 So we're happy to start our morning session with Chris Huckers who will tell us he'll give us the last lecture about black holes and quantum error corrections. Thank you. Okay. This is possibly going to be my favorite of the four lectures because it brings us up to very modern material in this topic. So what we learned in the first two lectures was about you know some historical ways of thinking about reconstruction and we saw that there was this theorem we called theorem one that told us there's this tight connection between the quantum extremal surface formula and reconstruction. So if you know given some say CFT subregion B possibly that's the whole CFT you can learn what you can reconstruct on B by looking at where is quantum extremal surfaces and they can reconstruct operators in the entanglement wedge is what it's called. But that theorem had this assumption of the fixed QES subspace which we learned last time we shouldn't be fully satisfied with because there are interesting questions to ask like about evaporating black holes that don't fit into these fixed QES subspaces. Different states in the subspace we're interested in have different quantum extremal surfaces. But then we dealt with this problem by seeing that there's a theorem two which generalizes theorem one by just lifting this assumption of the fixed QES subspace and so this the existence of theorem two confirms for us that we can still use the quantum extremal surface formula to learn in principle what is and is not reconstructable. I didn't emphasize this point but it's important that theorem two just like theorem one has this if and only if between this quantum extremal surface and this reconstruction. So it's not just telling you you can reconstruct all the stuff in the entanglement wedge it's telling you you can't do more. So you know if you could reconstruct something that it's not telling you could reconstruct something outside the entanglement wedge that would be saying like one that condition one has a very large little b. The theorem would tell you that therefore that larger little b should be the region that we're calling the entanglement wedge. So theorem two told us exactly what you can and can't reconstruct but it was a little hard to just read off all the interesting physics from it because the operators you could reconstruct were a little obscure. They really use like product unitaries that worked on the state psi but that's life that's the that's the theorem about reconstruction that we have so far and so in principle the program now would be take this theorem and say compute the quantum extremal surface of various regions like the CFT when the black hole is in various states and knowing that information combining it with theorem two you could learn exactly the operators that you can and can't reconstruct on the CFT and I would like to be able to tell you exactly that data now but it's very hard and no one has done it so that's not the story that I'm about to tell you but even though that's hard there's something qualitative and very important that we have learned since then from this line of thought and that's the following so first let me emphasize something that we learned from theorem two that's that's very nice so as we saw last time we were talking about the evaporating black hole this is an evaporating black hole in ADS and we might draw its Penner's diagram like so you know maybe there's some dust ball that collapsed to form this black hole remember this is r equals zero on the left and this is the asymptotic boundary and then this is the singularity the dash lines the horizon this goes down to minus infinity in time and we coupled it to a reservoir which I could draw I could draw like this this is often how it's drawn where it's literally just draw like some it's a non-gravitational flat space region so it's in flat space this is its Penner's diagram it's like this triangle and I'm just drawing it sort of already glued to the asymptotic boundary so this really emphasizes this idea that some mode passing radially outwards and hitting the boundary would just sort of pass through and enter this flat space region so this way of drawing it is just reminding us that they're sort of defined with boundary conditions that relate them in this way and remember this allowed the black hole to evaporate because these outside Hawking modes just pass through don't get reflected back into the black hole and therefore the black hole loses energy over time and what we saw was that if we say take some state so by this orange dot I mean this I want to talk about a time slice of the CFT at a late time so after the black hole has been evaporating for a long time and you know maybe there's a time slice of the bulk that I might consider that it's like this orange line that's sort of a dual on this orange line there's a dual state to this CFT state we saw that if I want to compute the von Neumann entropy of the CFT at this time it's given by of course the quantum extremal surface formula a gamma that changes over time so at early times so I could draw I could draw this probably better with colors so say if you tried to compute the von Neumann entropy of blue the blue time and the CFT it was given by this quantum extremal surface that's like the trivial quantum extremal surface gamma didn't exist and then the only contribution came from this SB term which really just picked up whatever Hawking modes are on this blue slice at this time but at this later time after the so-called page time we're now called orange B threat to represent that we're talking about the CFT at this time we saw that I mean I just claimed to you I didn't prove it that there is a different quantum extremal surface that nucleates essentially here right behind the