it would take ten to the sixty seven years

Channel: YaleUniversity

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Keywords: Yale, physics, quantum information, quantum gravity, quantum entanglement, seminar, John Preskill

Id: MuklWupCvWU

Length: 68min 28sec (4108 seconds)

Published: Tue May 17 2016