BSS2023, Sagar Vijay, Random Quantum Circuits II, July 5

thank you for that song Let's see yeah it is green light yes all right excellent okay thanks but I think everyone was like yeah nothing oh yeah all right thank you again oh yeah thank you yeah do you mind just uh testing the yeah okay great thanks yeah out okay um good morning everybody um and we're also fresh start on Wednesday with your father's second on random circuits okay uh great so um so in in the last lecture we really assembled a lot of the uh mathematical tools uh using you know Computing the tar average of uh you know random unitary Gates and things like this uh and from here on out we're just going to be using some of those tools uh to understand the behavior of um you know certain Quantum correlations and how they change as we continue to evolve a Quantum anybody system specifically quantities like uh quantum entanglement that as we heard yesterday are also intimately related to the difficulty in storing a mini body weight function uh other measures um that uh you know quantify operator growth which we also heard about yesterday and uh maybe I can wait just a few seconds for you oh really I see foreign [Laughter] okay so everyone seems a little bit more settled um okay so just uh so I'm going to study the brickwork car random circuit today uh and uh present you know some of the behavior of quantum correlations like entanglement in the heart random circuit when we start from a product initial state which is uh initially unentangled so the picture that I have in mind uh is that I have uh some unitary Evolution uh I'll call it U of T starting from a product initial State uh and here I'm imagining now that I have uh n cudits uh so a q did is just a generalization of a qubit where on each site now I have a Q level system instead of a two level system uh and we'll work with uh cudits with uh so there's Q States uh system uh on each uh lattice site so uh I have these states here are these sites here hosting this these um Q State systems and then uh the brickwork car random circuit that I have in mind has the following geometry so I use my my River Stone or whatever David was calling them yesterday uh you have non-brick but aesthetically pleasing uh you know uh brick so so what I'm imagining so this is a part of an infinite system I'm only drawing part of it so what I'm imagining is that a single time step of this brickwork our random circuit is going to involve these two now cudit unitary Gates uh and each site is one of these cudits uh and uh just to avoid the confusion that arose last time I'll call these unitary Gates I'll label them V1 V2 V3 et cetera so when I say when I use the notation V I'm referring to uh a two cuded gate and and U of T is just referring to the composite Evolution the quantum circuit that's obtained by repeating this architecture over and over again now when I say repeated over and over again it's not the same two cued at unitaries they're um so each uh each VI is independently chosen from uh the uh um from the heart measure over the unitary group over uh so Q squared by Q squared unitaries right since they're two cubic cudit Gates so uh so this is what I will refer to as the brickwork car random circuit U of T is obtained when I uh when I repeat this two-layer structure T times so so here I'm imagining that you know this has been repeated T times to obtain the circuit uh um U of T okay and we want to quantify things like the growth of entanglement um on the approach to some kind of equilibrium in the absence of any conserved quantities uh and what I want to sort of the big sort of takeaway from today which I'll just write in the corner here is that um the sort of statistical mechanics of uh a pairing field so this is a field which I'll write as Sigma r uh T which takes values and and say the permutation group uh I won't specify what n actually is the statistical mechanics of an emergent pairing field that comes about when we do the har average in the calculation of quantities of Interest like entanglement is really what governs the universal structures uh of um uh Universal let me say Universal structures uh in the random circuit uh uh and this is really um uh you know for example uh the growth of entanglement um um say starting from a product initial state so if I imagined that I had an initially disentangled State you might imagine that quantum entanglement will increase as I continue to apply these two cubic Gates and how to understand this and other measures will require studying the statistical mechanics of a field that lives in this sort of space-time one plus one-dimensional space-time uh and there are some interesting symmetries that this statistical mechanics will have that are really a consequence uh of the you know par average and of of this random circuit structure that I'll discuss okay so this is going to be the big kind of takeaway today uh and um when we talk about entanglement we've already heard about how to quantify it uh so then you know I'll um just uh as a recap um just say that so if we want to quantify bipartite entanglement um so just to review from from Frank's lecture also uh you know we were interested in discussing um uh you know interested in in taking a bipartition of say some pure state so um I might have a wave function for some cudit chain um sorry to use that color there for a Cuda chain and I might have some subsystem which I'll call a and we know that given a pure state of this cudit chain which I've kind of written in the Continuum uh I um am interested maybe in in understanding if I want to understand correlations of all local operators within a I'd be interested and say you know that's all encoded in the reduced density Matrix on a right so rho of a which is the reduced density Matrix on a is obtained by tracing out the complement uh where everything outside of the red region is the complement of a right and um uh a measure of how entangled a is with with a bar is given by the Von Neumann entanglement entropy so actually before I say that very quickly um so row a obviously has unit Trace um and another feature of uh and is her Mission and has non-negative eigenvalues uh and the um uh so the so the Von Neumann entropy uh which I'll call S of row of a is uh defined as minus Trace row blog row okay so this is uh manifestly a uh let me call this equation one just a few comments just to add to what Frank was saying last time so I I'll call this um this is a manifestly basis independent measure of quantum correlations within a in that I can do a unitary transformation on Rove a and it doesn't affect the volumen entropy obviously um right uh but uh one other property of the monument entropy I want to just emphasize is that uh the monomial entropy is always you know it's not negative uh but it's maximal say if this is a cudit chain uh it takes it takes its maximum value to be n a Times log of Q where n a uh uh is the number of cudits within this a subsystem okay so that's the maximum value that the one element entropy can take and this maximum value is specifically attained right so uh this maximum value is uh obtained if and only if the reduced density Matrix itself is proportional to the identity on a right so it's well it's really the identity on a divided by the Hilbert space Dimension to guarantee that it has unit Trace but it attains this maximum value if and only if you have this infinite temperature State okay all right yeah so so that's what I would call maximally entangled yeah um right and volume one entanglement would be any scaling of the entropy with sub-region size that grows with the number of cudits uh not necessarily just the maximum value right okay so uh just to add to what Frank was saying so there are also reny entanglement entropies that we might care about the reason we care about the vulnerable and entanglement entropy generally is because uh it enjoys certain properties that allow us to give it an information theoretic interpretation so I'm not going to discuss that uh so the volumen entropy itself satisfies sub-additivity and strong sub additivity uh which the Randy entropies don't satisfy uh so that that's important to to understand for instance how the Von Neumann entropy is related to the amount of say classical information you can transmit if I were transmitting pure States from some Ensemble to to someone else uh or um like Frank was saying if you know um yesterday the Renny entropies for n less than one uh which uh I always thought it also included one um uh are important in in understanding whether uh you know there's an efficient MPS representation of a one-dimensional uh Cuda chain right or a Quantum anybody State okay um so let me just also add that uh and introduce one final thing which is uh so so for calculational convenience we may consider and also since these are also valid measures of of uh um entanglement insofar as they are quantifying how far away a density Matrix is from being pure for calculational convenience we may consider the Renny entropies so these are defined as I'll use the superscript N uh of row of a which is defined as one or one minus n log Trace row a H to the nth power okay um and uh the Von Neumann entropy is formally the limit and N goes to one of the Rainy entropies right um uh and also there's this other quantity which is in the literature it's sometimes referred to as the Hartley entropy which is formerly s uh s of zero so uh this is