horizon and so the the little b of this orange region is between this dot and this x so this is little b whereas down here little b was everything and all of these Hawking partners are excluded so the interior guys which had a lot of entropy are no longer contributing to S of B and therefore this term is very small and then this term was approximately the area of black hole over 4g Newton and so there was this transition as time went on from being given by this term to being given by this term and then that is why you'll see plots called the page curve that look something like this that the entropy is going up and then falling down with time and then this time is often called the page time that's a reminder so what I want to emphasize is there's some other cool facts you can point out about this situation because you know as the black hole is radiating you know more and more modes accumulate in this reservoir so so an early time slice of the reservoir might look like this a later time slice might look like this so these later time slices have more modes on them and so therefore you know that's that's just because the radiation is accumulating in the reservoir and we can ask about the entropy of the radiation and the upshot is that the entropy of the you know the CFT and the radiation are in a pure state together say if we started this whole thing out in the pure state so we drew this picture which sort of combines all of these systems there was this gravity system and I was drawing it's dual CFT in the same picture with the reservoir but we should remember that you know really there's two descriptions going on there's the one that's best given by this drawing and then there's another which might call the CFT description or the mental description I'm gonna call it the fundamental description because it's you know using the CFT which is in some sense a more fundamental description of the physics than semi-classical gravity which is an effective theory and in this picture I wouldn't draw this in the CFT description we would just draw there's a CFT so this line is just exactly this line I'm still suppressing the space dimensions and then the reservoir is of course something I would draw like this but this is still CFT and this is and this is this whole triangle is the reservoir so the physics as described in this picture is that this you know CFT is sort of emitting quanta over time and of course since the whole thing is in a pure state it's all in some big state psi say this is the state of the CFT and reservoir if you at any time compute the entropy of the CFT which we're calling be here that will always equal the entropy of the reservoir I don't know if you've seen this fact before but this is a general fact about von Neumann entropy that if you have some pure state that's bipartite just has two parts the von Neumann entropy of one side always equals von Neumann entropy of the other side you could prove this say using the so-called Schmidt decomposition but we won't try to prove it here so what this tells you is all is that let's say you were to compute the entropy of R it would follow the same exact curve and well I don't want to be too pedantic about it the upshot is that S of R is also given by this QES formula this is one argument for it there's independent arguments that you should consider S of R to be given by the QES formula as well but its entanglement wedge is sort of the complementary one so at early times at early times when we were considering some CFT state and what entanglement wedge was it was the entire time slice of the ADS at late times whenever the CFT loses access to this part so now it's entanglement wedge is just this outside part where does this go it doesn't just disappear from all entanglement wedges it becomes the entanglement wedge of the radiation I found this very surprising when I first heard about it because R doesn't have to be a CFT or anything it's just some system some quantum system entangled with this holographic CFT and the claim is that you you can also use the QES formula to compute its entropy and it can have entanglement wedge that includes part of this ADS space time might seem surprising let me not try and argue this too carefully I just want to tell you this so I can move on to some puzzle that this informs yes yeah here evaporating black hole yeah so should I think of the CFT as diminishing in time yeah it's losing information and it's losing energy with time yeah also not number of degrees of is it a changing CFT or is just a fixed CFT it's just a fixed CFT and it's it's coupled to the reservoir by some by local coupling I'm at some operator madding to the Lagrangian of the two systems that just has it's like little g times o a CFT o reservoir and so it's just coupling them and the reservoir is very big and very cold so the CFT is losing energy to the reservoir by this coupling so maybe this is a very very simple answer but regarding the page curve that you do so my understanding is that that's one of the great motivations behind all of this because yes basically the starting state and the final state are both pure because they have von Neumann entropy equal to zero and this is very nice but what about all the in between times is it is it fine that the von Neumann entropy is fine so that the state is entangled during the process yes yes so this non-zero entropy in the in between times it is coming from the entanglement between the CFT and the reservoir if you were to compute the entropy of the total system like see CFT union reservoir that's function of time that would always be zero yeah great so the so keep this in mind we're gonna come back to it and maybe it'll even make more sense why I was emphasizing this point or what I meant by it what I want to get to now is that there's sort of a puzzle that's been implicit in this discussion of evaporating black holes and what we can reconstruct I want to make it explicit it was very perplexing puzzle bothered me for many years and I think now we have a better understanding of what's going on so the puzzle is this so we said just now that there are sort of two descriptions of the physics of this evaporating black hole one is given by this picture where you have some ADS spacetime and the boundary conditions are such that it can lose Hawking modes into the reservoir the other one didn't involve gravity it was just a CFT couple to the reservoir these are just two descriptions that are related by the ADS CFT dictionary and so you know as before as always we imagine we have some linear map V which maps ADS states to CFT states that was that this is sort of our starting point but what's happening is that a very important claim I mentioned it in the discussion section yesterday because there was a question about it but I need to say it now too so the claim is that if you think about these two pictures and what's going on with this evaporation process we're going to learn something very surprising about V and about reconstruction the claim is that you know at least at late times and it's evaporating black hole process there are more states I'll say this way there are more yeah there's a larger there's more degrees of freedom the black hole interior than the CFT is using to describe that state this is the claim and so a little more concretely well I'll write an equation in a second what I mean is at this time let me write some equation so what I mean is if you were to look at the CFT state as a function of time you know this is some CFT state and in the bulk there's some black hole and you can ask you know maybe the CFT state is something that looks thermalized and you know CFT's have infinitely many degrees of freedom but the the CFT Hilbert space is of course infinite but it has support not on the entire Hilbert space you know it some thermal states perhaps has support on the entire Hilbert space but you can not you can truncate that to some finite subspace of the CFT without changing the state very much or alternatively we could just consider some say micro canonical black hole which literally just has support on some finite subspace finite dimension subspace of the CFT the idea is that the CFT black hole states in the CFT use a subspace such that the dimension is approximately exponential in the area of the black hole over forge Newton so this is the dimension of the Hilbert space that describes the physics of the black hole is the claim so I mean this is this is like the usual lore that existed before ADS CFT that according to some outside observer or yeah the a black hole of area a is just well described as a quantum system with a Hilbert space that's very large Hilbert space of area of dimension e to the a over 4g and ADS CFT makes this very rigorous because the subspace that you need of the CFT Hilbert space to have a black hole state that you know is well described by normal say classical physics you know some micro canonical black hole say will literally have support and a subspace of the CFT Hilbert space of this dimension but meanwhile Hilbert space of the interior this is the Hilbert space that you might use to describe the modes the low energy modes in the interior of the black hole in the semi-classical gravity description this is what I mean by h interior so h interior is describing said the low energy modes on this time slice in this description so this is a right here it's a semi-classical it's described it's a Hilbert space of semi-classical degrees of freedom we need more and more of these this is this is growing with time this is growing with time because the of the Hawking process so as time goes on you have more and more modes outgoing modes so if you were to picture it these are modes just inside the black hole horizon that are trying to say escape the black hole they're moving outwards but they're getting redshifted by the potential of the black hole so that they're getting redshifted they're getting brought down from the UV to be long wavelength modes and on this diet on this picture those are just accumulating so the number of low energy modes is just growing with time I mean and the Hawking process relies on this fact because it's these low energy modes in the interior that are the ones that are entangled with these effective field theory Hawking quanta outside and so what happens is this is just growing with time and the black hole decreases with time let's say the dimension of this or to make it more comparable to this the log of the dimension of the interior modes and so at some point which is exactly at the page time the number of interior modes so that the Hilbert space of semi-classical gravity describing the interior degrees of freedom is larger than the physical Hilbert space of the black hole and so the conclusion is that at the at this time and every time afterwards this v must be non-isometric it must be mapping a larger Hilbert space into a smaller Hilbert space so it's mapping the state of more degrees of freedom into a state of fewer degrees of freedom and so let me write this out so after the page time so once you have an old black hole this v is what we call non-isometric and let me remind you what an isometry is so you know what I mean I say it's not that so an isometry v is maybe I'll just use a different letter so my geometry k is which maps from say a Hilbert space a to Hilbert space c is a linear map such that k dagger k equals the identity so maps a to c and then c back to a and it's just the identity if it does that so you can prove it's a linear algebra proof that