just the log of the rank of Rove okay um I don't um maybe I can discuss that offline it's not going to be directly related to what I'm going to talk about there's a well-known result in Quantum information that many of you might be familiar with known as Schumacher compression which is like Shannon's Source coding theorem except if instead of sending classical bits you were sending Quantum States you can ask if I'm sending you um you know any Quantum States chose from some Ensemble chosen from some Ensemble uh then you know how many sort of classical words am I able to transmit to you by sending those Quantum States and that is Quantified by the entanglement entropy of The Ensemble that I'm drawing those States from uh so I can I can come up with a reduced density Matrix that's like The Ensemble of those States and then um so so that's that's actually it's it's much more precise than that uh literally the number of messages I can send distinct messages is is in some Optimum way exactly given by the monument entropy but I'm not going to be using that fact okay okay so the Hartley entropy is the sort of end going to zero limit of this quantity I'll use that shortly and finally uh um so the one other um point to S of M plus one uh uh the M plus one through any entropy it's always less than or equal to SM and this just has to do with the fact that row a has non-negative eigenvalues that are all less than one right uh so um so this is always true uh for all M okay so the the Hartley entropy is the largest of of the series of running entropy so we're writing down okay okay so so this is just a overview on how we quantify bipartite entanglement um actually maybe if there are other questions uh so far this is more more or less background yeah yes uh yeah that's right yeah it's also true for for the volume entropy yeah right oh yeah sure no that's great yeah I will um yeah I'll try to I'll try to write 1.5 to two times larger yeah but but actually that that is very helpful like please yell if you if you aren't able to see my handwriting um okay Okay so okay so now let's go back to studying the entanglement of the brickwork circuit uh entanglement growth in the brickwork circuit starting from a product State and uh I'm going to first be interested in uh this quantity which is known as the Purity uh uh of a uh subsystem uh which is given at time T in The brickwork Circuit by this this quantity so this is formally just the exponential of the second Randy entropy of rho a of T right so this is a purity uh um and uh the reason it's well reasons known as I mean all of the monument entropies I guess measure in some sense the purity of roebay but um rho a squared is equal to itself if and only if Rove is pure right which means that row a is really just described by a single wave function so I'm going to be interested in this quantity and we'll use this to gain a kind of foothold to understand how some of the other running entropies behave in the brickwork circuit okay so um all right so let's uh so let me use this tensor Network kind of notation so I have U of T right which was this you know Quantum circuit that involves T applications of uh you know odd Bond and even Bond two q that unitary Gates so I I have this kind of tensor Network representation of you and what I'm going to do now uh is I'm going to just group the legs of this of this sensor into two disjoint uh subsets right uh so what I want to do is um I Want U of T uh so so the the white leg that's shown over here is uh going to be um labeling the Hilbert space uh of a region which I'll call a and this is a Hilbert space uh which is the complement which is a bar and uh these are just contiguous regions so you can really imagine that a is like if I go back here a is like all the sites to the left of you know where my hand is and a bar is everything to the right okay so these are contiguous regions and this bond is like you know the bond Dimension is the Hilbert space dimension of that sub-region uh and um so so this is uh so this is uh uh so this is the brickwork circuit uh uh at time t right and uh um what I want to do is to again use this graphical calculus to represent what what the purity of a subsystem is going to look like at time T right uh and there's um there's something simple that I can I can say which is that uh um actually let me move to a different board in the interest of keeping my handwriting one to whatever two times larger um uh so uh so rho a of T right is so row a of T um if I start with so I'm imagining starting with a product initial state so I'm going to trace over a bar and I'm going to apply this Quantum circuit uh to a state which I'll denote uh um as follows so this is so I'm going to start with a a product State on the N cudits which I'll call the you know zero State we'll see eventually that it doesn't matter what state I start with because U is going to be given by a product of heart random unitaries and it doesn't really care about the basis of the initial product State at least as far as you know calculating the growth of entanglement uh in some average sense so so I have no this so this is my uh reduced density Matrix right and uh graphically how would I represent this well um so I have rho a t so this just has carries the legs of the a Hilbert space and this is equal right to uh okay I'll switch back to my River Stone um Okay so so this is equal to the following uh graphical you know representation so I have uh the zero State on uh on n a qubits and then the zero State on n a bar qubits I'm just going to my handwriting is getting small already so I'm just going to represent it in this way uh and similarly the zero State here and the zero State here right so that's you and you dagger evolving this initial State uh and I have to do is take the trace over the a bar subsystem and that Trace just means that I can track this Index right that's my graphical representation of the density Matrix okay are there questions about this so it's convenient to actually uh sort of you know uh move things around so that instead of talking about an operator we actually talk about row aft as a wave function on a double tilbert space uh so what we can do is um so it's possible to represent um so row a of T no we we represent like this uh but I could just bend the legs backwards and this was I think discussed uh yesterday as well right um so I have you know I and J maybe labeling some Matrix elements uh and I could just bend the likes of row a of T backwards so that this looks like a wave function in a Hilbert space that's uh twice as big right um uh or that is there twice as many cudits in it so uh so formally what I would say is that you know if I had to write this out this is an operator so this is row a t i j uh and uh um say I they uh and here you know that's that's what this this quantity would be uh whereas this quantity here uh is really uh a wave function and unnormalized wave function which is row a of t i j I tensor J right so this is uh the quantum information literature this is sometimes known as the Troy isomorphism but uh okay so the important thing is that while I am writing this is a wave function an important thing is that while I am writing the operator now as the wave function in the doubled Hilbert space it's not normalized right um yeah yeah that question sorry yeah let me repeat the question so Anisha was asking uh so row a is a you know non-negative uh hermitian operator right so is there something special about this wave function uh that it's a structure that it needs to have given that row a has has that structure um yeah so so hermeticity imposes so hermiticity of the operator actually is a very simple is very simple to think about it's a it's a symmetry um uh and that that much is is clear it's kind of like uh um it's kind of like time reversal um or it's sorry not it's not like time reversal I guess it's like swapping these two copies followed by complex conjugation so it's not it's like anti-unitary um but the positivity I don't think there's an easy interpretation uh of at least I know of um uh not only is it um not only is it positivity but okay this is getting off track but where I evolve row a into another density Matrix I need its Trace to be equal to one always and that's the statement that this wave function has unit overlap with like the Bell State always right which is kind of an awkward statement to to be making uh so yeah I'll leave it at that yes so when you already wrote you know in this right panel you've already done that right lucky you wrote zero zero you wrote two cats rather than the uh oh no is that already you mean this year on this Arrow no that's zero zero that's fine yeah sure well I I yeah but what I'm wondering is does J now evolve should I think of it now it's evolving according to you now you dab it yeah so so Jay now evolves uh you know um Jay now evolves according to you star I guess yeah right that's the way you would write it um that's why you were considering yeah right so here I've written it in such a way that this is just the straightforward matrix multiplication of you and you dagger that would happen where I to contract these indices um and yeah I've just sort of copied what I wrote here but yes if I fold the leg backwards yeah oh sorry yes I should have yeah I should write this as a I guess I would write it like this yeah sorry about that yeah are there any other questions about the notation right