this is only possible if the dimension of a is at least as big sorry dimension of c is at least as big as the dimension of a otherwise you know in going from a to c you have to have some kernel some subspace of a that's destroyed and then you can't get that back by acting k dagger and so the claim here is that the adse of t-map must be non-isometric sometimes you know there's some subspace of h8s on which it's acting non-isometrically and the argument was this one about evaporating it must be non-isometric here so it's mapping from a larger to a smaller Hilbert space and this is perplexing because it therefore must be destroying some information because the must destroy information right so whatever the you know it has to have some kernel some some subspace of h interior that it just annihilates in the map to the CFT Hilbert space and so whatever information was stored in that subspace is just gone to your treatable which is perplexing because you know we we've been saying or we know the hope is that information is not lost in gravity and the CFT is the full non-perturbative description of what of the physics here is the idea so how can there be some information that's that's not in the CFT it would have to be some information that we should perhaps just think of as unphysical it doesn't really exist it's just sort of like fictitious information that's present in the semi-classical gravity theory that's somehow not physical because it's not information that makes it to the CFT under this map that might be the word you say but but it's a little worrisome because what exactly is the information that's lost in this map like is it if I just like throw my diary into a black hole is that gonna be part of the information that's lost in this map we would hope not because we hope that this sort of information that I die the information I just throw into a black hole makes it out into the radiation that's the expectation so this is sort of the puzzle and the let me just tell you thinking about how this puzzle is resolved so this is there's a sort of proposed resolution to this puzzle that combines a lot of work from the previous four years or really going back 15 years or so and this is so this is a resolution that is sort of a conglomerate of a bunch of ideas but all put together in this paper by me and Neta Engelhardt's and the proposed resolution is that you know maybe even though this weird thing is happening by this ADS to CFT map V that some information is getting lost maybe it's the maybe all the information we care about is not getting lost and only only information we don't care about is getting lost so so the proposal is the following that yes there exists a possibly large set of null states or some kernel to V kernel being the subspace that's annihilated my V but V acts isometrically I should say approximately isometrically up to non-perturbatively small errors and genuinen this set on it on the set of states we care about which are the set of computationally raw yeah the set of simple states so that is to say that if if you had some states psi of the ADS space you wondered if it was highlighted by this map or to say this better if you wondered if its inner product was approximately preserved by this map V it would be if psi is in this some set of and the simple states are defined in the sense of computational complexity so it's it's like you have to pick some reference state maybe it's the vacuum state and the set of simple states are all of those that you can do using some number of basic operations like adding a particle here flipping a spin there it's simple only if the number of basic operations you need is polynomial rather than exponential and I can tell you polynomial and what and that's the entropy of the black hole that's present so if there's some black hole around and there's some experiment you want to do say on the Hawking radiation or on the black hole itself and the experiment only requires if the experiment requires that you do an exponential number of operations exponential in the black hole entropy then all bets are off as far as the encoding of that information that you're probing meaning that you should not trust the semi-classical description so if you were to use this picture that we've drawn here that has the gravity description in which say in this picture these modes are very entangled with these guys if you were to do some exponentially complicated operation like test what the state is of these guys say these outside Hawking partners because the the operations exponentially complicated the claim is it's not guaranteed that this picture to the information that the information you're probing in this picture will make it to the fundamental description and so your experiments won't necessarily be well described by this picture instead you have to go use the fundamental description and the converse of this is that if you're doing anything anything simple a simple experiment where you just want to say like measure the state of seven Hawking quanta that's an experiment that's not exponentially complicated and the entropy of the black hole and those experiments would be well described by this picture because the information that you're probing will make it to the fundamental description so therefore it is physical so let me just wrap up by saying this is the proposal and the idea yeah so given more time what I would want to do is is show you evidence for this proposal that this proposal can work by demonstrating to you explicit models this is realized so so I'm not gonna write a whole bunch of stuff given this limited time but I will say this journey of trying to figure out how reconstruction works in ADS CFT let us in particular to hear where we realized there was something we