it's not a wave function only very special way functions can be it can be density matrices once I kind of rewrite them in this way yeah or hermitian for that matter right um yeah okay this is just a convenient change of notation uh because we'll be working with this double filbert space for you know some of you can probably already see how this is going to relate to what we discussed last time um okay so now let's let's just look at the Purity so trace of rho a t squared right um okay so uh actually before I do that very quickly let me just say now that uh now this is a point that Leo just made so now row a you know if I were to write it like this uh right in a as a wave function in a double Dilbert space and uh this would be written as U of T uh have uh so let me kind of write it in this fall anyway you star of teeth uh and I have uh these legs acting on the Zero State uh I have these a subsystem legs that are kind of poking out uh and um I also have this um sorry this that's happening right so the the trace over the complement is now you know represented in this way and and by sort of flipping you dagger uh up I've transposed the Matrix until I would write it as U star right okay so this is how I would write Rove a as a wave function on a double silver face um and this is all just to say that uh so so um so trace of rho a of t squared is now uh an amplitude uh in a uh replicated uh or let's say multiplicated uh over space yeah it's well it's not that it will be more than doubled uh so um so so formally right uh oh how would I write this um graphically well I would take two copies of this guy and then do the matrix multiplication and the matrix multiplication involves the contraction of these legs right uh but this is like the you know the forward leg and this is the backward leg of row uh so this backward leg has to contract with kind of the forward leg of the second copy Etc so so fine finally what I would get is is something that looks like this uh so I'm gonna write it in a form where um the unitaries are kind of they look like they're stacked on top of each other and this is just uh um it will become clear maybe later why why so it you know you imagine that this uh in this brickwork this um tensor Network representation I'm just drawing them one on top of the other um so uh okay so we started off with uh two copies of the density Matrix so what does that mean that means I have two copies of a state that looks like this uh so so just so this is this zero state okay so we started with with two copies of the density Matrix and now there's this leg that's poking out that we we haven't contracted right uh so and I I haven't drawn the other the other leg uh for for each of the matrices they're kind of hidden uh but you can see that they are contracted with the product State and now what do I do well I do matrix multiplication of row a and row a and then I take the trace right so the matrix multiplication corresponds to Contracting these two legs right so this is my representation of rho a squared and then the trace is just Contracting the remaining index so so this is just given by something that looks like this okay is this I guess something everyone can see or is there any confusion about this okay so when I say row a squared is now an amplitude in a replicated Hilbert space hopefully it's clear so this is you you star U and U star of T right so it's as if I've taken four copies of my system evolved it according to this tensor product of unitaries and then done a contraction but now with an unnormalized state right that unnormalized state is like some very particular Bell State across these multiple copies okay great um so this is the structure I mean we haven't used heart Randomness or anything yet right so this is just the structure of how the Purity looks if I write it as an amplitude in a doubled Hilbert space for any Quantum circuit or any you know Evolution for that matter U of T could even be hamiltonian Evolution so every other question there okay no no yeah I just want to make sure everything there are no typos or anything um okay so now in the brickwork circuit uh let's um uh calculate uh the expectation value I should write larger maybe the expectation value uh for all of these VI unitaries independently chosen from the heart measure over uq squared of trace of rho a of t squared where a is just some sub-region okay uh and um so this is going to be my average our average Purity for a subsystem right uh and in the process of doing this calculation I'm doing this uh not just to show how you could might do calculations using random circuits but more importantly to show you that there's some Universal structures that emerge in this calculation that we might want to look for in a more general setting that's not just our random unitaries right uh okay so let's calculate this quantity uh and each of these two qubit cudit now unitary Gates is independently chosen from the heart measure so so I have you know trace rho a squared which looks something like this I mean it looks exactly like this um and if you imagine that I stack u u star U and U star exactly on top of each other which I haven't done to sort of a caricature of that then what I would see really is that uh at each space time Point uh you know at each space time point I'd see something like you know V1 V1 star E1 V1 Star right right so this is you know this V1 star comes from this copy V1 comes from this copy V1 star V1 Etc right so it's as if you know I've I can kind of Stack these on top of each other and do the heart average independently for each of those unitaries right and then we can just sort of Transport the Machinery of doing the heart average for a tensor product of unitaries that we already introduced last time right so we'll be calculating quantities so so um so we'll really need to calculate this quantity which I called P2 last time uh which is defined as the expectation value uh for V which is uh you know a unitary on Q squared a q squared by Q squared unitary right so what if we do that then whatever P2 is we just kind of substitute it in into the space-time structure and we obtain a tensor Network that if we contract it we get the Purity right does that make sense this is all a visual calculus I think it's much easier than writing out everything in with indices and stuff so hopefully it makes sense okay um so um so let me uh you know again write this uh P2 let me write it as a as a tensor now okay so uh I'm writing it as a tensor uh but uh there's I don't want to use too many colors but P2 now acts on a Hilbert space which is four times the size right uh you know it involves eight qubit cudets instead of just two right so it acts on that Hilbert space of eight units right so uh each of these legs is now representing a four cued at Hilbert space that that the the unitary or sorry the our average quantity is acting up right so I'll say that this has uh this has Bond dimension of Q to the fourth right so that's for each of those legs right uh and this is of course equal to this expectation value uh of uh the so um where each of these legs has Bond dimension as Bond Dimension Q right so this is this is what I I want to calculate um and last time we already you know we already figured out and at the very end of the last lecture uh we already wrote down what uh what kind of projection operator P2 is going to be right um yes all different pairings across the space that I've learned so they can't uh they won't and the reason is that each unitary in the circuit is independently chosen from the harm measure so they're independent random variables that I'm doing in average over yeah exactly so so they're uncorrelated in different points in space time right now that that's an important point to Leo with asking just to reiterate the question why there aren't like you know other tensor products of unitaries or why this doesn't pair different points in space-time and the reason is that the the circuit has individual two-cute unitaries that are independently chosen from the heart measure but that also suggests that you know were you to have a 4K version of this circuit where you repeat the same say hard random unitary um at each point uh in space at different points in time then you will have these pairings that go across you know space time and we'll actually get to that in the next lecture uh um okay okay great so we know that uh so P2 uh is a uh projector right and uh it's a projector onto the Hilbert space that's spanned by uh kind of symmetric states that live within this uh you know whatever queue to the uh well whatever eight cued at Hilbert space right uh so um so it's a projection operator and the simplest way to write that projection operator uh is you know in terms of these states um so I'm now referring to if I refer to these two indices as some Composite Index which I'll call say Alpha I'll use maybe Greek indices Alpha Beta Gamma Delta then P2 is a projector onto uh the states uh this which we Define last time which is now defined uh as uh alpha alpha beta beta where Alpha and beta go from one to Q squared uh and then this down state uh which is defined uh as Alpha Beta beta alpha alpha beta let's go from one to Q squared okay so um this just comes straightforwardly from doing the higher average as we did in the first lecture uh and this projection operator though is not like the most straightforward thing to write just because these two states are not orthogonal right they're unnormalized and they're not orthogonal right we we discussed this last time so so uh yes it's a real it's actually a projector yes yeah so so P2 it's not just up up you know bra cat plus down down right which is what you would write if these were normalized and or orthonormal states right um so so how do we write this well we we can easily and I leave this as an exercise you can easily figure out you know what states within the Subspace are orthogonal and then just write P2 as a projector using those States right that are apply that give you an orthonormal basis within the Hilbert space spanned by these two wave functions okay I'll give you the answer um so the answer is just that uh so um so the exercise is to show that uh T2 is equal to the following uh that's one over Q to the fourth minus one okay so this is this is the answer that you would get where you do this calculation and this follows very straightforwardly by demanding that P2 is a projector onto this Hilbert space and using just the inner products of these wave functions okay with each other okay yeah onto the Hilbert space span by these two states delivered these are paired wave functions in this complicated well it's a projector um well it's sorry yeah P2 squared is P2 so it's weight one right it's a projector so yeah um right okay so this is an exercise uh and it's it I think it's fairly straightforward to do it involves you know some calculations which um and I'm more than happy to discuss it if you guys have any questions about it but um uh and um here I actually just want to make a comment uh maybe it's a little bit out of place but just as an aside because this question came up last time and I don't think I gave it a very satisfactory answer to it uh which is um we discussed so we discussed so just as an aside uh so we discussed last time that P1 uh say um which is uh let me um write it as uh this our average um where V is so we wrote it down like this so V is a unitary uh d by D unitary this uh we said was equal to one over D summation over i j going from one to d uh of I I J J right so it was a projector onto this paired Subspace uh of uh this D Squared dimensional Hilbert space right and uh so there was a question last time that uh Y is P1 uh why is uh this operator right uh why is it not equal to just this right okay uh I forget who asked this question but if you if you don't remember the question it's fine but if you do remember the question then I'll give you maybe a more satisfactory answer so the reason this this can't be true is because we demand uh that uh whatever whenever we do the heart average we demand that the quantity we obtain is invariant under conjugation by any unitary in UD right so were I to multiply P1 by some you know conjugate and say this is uh um uh like right multiplication of V by W so then this P1 gets multiplied by W and W star well that's equal to this quantity which is clearly not equal to itself right so under conjugation I don't get the same quantity back right so it's clearly not the right answer but you can check that the basis States or not basis States but the the states that whose span gives me the Hilbert space that we're projecting onto um allow you to sort of manifestly satisfy this sort of invariance condition that is if I start with paired States well it's clear that you know W contracts with W star to just give you the identity uh and that prop this property is satisfied for the correct expression for P1 okay great yeah that's because it was that was the wave function app that won at that place was it that he wanted so yeah so so P1 so P1 in this notation is one over d times this okay uh yeah so so if I If I multiply this by W and W star that's a contraction and that just gives me so w w star contracted like this is equal to this right because that's just a w w dagger right uh so so that leaves P1 invariant yeah this time I'm not conjugating this time I'm using the right and variance of the harm measure or something so that's actually turns out to be equivalent to invariance under conjugation but that's not something I talked about uh well if I conjugate it by V so then I would or by W so I would multiply by w w dagger then yes that's true but the par measure has a stronger condition which is I can multiply it on the left and the right so maybe conjugation isn't the word I should use yeah together yeah that's a great question so what I've done is uh I've stacked you you star you you start kind of on top of each other and I'm I'm I'm cup I'm grouping this left Bond on each of these unitaries into one Bond on P2 okay so so the you know um you can identify this sort of same cudit on each copy of the Hilbert space and it's uh the leg that acts on that cudit on each of those copies have grouped into one like okay so this is an aside um this wasn't a point of confusion for you then I would ignore it um okay so now uh we do this for average right and and we do the high average independently for each of these two queued unitaries so each space time point we then end up with one of these projection operators P2 right so uh but um but before I get there I just want to emphasize something kind of uh you know which I think is sort of fundamental and interesting which is that uh um so this is not to do with the aside but let's say we're given this expression for P2 um and note that uh P2 is invariant uh under this transformation where I take the upstate and I exchange it with the down statement right so that weight is completely invariant under that transformation right so this is like an say a sort of a Z2 symmetry and whenever we see a symmetry we should probably ask where it comes from uh this icing symmetry is um naturally implemented uh by the following operation so you know if I just take um if I just take this state here right and I swap the second and third uh second and fourth copies rather with each other I get the down state right so I can I'll write that as uh you know so note that so if I write a swap operator and the swap operator acts in this you know replicated Hilbert space so this is like uh acting in this Hilbert space of eight cudits that P2 is acting on so if I if I write down the swap say between the um uh the second and the fourth uh kind of indices here right so I'll call it swap two four of up that gives me down okay so uh sort of in the diagrammatic notation what I would write is the upstate looks something like this uh I've um so I've the up state looks like this where this is the alpha index and this is the beta Index right uh so now this is this Composite index that labels the Q squared dimensional filbert space uh that um for each of these guys yes yeah so so last time we were doing the har average uh over um the unitary group U of D now D is Q squared because these are two cudit unitaries right and they're chosen uniformly over the unitary group uh you have q squared but they act you know they um I want to retain this index structure they act on two cudits right each of these unitaries um so I'm just saying that the you know if we um if we do the har average over so this is over u q squared then what we know is that these states have to be perfectly paired with these states or and these states have to be or sorry these two states have to be paired and these two have to be paired or the other pairing has to happen right uh and that requires that we have a you know we contract these indices in some particular way so for instance I can contract this index with this one this one with this one uh that's the same as you know fixing this composite Dilbert space to be in the same state as this one so it was just easier to notationally write it this way um if it's confusing to you you can also replace Alpha by you know separate indices that label the two-dimensional Hobart spaces and it's the same thing right um you'd get the same sort of pairing structure yeah um these are just some orthonormal states in this Q squared dimensional overt space that you know defines you know uh that's you know spans the space of states of two cudits right maybe another way of writing this is uh so if I were to write you know last time we showed that uh you know um so let's say I have uh V um so let's say I have this far average that I'm doing over uh over these you know four unitaries right and if this is a d dimensional hybrid space so Bond Dimension D runs to this leg then we know that the this is a projector onto the states which which pair these legs together right so there's this state uh or the other state which looks like this right uh and those states are naturally written by you know writing down kind of bell States using the D dimensional Hilbert space but you know I could also here D is Q squared uh and that's kind of where it all comes in right uh so D is I've retained this two-like structure just because they're too cute at unitaries but if it's confusing to you can just group them immediately into one leg and just do the power average yeah welcome yeah the swap of one three also we'll get to that in a moment that uh that technically is another symmetry but in this uh but for P2 it's actually the same uh as this this symmetry operation okay so when I when I say swap two four what that does is it takes the second copy and it swaps it with the fourth copy right and so this is exactly equal to this right so this is swap of the upstate and this is this is precisely the