couldn't reconstruct all operators in the interior of a black hole using the CFT or using radiation or even using both and we want to know what we can reconstruct what we can't and this is actually a subtle question because the map is this non isometric one and that might worry you because if it's not isometric clearly some things can be reconstructed some things can't because some information is lost have enough control yet over the actual ADS CFT map to figure out what information is lost and what's not but we can prove that it's at least mathematically consistent for so-called simple information to be preserved and not necessarily complicated information the proof involves some models explicitly demonstrate this and this you could tie this back to the black information paradox and say this actually nicely ties together a lot of the lessons we've learned over the past few years sorry so can we say that there are also some very complex states in the CFT part that we also don't have access to and if we have access to all the very complex system complex states on both sides the isometry would not be approximate in this time and it would be precise so exact isometry so yeah unfortunately so no it would it's yeah the idea is that even if you have access to all states in the CFT the map is still non isometric and this is from this counting of Hilbert space dimensions so it's like there must be some states annihilated by this holographic map and the game we're trying to play with this proposal is saying given that is there some way that physics can still work out right and the answer seems to be yes sorry I'm a bit confused on the fact that so if I'm saying correctly this picture is somehow then kind of like semi-classical yeah and however the fact that the Hilbert space of a black hole should be e to the area over four should still hold in any microscopic theory yeah so why are we insisting on saying that there is an Hilbert space on the interior of the black hole and that we have to do to find an isometric map instead of just saying okay then probably like you know this counting is just wrong and yeah good each interior is not yeah so I think what you're getting at is a very important possibility to consider which is that maybe we were wrong just in the level of the semi-classical theory in the way we were counting the number of degrees of freedom I think that that turns out not to be the case I think if you are careful about it that you do conclude that the number of degrees of freedom in a semi-classical theory is growing with time and the the basic argument is that there's a number of ways you could argue it but but one of them is that you can look at in the semi-classical description right we you can do this setup where we extract the modes into the reservoir and the reservoir you know is isolated from ADS so we have no trouble looking at say and it's non-gravitational so we can look at the states of the degrees of freedom there and the density matrix there looks like max way entangled with these guys and it looks so maximally entangled that to be purified by these guys these guys have to have a very very large dimension much larger than e today over 4g yeah good so the page curve yeah you so your question what I'll repeat it because you didn't have the mic was isn't what I just said in contradiction with the page curve so the page curve is the physical entropy and so that's the entropy and what I would call the fundamental description so like the fundamental true entropy of this radiation is following this page curve but the state of these modes in the semi-classical description or what I was would call the effective description is this very entangled one so so there's yeah I want to emphasize there's sort of two descriptions one is what I would call the effective description and one is the fundamental description and in the effective description you can ask what the state is of these guys and their state is such that their entropy in this effective description follows the Hawking curve not the page that comes down we know that can't be a true physical answer from many arguments true physical answer has to follow the page curve and indeed it does so it's in the fundamental description where these guys have a different state that state follows the page curve maybe let me make this comment because I don't think I said this and maybe this is what you're getting at this V is a map from ADS states to CFT states and so it acts trivially on this reservoir so you might worry that you know how can everything I'm saying be consistent where in the effective description these guys look very entropic and in the fundamental description they look like they have much less entropy because the map between the descriptions just acted like the identity on this Hilbert space so it shouldn't it have not changed the state and so if it's entropic in one it should be entropic in the other and the answer is surprisingly no because the map from ADS to CFT is non-isometric it can change the state of any system that was entangled with it so what's effectively happening is it's quantum teleporting the information that was in the interior outside so yeah I would be happy with more time maybe during the discussion to explain what I mean by this but this is very nice story like realizing that the map is non-isometric can help explain why in one description this radiation could have a different states than another description the answer is that the non-isometricness actually is crucial to the information getting out in the fundamental description well there is still 25 minutes so 25 minutes good okay so what I will do is explain to you one of these models and I think this will help so the idea is the following so we want to define a V