down stake okay okay so what is the Symmetry uh why do we have it um it's very simple so uh the reason we have this symmetry is because uh so um so let me just rewrite this expression again um okay so the the swap operator right acts on you know these sort of composite legs so it acts on uh the second leg here right and the fourth leg here right that's how I defined it um and basically this this swap symmetry in P2 is just saying something very very trivial which is that uh if you do this R average um it doesn't matter if you know if V Star is here or here like you can exchange these two copies of V Star and the Heart average is left invariant right so if I conjugate P2 by this swath operation that's like exchanging these two legs you know this copy of V Star with this copy of V Star right so that leaves the entire quantity invariant does that make sense so manifestly whatever we obtain at the end has also been variant under that symmetry which is the reason why up down can be exchanged right so that's the origin of the sizing Symmetry and it's it's a very intuitive symmetry right like if I neglect the boundary conditions on Trace squared I'll call them boundary conditions but they're like the contractions that this replicated operator has to undergo at the edges if I neglect that then this is actually a symmetry of this composite Quantum circuit I can exchange U star the two different copies are U the two different copies of it and um it's you know exactly the same right uh it's exactly the same for every you uh not just after you do the r average yeah was that your question yeah okay like the picture seems like significant you can fix it yeah uh no even if I even before I do the har average this tensor product is invariant under exchanging V Star with itself right for a given choice of I mean it's it's exactly the same expression right I mean sort of trivially so right I I mean it's so it's almost so trivial you might not see it like kind of a thing right where you're exchanging V Star with v starring you get exactly the same thing right um but as an operator that's non-trivial that operator is performing a swap of these two Hilbert spaces right yeah no so so sorry remember that the swap acts like this on a state when it acts on this operator it conjugates it right so I'm acting with swap I'm conjugating P2 by the swap operator I'm exchanging up with down for both the cats and the bras in P2 right so that's to conjugate P2 so so the statement really is that Swap this swap Operator 2 4 P2 swap two four is equal to P2 and this is trivially true for every unitary that I choose that appears in this our average rate okay yeah foreign I'll get to that yeah that's a really good question yeah I'm gonna get to that eventually so so uh I think the question that was asked was basically this Purity circuit is actually invariant even if you include the boundary conditions under this swap operation applied globally and the reason for that which I'll discuss in a moment is that that converts trace of rho a squared to trace of rho a bar squared and in a pure State those two are the same uh but in general the boundary conditions can break this this swap symmetry um right so for instance If This Were coupled to some external environment and then Trace Roy squared and Roy bar squared aren't the same then then obviously this is not a global symmetry it's broken at the boundaries but it's a local symmetry there's a meaningful sense in which in the bulk of this unitary circuit I can do this Swap and it's in you know everything is invariant right okay okay so let me move to to that board um so this this uh swap symmetry is more General when we have more you know copies of the heart random unitaries but it's uh we'll get to that later it's not important for now but let's now look at just this uh our average of Trace row a of t squared right which we uh had written down uh earlier on the board uh or on the board that's kind of hidden right now so this uh we're right to write this um as a tensor Network I would now replace at every so I would get a two-dimensional tensor Network where at every space time Point uh I replace the copies of V V Star V and V Star by just this projector P2 right and that projector acts on this replicated Hilbert space so what this looks like is really like its own kind of brickwork circuit which isn't unitary uh but it's brickwork because of the original architecture uh no um etc etc and for now I'm neglecting the boundary conditions this is just the way that this circuit will look in the bulk and at the boundaries it'll be contracted in some way right yes I mean so yeah exactly that's exactly right but now each line of those is four lines yeah I was just going to get to that exactly yeah yeah so so each line of these as I've already fixed the notation here whenever I write P2 with these two legs each line here is is has Bond dimension Q to the fourth right because there's there's really four cudits that are being acted upon by this one Composite Index right and that that labels the action of the V V Star V and V Star on the same cuded on the different space-time different copies of the Dynamics right um but we also know now that there's a an easy way to do this contraction uh which is okay we have an expression for P2 which we can just substitute in and when we do the contraction we obtain what looks like a complicated sum uh of these states uh their overlaps have to enter when I contract the different legs of P2 et cetera et cetera but um you know the different terms in this tensor Network contraction can be organized in a very simple way there are two spin States right per P2 that can arise two eising spins and those two ising spin States label whether I've chosen the upstate for this cat and the down State for the bra or you know whatever spin State I choose right and then there's some contraction that has to happen across the legs so sort of schematically I can write this right as um so every P2 say you know this P2 in the middle gets replaced Now by two eising spin degrees of freedom so Sigma and Tau uh are can be up or down and that labels my choice of whether which term I choose say in this in this tensor right when I when I expand out this complicated tensor Network that's being contracted in in a sum of of these amplitudes that are you know and overlaps right so this is a I'm being very schematic here this on the left hand side this expression is precise I'm Doing intensor Network contraction here what I'm saying is I pick Sigma and Tau and whenever I pick a sigma and Tau to be up or down there's a corresponding weight that shows up in P2 that I have to choose right and then there's a contraction that has to happen across these likes right the contraction has to do basically with the fact that you can show the sigma of Tau and Tau the overlap okay uh yes basically the time evolution of row a square something like this right we have so many copies of two oh because remember we're doing this average for every gate independently in the quantum circuit this is for every so remember V my notation is that V is just a two cubed gate and whenever I use capital u I'm referring to the quantum circuit this is for yeah this is one space time point and it's one gate that was independently drawn from the higher ensemble yeah okay so this this overlap you can um uh you can compute and uh this would be Q to the fourth if uh um Sigma is equal to Tau and uh Q squared of Sigma is not equal to Tau and that's something you can determine yourself right so the weights here are ising symmetric the overlaps are also eising symmetric so this looks like a partition sum for some kind of izing model where we have an ising symmetry in the bulk right which just has to do with the swap symmetry that I wrote down over there right the boundary conditions might break that symmetry but the boundary conditions you know can't really change even if they break the Symmetry they can't change the bulk physics so much if the spark of the system yeah yes oh sorry yes um on each leg they should be Q squared and Q yeah yeah that's because I've broken them up into two um okay let me let me just say let me rewrite this um the up down States as I wrote them there have overlap Q to the 14 Q squared but um that's technical Point let me just let me just say that there's some Oak left you know let me call this Sigma Prime there's some between these two spins that depends on their relative State uh not on you know just whether or not they're equal or different and another weight here that depends on P2 that I've written over there yes okay so you can ask me about that later so this looks like some eising partition function and uh the icing symmetry just comes from this bulk swap symmetry right um and here we're just going to argue now by thinking about the boundary conditions that the Purity is really probing the you know bulk ordering of this icing of these izing spins uh so that's really the entanglement entropy and its growth in this circuit is going to be related to the free energy cost of a domain wall in the ordered phase of an ising spin system okay yeah yeah which one up yeah so um expertise yeah so so that's a great question when I give you a tensor Network that looks like this naturally you might say you can write down eising spin states that live on bonds and then this looks like a transfer Matrix that you're evaluating right um but I'm doing the the