that maps from a larger Hilbert space to a smaller one so what it's going to do is we're going to take we're going to imagine we have some number of qubits it's called them let's call these qubits V is supposed to map the Hilbert space of little l to the Hilbert space of capital B and little l and this is going to be the dimension it's going to be much bigger than Hilbert space of B so we're going to define a map that does this and we're going to see the properties that it has and which will which I think will be surprising so so we have these these guys L and then we're going to have some qubits here that will call B and so the map is going to sort of act on these guys it's sort of going upwards so time you know time is going up so we start with these guys then the first thing we do in this map is you can tack on some extra system that we'll call F and it's going to be in some some initial states we'll call size zero F we're going to act a big unitary and this unitary will be drawn at random so it's chosen at random so there's a rigorous way to do this so if you give me some large Hilbert space you can consider the set of unitaries that act on that Hilbert space and you can just draw one at random so you're sort of choosing a random from the so-called harm measure the details of this choice aren't important but it's a it's a it's a very convenient way to choose things a random and it actually is well understood how to calculate things like properties of states you get by starting with some state and then acting around unitary but because it's unitary and it's acting on all of these qubits it outputs the same number so we had three four five six seven eight nine ten that puts the same number and B as much you know it's just these three so we have to decide what's going on with these and what we're going to do is we're going to post select them on to say the zero state so we're going to call these qubits P so sorry if you can't see this this is these qubits are called I will call them capital P those say seven qubits in this drawing and this triangle here means that we're post-selecting them so so far what we have is we started with some states so we started with some states Phi on this Hilbert space L and we wanted to act V on it and what that looks like is we took Phi just on L we brought in this extra system F that was just in some state psi not any state then we acted some unitary which acted on L and F together so like mix them up and then afterwards we're acting with this bra some number of those qubits P so P is you know it's not just made of qubits from L or made of qubits from F it's made of some combination so this is this is a non-isometric map because it's mapping a state from say a large number of qubits in L to a smaller number in B but so far it's not yet the the full thing that we want because it would typically map normalized states in L to sub normalized states in B so to fix that we need to just multiply by a number that's part of the definition and it turns out the right number is just the square root of the dimension of P this is the this is the Hilbert space dimension of P and we're taking the square root and if you do this so that's I should write that here the state you you get defined this way will be very close to normalized typically so this is this is the model and let me tell you some nice properties so the first nice property over here right is that is that inner products are preserved so there's this this this is an equation or inequality that you can drive so here we're integrating over all you so this is some well-defined thing you're integrating over the harm measure on unitaries this is given some compact group there's a well-defined left and right invariant way to measure and to integrate over it but those details won't be important so it's just some measure on the group of unitaries that shows up here in the integrand because these V's are defined that way and depend on you so let me write this and then explain it so the claim here is that this inequality is true with something you can derive using this V we have written down and so here psi 1 and psi 2 are both states here we have tensored on this extra system R is capital R sort of thinking of a similar setup to here where we don't want to just talk about starting with states that are in L sorry I just lied to you I said this is a state in HL the state in HL tensor ours some arbitrary reference or reservoir system that we're adding so this is a state in LR that we're including for generality so you can include this extra reference system and this inequality is still holds and we just have to remember that when you act V on the states it's acting you know like tensor product with the identity on the R part and so it's mapped both psi 1 and psi 2 with V it looks like this so this is so here we have the inner product between these two states psi 1 and psi 2 here we have their inner product that you get when you first map them into the B Hilbert space or the BR Hilbert space and then take their inner product and the claim is that if you forget about the integral per second the claim is that the inner products are very close right because this is the difference in their inner products so the inner product you would get on the LR system and the inner product you would get on the capital BR system and the difference is less than this very small number it's very small because it's you know one one over the square root of the dimension of capital B and even though B was smaller than L it's still some Hilbert space with possibly an extremely large dimension and so morally actually it's like a black hole over 4g so this would be like the difference in the inner product of the states before and after mapping it suppressed by this amounts like e to the minus a over 4g and then this average is just saying this is it's very easy to compute these things with these integrals here there's some known technology using one garden functions and this is just telling you that in some sense for almost yes so literally this is just saying this is true on average over you for any two fixed states psi 1 and psi 2 and then the meat the real thing that you get out is that this implies with some more work that given any fixed you that this is true for almost for a very large number of psi 1 and psi 2 but let me actually come back to this that point momentarily what I want to emphasize right now is that this inequality suggests that V is likely to preserve the psi 1 psi 2 dimension of H L is much much bigger than the dimension of H B this is the claim and then I will explain this in more detail when do you saturate the inequality good yeah there will just typically be some states that are close to saturating it actually I'm not sure exactly yeah it'll be I'm not sure how close you typically get to saturation that's a good question some states will just happen to do better than others so some states could actually be preserved very well because we're sort of choosing if you just choose you at random see if it one of these not doing a whole integral some states will do very well and have their inner products almost exactly preserved and some states will do very badly yeah the claim is that some states could even do horribly here and have the inner product be different by some very large amount but those states are so few and far between that's like measure zero so they contribute very little to this integral but the ones that saturate the bounds are hard to characterize yeah yeah if you yeah V dagger would be something you can read off from this where you just sort of like add the cat pee and then do you dagger but this would not be a yeah so but this V dagger wouldn't return you to the states that you started with typically it's typically yeah you would have supports on yeah a typical bulk state you put in here would come out with some support on P that wasn't just the zero state but like had many other states on these guys but those would all get killed and then doing the V dagger would not put those back in and so you wouldn't get out the state you started with so yeah if you want to do the inverse map starting from the CFT and figuring out the ADS states V dagger won't do it for you and given V is V dagger uniquely defined or you have a different option like could I choose state 1 in P instead of 0 in P and call it V dagger yeah given this V that I've written down V dagger will be defined because this V is just some matrix and so V dagger is just the so this V is a rectangular matrix and so V dagger is just you know whatever you get by transposing and complex conjugating but you could define other V's that look like this but with different post-selections yeah I think what I would like to do is give you some intuition that I think this inequality is a little hard to parse and where I want to go with it is a little hard to parse unless I give you some intuition on what it can mean to for V to approximately preserve the inner product of two states even if the input is much larger than the output and there's a good model for this which I'll call the phase model you can ask a question yeah here the complexity of the states didn't play a role right yeah exactly it did not play a role so the the complexity of the states would come in when I would say yeah this was some statement that was averaged over these then you can ask what are the properties of a typical V so if I just picked one of them at random and in fact you can use this plus some other information some theorems about measure concentration to argue what the properties of these typical V's are and then the the answer is that they will preserve the inner products between a large number of states with extremely high probability not all states because that would be impossible but a large number and then you can argue that the number of low complexity states is that that's the set of states whose inner product is preserved by a typical V includes the low complexity states would be the argument this example is the following so let's say we have our hopper space little l and it's labeled by some index n and I'm going to define a different map not this one it's a simpler one but it's going to illustrate some of the qualitative behavior of these non isometric codes so I call this new V V phase and what it does is it just maps it to capital B this is the definition just maps this state like this basis of states indexed by n to these states of B where here we have M and these are some random phases so these status are just random numbers generated from between 0 and 2 pi and a random is a function of n and m so we're just choosing the set faders at random and then now you given this you can just go ahead and compute as an exercise what is the inner product between say two of these states for n and if you write it out you'll say okay well I can just go ahead and use this definition to sorry this not be a dagger yeah and this you can evaluate this and that one and it comes out to be one over the dimension of B times the square root of m e to the i theta n comma m minus i theta the m's are the same because you got a chronic or Delta whenever you did this mapping and then took the inner product in the B space and then this is a sum that you can just you can try and do and the trick is to realize that because these status are random as a function of in an M that this is essentially a random walk and you know and you we know how to sum terms that form a random walk what you get is first of all if in an