contraction in a slightly different way um the reason for that is because uh if the if the icing spin States live on bonds it's a technical reason the expanding the boundary conditions in terms of those States actually becomes quite awkward uh it's it's for a technical reason that we do this um but uh yes um so this is this is different but you know uh it is you can convince yourself that it's really kind of the same thing um so here's Sigma and how label the states you're right they they label or sorry the states that we're projecting onto right um so um yeah yeah quantities of partition functions with this.net or what is that quantity yeah I haven't specified the boundary conditions so yes it is a partition sum with some specific boundary conditions uh in this sizing application function yes it is a partition function with some specific boundary conditions in this sizing magnet yeah the boundary conditions are fixed by the fact that we're calculating the parity right maybe that is clear to everyone already here I've just drawn what the bulk unitary circuit looks like when I replicated and do the har average but it's the boundary conditions that tell me that it's the Purity that I'm calculating right the way you connect the big use right yeah the way I take the trace or you know what initial state I'm acting on hasn't even appeared yet and that appears at some boundary right kind of isomastic um yeah it is uh yeah exactly right um I think you need to make sure like here if I specify the states on every Bond it's like unambiguous what the tensor Network contraction is doing but here I'm specifying what this operator looks like and then I still need to do a contraction which I haven't told you about yet right yeah oh okay uh okay that might be but I'll let's discuss that offline in the interest of moving forward yeah a diagonal yeah that's a great question why wouldn't you work in that basis yeah the reason I don't want to work in that basis is because the Symmetry is not manifest in that basis the Symmetry under exchanging U star the two copies is really manifest if I work in the up down basis and then there's a lot of intuition from the statistical mechanics of systems with these symmetries that I can import to understand this physics right oh no no it's uh it's not manifest because it's not going to exchange the two states that are orthogonal yeah but it'll be some com like symmetry action will just be much more complicated in that language right um but I'm just saying locally if I label these states by the states which diagonalize P2 the Symmetry action is not going to be as Manifest right here that symmetry action is Sigma goes to minus Sigma and there the Symmetry action will be much more complicated it may relate the two states together in some some confusing way and this is much much clearer yeah yeah that's a technical point but yeah yeah because yeah it's basically because uh when I do um so uh when I expand out this sensor Network maybe this is the easiest way that I would think about it I can choose uh I can basically choose different terms that appear in the expansion of P2 that I wrote down right so P2 has terms that you know um you know up up type terms up down down Etc um and one way to organize this summation is to just say okay let me replace P2 with two spin States and let me sum over all of the states so here's a I mean so I'm going to sum over all the states of the spins whenever Sigma and Tau appear here I'm going to choose the weight that appears in P2 corresponding to whether you know if Sigma corresponds to like the kept state in P2 Tau corresponds to the bra and so if I choose Sigma to be up uh and Tau to to be to be down that to me is equivalent to choosing the sort of up term in the projector and I would get a weight which is 1 over 2 to the four minus one right which is what I had written down earlier uh now the tensors are also contracted which is something we haven't discussed right these These tensors are being multiplied together just because the quantum circuit involves products of the different two-site unitary Gates that contraction just involves the overlap of these different paired States and that overlap is something you can easily compute it's just they're not orthogonal like we mentioned and they're not normalized States so it's a non-trivial overlap yeah does that help to clarify I can maybe I can discuss the um well um yeah does that help to clarify what I'm doing it's like each term that appears in expansion of P2 that I wrote down is denoted on the right hand side by some configuration of spins to which I assign a weight right yeah and that weight is determined not just by P2 but by some complicated overlap now the the details of what those overlaps are I don't want to fixate on I just want to fixate on the symmetry which is this is an ising magnet right it has a globalizing symmetry uh and we're going to probe that by looking at the entanglement structure of this evolving state are there any other questions I think Leo yeah sure but can you summarize like this is a nice partition of a certain kind of IC model yeah but with the two types of exchangers uh those come from the P2 and then there are these overlaps you know up without two types of Weights you mean well exchanges oh oh yeah yeah exactly yes yeah well is that how I should think of it yeah I'll get to that in a second yeah uh that's a that may be a technical detail that I don't want to cover because it'll it'll take a while to go through that calculation but yes really we have a weight assigned to spins on bonds that's it right uh so given a spin configuration there's a weight that I will assign independently to the spins on buns you can think of that as some kind of exchange type interaction it is pairwise there's some more complicated structure that I I don't want to get into I'll get into in a moment but um we can discuss more offline like I'm being a little bit I'm being a little bit facetious by just saying that this is an ising magnet because one thing that we haven't gone over is the fact that the Wine Garden function actually has negative weights so there's some negative weights that show up in this partition sum but it can be reorganized into something that is manifestly a classicalizing magnet so it has a boltzmann weight associated with interactions between the spins yeah and some other representation yeah um this kind of thing happens in like percolation and stuff too right where it's really yeah yeah exactly but well no no it's not an it's not a local symmetry that's a great question so it's a uh yeah so when I contract these unitaries together it it's no longer a local symmetry it has to be done on every every unitary in the circuit that Global Z2 symmetry can be broken by the boundary conditions in this specific example it's not broken even by the boundary conditions yeah okay book oh I think it's not a local symmetry it's like it's a your question why it is or it isn't uh are you yeah yeah oh because uh um when when you do that swap locally at a space time point that intertwines the different copies of you and U star right um it comes from yeah so it intertwines the different copies so really it's just the global swap of of all of those copies that is a symmetry but that can be implemented by a local generator applied to every site and that local Z2 symmetry generator is this swap operation applied to every space time point right yes oh um yeah so it'll uh um oh right right sorry uh yeah they're they're contracted though like if I replace so okay if I change the upstate to the down state here I have to do that everywhere at the same weight because of this contraction which we haven't discussed as like as an operator level on this way I expecting that is um no because remember this the swap operator acts here and here right so it leaves this P2 invariant but it changes this it backs non-trivially on this like right and on this leg so it's not a symmetry of you know it changes this operator right it's not a you know once I start Contracting these guys it's no longer a symmetry unless I apply it everywhere I yeah go ahead does it help you think of each Bond as like four bonds radiant when you're swapping yeah yeah exactly right um yeah so that swap that I wrote is really you know if you can think of it as a tensor product of swaps on these individual bonds right because it's swapping the second copy with the fourth copy in this replicated space but it does that independently for these cudits so it's a tensor so it's get supplied here here and here uh but that's not a symmetry of this part you know of this that doesn't keep this weight unchanged because you know it'll not act non-trivially on this P2 but if I apply it everywhere it's a swap everywhere right um so if you have any more questions about that I'll maybe we can discuss it offline just in the interest of kind of moving moving on and in the hope that uh enough confusion has been resolved but um okay so I I just want to very briefly discuss the boundary conditions so um and then for this uh eising partition sum so this Z icing that I'm writing down is is exactly this expectation value taken um with a product initial state right okay so so what are these boundary conditions so uh the boundary conditions are actually um pretty straightforward to think about uh if we remember so I'm just going to write down what this ising magnet kind of looks like uh