n prime were the same then these would be the same so this would be one for every term and so you just sum up dimension of B terms all equal to one and I would cancel this one over B and so you would just get one but in is not equal to n prime now you have the random walk because these don't in general cancel instead they they're just you know both independently random and so you're summing up random phases for in for this number of terms and that means that you get something of typical magnitude square root of the dimension of B that's the answer for random walk and so the square root of the dimension of B over dimension of B gives you something that's of order one over the square root of the dimension of B and so we see something that is perhaps surprising to us but it's very it's very true that you know here we didn't assume anything about little l being smaller than capital B so in fact I want to consider the case where little l has a much larger dimension than capital B this is still true this v phase is in a sense acting approximately isometrically on it right even though even though it's technically not an isometry it's embedding a larger Hilbert space into a smaller one it seems to act approximately isometrically because it preserves the inner product between two states if they were the same these basis states and if they were different then okay it's no longer doesn't preserve the inner product exactly because it's not this isn't zero but it's very small it's one over this like square root of the dimension of capital B so if that was very large and this would be approximately zero okay so so the idea is that you can find part of this I guess you can find the dimension of HL states approximately whose inner product is approximately preserves let me say it that way preserved even if now there is a limit to this l can't be arbitrarily large and have this still be true because then the random walk argument starts to break down you get these intersecting paths and the limit is that if HL is doubly when it's when it's like exponential and the dimension of the Hilbert space there starts to be a problem so if it's larger than about exponential the dimension of capital B this argument that we used to drive this starts to break down and indeed that's a very general phenomenon so you can embed a larger Hilbert space into a smaller one and it's approximately isometric way but not something that's exponentially so this phase model is supposed to be intuition for how you can embed in some sense a larger into a smaller Hilbert space and preserve inner products and in approximate way so in by saying if you go back to the static model it's doing something morally very similar and the idea is that perhaps ADS CFT its map is doing something also morally similar and the static model if you use this inequality plus some extra arguments with measure concentration you can argue the following I'll write this down and then I'll leave it there I'm calling the static model as the mom the V I wrote down that had the random you so let me just say this is called the static model because there's also a dynamical model in this paper which has it does something very similar but has some dynamics in it this was some inequality that helped the held over you and then you can drop the on average this gamma is some order one number maybe like a half or something it's less than a half but and then this is true you can prove as a supremum over psi one and psi two you look at all simple states psi one and psi two all computationally simple states with extremely high probability over the choice of you and the model we wrote down this inequality can use to hold so all simple states have their inner product approximately preserved by that model is the upshot and so this model isn't is a proof of principle that the principle the proposal I wrote down is mathematically consistent you could have all simple states have their inner product preserved and so a lot of the information the information you care about in ADS can be successfully mapped to the CFT even though a lot of the information a large part of the Hilbert space is annihilated so let me leave it there thank you for your attention in these lectures it's been very fun I'm not sure about what I'm saying but I have a feeling that in both cases the average must be zero and it what you wrote is actually the standard deviation this is the average with the absolute value so what what's the measure it's yeah because it can't be zero because this isn't all is a there are many terms where the difference is finite and then we're just sort of integrated summing up a bunch of positive terms he's in the square squared and then square root of all this or no it's a it's this one yeah what about this one is it because in random walk the average is zero but yeah this is under deviation it's square root of yeah this is if you were to sum this up for different ends and m primes sorry I think this is right because this is right this is a random walk in this complex plane and so right so we're not averaging over all choices of theta it's like we've picked some Theta's some Theta as a function of n and m and so we're you know this is some definite finite number of terms we're adding up and so the magnitude will be square root b it could have some phase it and so different for different in and in prime it would have a different phase and so if you added those all up you get more cancellations but for some but you know in equals seven in prime equals 13 this is going to be something that has a magnitude of order order square root b and so this is the answer you get when you divide it but yeah so I believe this is right but I'd be happy to talk about it more not some increase for the great set of lectures