near its boundaries uh okay so let's say this is the lower boundary of the ising magnet and this is the upper boundary and let's just remember that uh this trace of rho a squared is taken in such a way that you know time runs upwards right so I have this uh you know the zero product State sitting on the lower bonds uh and then I have this uh you know uh tensor contraction that's uh setting up here right okay so um okay so uh what this means is is very simple so the initial state is dictating the lower boundary conditions for the sizing magnet and the contraction up here is dictating some boundary conditions that set up here that tell me how the different copies are related together right Okay the reason for that uh is um so so at the bottom of this ising magnet for example what do I have well I have you know this zero State tensor to the N so I have that on N bonds so that's the initial state that I that I start off with of course there are four copies of that initial state each uh of these bonds here right represents four cubits right it's a four acute it filbert space that is being acted upon so each of these bonds has to be connected with four copies of the initial state right so this is an overlap that I'm now evaluating on this bond between these states and the isingspin that sits over here on the upper Bond the situation is a little bit different so in the upper Bond um remember that I'm not color coding this anymore but uh AR where I'm sorry this is a which I uh and this is a bar right um so I trace over a bar and then I do the matrix multiplication on a to get the trace over a squared but um these legs so which represents a and a bar are really just giving me the you know a contraction of the upstate and of the downstate in those respective regions right remember that uh the um so remember that this this up State uh is represented by just you know this kind of pairing so it pairs the these two copies and these two copies together whereas the downstate will pair you know the first and fourth and second and third copies together right so I'm I'll uh kind of be schematic maybe but if a bar extends is uh so let's say uh let's say the a bar Hilbert space lives on the left and the a Hilbert space lives on the right then the boundary conditions that I have are that I have to contract with the upstate on this side and I have to contract with the down state on this side okay so there's an ising domain wall that is sort of nucleated depending on which you know subsystem I'm choosing to calculate the Purity and then there's some initial conditions of the state that I start off with that determine what happens here in the ising magnet okay okay now now we can kind of return to one of the statements that was made earlier the icing symmetry if we apply it globally it looks like it the boundary conditions might you know might break it the upstate goes to the down state and the downstate goes to the upstate right but of course that just means that I've chosen a bar to be my a subsystem and vice versa and of course the purity of the complement is the same as the purity of a because the entire system is pure right now that also suggests something which I'll get to in a moment which is that if you couple the system to an external environment like if the global state of the system isn't pure then that coupling will act like a symmetry breaking field in the sizing language right okay um yeah uh I'll get to that this the sign of that field is normally uh fixed just because your steady state when you couple to the environment your your system as it repeatedly interacts with the environment will eventually become maximally mixed right so it'll become like the identity uh and that uh that means that the sign of the field has to be um up so it'll prefer the upstate yeah right so the the upstate is like the uh is is the identity density Matrix if I were to unfold you know if I well the op state is like the um yeah is is related to this infinite temperature density Matrix yeah if you unfold the legs maybe make sense yeah I was there another question here yeah um yeah uh so I'm I'm going to get to the symmetries and their interplay with this like how you would break these symmetries and stuff in in just a moment so let me return to that later in like and at the end of lecture if you still have questions just uh yeah okay yeah legs the white legs are full of legs okay yeah so each of these uh these legs um this is why I said this is schematic because the this internal leg has no meaning to it right it's it's just uh yes these days yeah these yeah that's a that's a great question so I I'm glossing over some technical details which I think are gone over in more detail in the notes but um this this Upstate here and this down state here you can think of as uh the corresponding states that you would write down in this queue to the four-dimensional over space right so you can write down the same paired States in this four Q to the four dimensional Hilbert space here and and here whereas previously I'd written it down in a even twice as large field of space right yeah yes I'm being I'm just saying that they're being contracted with uh with that state so yes sure you would write them as always making me do extra work thank you actually you know this is you're probably the only person who can read this so given your position in the front of the room okay so um let me move on um so and yeah I'm I'm actually just going to mention that so um let me actually just state it here in some I've already written down the boundary conditions so the boundary condition here involves uh a contraction of this state zero you know uh with you know the upstate or the downstate right uh and you can convince yourself that that uh that overlap is actually independent of whether you choose up or down so uh you can convince yourself that this Upstate overlapped with zero tensor to four is equal to down which is equal to one okay so it's actually true for any product State you choose this overlap is one so what do we have so this is in the ising language these are free boundary conditions whereas up here we have a domain wall between a and a bar as the boundary conditions okay all right now one last thing I want to mention is uh that um in this sizing magnet so in this ising magnet uh say um so say I'm I'm drawing everything kind of in the Continuum but you should you know for the calculation of the trace of the Purity we have this domain wall boundary condition and the free boundary condition down here but we should you know we might want to ask like what is the nature of the bulk Isaac spins are they in the ordered State and the disordered days Etc uh because ultimately we're only going to be changing boundary conditions when we probe the Purity versus any other observable in this two replicated kind of problem and uh there's a simple reason that you should see that you might argue that the bulk is in the ising ordered phase it's deep within the ordered phase and the reason for this is that uh the um in any unitary evolution uh the uh the identity mixed state on on all qubits so now I'm imagining that the full system is in this uh you know in uh at infinite temperature is a is a fixed point right is uh is sort of left unchanged okay um right so uh so what that means is that if I you know for example if I if I mess with the boundary conditions so instead of having domain wall boundary conditions up here in free boundary conditions here let's say I made the boundary conditions down here that all the spins were pointing in the up Direction so the UP State for all the spins here the upstate you know tensor to you know the this state over here you can think of as a tensor product of you know it's the state representation of a tensor product of the identity uh right um and so it's really if I choose the upstate down here it's like the initial state that I've chosen is the infinite temperature State except two copies of it right um the reason is that I'm perfectly pairing the sort of forward and backward kind of states of U and U star um but if I do this right so if I fix the boundary spins to be up then what this is saying is that this date at every time will just be in the upstate right so it'll it'll perfectly propagate upwards and all the spins will be perfectly aligned right so the you know let me neglect the boundary condition up here but you know it'll just perfectly continue to propagate upwards neglecting the final boundary condition the same is true if I flip up to down that's a little bit more mysterious uh why that's the case um but uh it's because of this globalizing symmetry so you might say you know we've only just changed the boundary conditions and in doing so we found that the bulk is like strongly ordered so you might just guess okay this ising magnet is deep within the ordered phase and what are we so what are we actually looking at so so another way of saying this is you know the expectation value uh um um it's no it's yeah it's not uh it's very close to zero temperature it's it's not exactly at zero temperature and the reason is that uh in the bulk well um yeah so actually let me say that uh um I I think let me let me say that I would think of this problem as being basically a zero temperature when I introduce an ising domain wall it has to you know it has to propagate and end somewhere in the system uh but there are no fluctuations of the spins are far away from where I've introduced The Domain wall so it really is at zero temperature right yeah yeah yeah no that's right um yeah sorry yeah if you have a finite time right I think the the issue is that it's not really exactly an eyezing uh partition function it was a negative ultimately no it is an ising partition function once we take care of the negative Bolson weights it's just that sure but I think that can be thought of as the problem being at zero temperature right yeah it's just exactly right so so the you know when you introduce a domain wall sure you know just because of topological constraints the domain one has to go somewhere and there are many configurations but that doesn't mean that the system isn't at zero temperature um I think that's maybe what David is saying okay so maybe another way of saying this is um uh Okay so so let me be sort of schematic um uh um actually let me in the interest of time so okay so let me uh just say the following so so this our average that we can do for trace of row a t squared uh is okay of course it's trivially equal to the following but I'm doing this just to um so of course if I take the trace first this Trace is always the identity the expectation value of the harm measure doesn't do anything this denominator is just one right uh but in the ising language what this corresponds to uh is this is kind of like saying so I haven't changed the initial state of the Dynamics here so I sell free boundary conditions down here but I've taken the subsystem to effectively be nothing right and uh or that maybe that's another way of or sorry I've um I've taken the complement to essentially be uh to not be important um here so it's as if I've taken all of the spins to be in the upstate instead of introducing a domain wall so this is like the ising magnet the denominator is like the ising magnet except I have all up boundary conditions up here and then I have free boundary conditions down here the second is like I've introduced a domain wall which is propagating so I have uh up here and down here in free boundary conditions here and I'm taking their ratio and this is exactly going to be equal to the e to the minus you know the free energy cost for introducing a domain wall in this magnet right so this is like the expectation value of a domain wall creation operator in the sizing magnet which is deeply in the ordered phase it's basically at zero temperature right so you would expect that the free energy cost should um you might expect that it scales so this is an infinite system the domain wall can't exit anywhere and these boundary conditions are frozen so the domain wall has to more or less be directed in time you should expect then that this free energy cost should scale more or less linearly in time right as the domain wall kind of extends further and further into the ball so this just gives a schematic description that the Purity should Decay exponentially in time in this heart random circuit but the reason is that we're probing you know some something like the domain wall free energy when we when we probe the entanglement okay there is I'll get to that uh yeah I've averaged it but um there's entropy to The Domain wall in that there are different configurations that you know the domain wall has a line tension so any configuration that has you know costs the same energy will will give me entropy for the domain wall right there are many configurations you need to sum over different domain wall configurations exactly yeah but overall you'll still expect this scaling yeah uh okay so that because of the lattice here there are actually different domain wall configurations that cut the same number of bonds yeah right they're all the same energy so it's just the domain wall length that enters and uh yeah I'll get to it in a second yeah yeah right um yeah sorry Anisha yeah um okay let me uh let me just write one one one thing I guess and then or maybe one or two things and then foreign yeah I'm being quantitative here because I uh because I think there's a lot that's been written in the literature where people are schematic about this because people know these calculations fairly well and we've known these for a while but I don't think there's you know there aren't too many papers where people go through this in more detail um so if this isn't helpful also let me know but um okay so uh let me just uh say one thing which is that um so these these weights are going to be invariant under the sizing symmetry transformation the bulk weights um and as a technical Point uh one thing you can do to convert this into um a sort of something that simplifies the calculation that has to be done here is if we integrate out every spin that lives on one of the two sub lattices of the honeycomb lattice in this partition sum so if I write out this partition function and uh I integrate out say what I call the sigma spins and I'm just left the Tau degrees of freedom then you obtain what is actually a model of you know classical statistical mechanics of these icing spins but you obtain these kind of three spin weights that live now on downward facing triangles that couple these spins together right this spin here when I integrate it out mediates the kind of three spin interaction between the spins on the neighbor it because they're only weights on bonds beforehand so as a technical Point um and I'll stop here uh so as a technical Point uh so integrating out this is an exercise uh one species uh say Sigma of spins uh uh gives um yeah I I'm referring to Sigma as the species living on one sub lattice and tau is the other so yes integrating out one sub lattice spins on one cell lattice um yeah let me say that explicitly uh gives this icing partition function which uh it's exactly the same but the the bulk weights now are on downward facing triangles which couple the remaining spins together right so the downward facing triangles live basically here right so they couple this the three spins here together once I integrate out the sigma Spin and those weights are izing symmetric so instead of having eight weights for every spin configuration they're really only four that depend on the domain wall configurations right so you can convince yourself that the the following weights you have the following weight so if I have Sigma one Sigma two and sigma three right if these are my spins then I have the following weights which I can write down in terms of domain wall so if there's no domain wall the weight is one if there's a single domain wall this weight is uh these are in the notes but there's something here that I would like you to think about which is if I have a domain wall coming in and out like this this is actually equal to zero and uh the two weights up here are actually very straightforward consequences of unitarity the fact that you cannot this is basically saying that uh you can't really nucleate domain walls in the in the bulk of the system uh and this is saying of course that you prefer the ordered state right so both of these are consequences of unitarity and I'd like you to think about this maybe before we we meet again next time um but uh even from this the calculation of the Purity proceeds trivially in that uh there's just a weight for different domain wall configurations of the same length and then you just multiply by some entropy or something so okay I'll stop here sure yeah there's a zero boltzmann rate for this yeah exactly yeah yeah right I would I would have all three domain wall configurations okay they're the same right well they're I'm just there are no fluctuations right is all that we're saying like the so that's that's a zero temperature problem uh the domain walls behave as if they're in an ising magnet that's at zero temperature they have to go somewhere um but um yeah yeah you don't exactly so so in fact uh you know the the answer for what the Purity is is like one line further from this right um it's I'll just I'll write it down but yeah Anusha you had a question um that's just the fact that you always expect Ed yeah I think um yeah one has to be careful so this this feature here is true for any Ensemble not just R I think um the fact that you can't create these domain walls and that this is equal to one um yeah that's sure sure yeah yeah uh yeah David yeah perhaps past diagonal interact the horizontal interactions and the diagonal interact just representation are both instruments yeah right if you were to try to write this as an ising model you would find that right sure yeah um yeah sure um yeah I well but sure yeah um I mean even even if I change the boundary conditions the lowest weight configuration would be one that looks that is one that you could easily compute in a zero temperature eising model right where you have bonds that are infinitely strong because you want to anyway we can we can discuss this later but you're right if you were to write this these weights down as you know e to the minus Theta H where yeah then you would have you know some signed interaction some of which are infinite and yeah yeah so uh it's still one thing so uh hour and a half I were considering then for these sort of ways um so now you have a boundary or no you have a top Android right now but is that these are you know finite energy right yeah fluctuations more energy right um yeah but uh I guess in the long time limit there really aren't right uh because in the long time limit you're kind of uh um yeah okay so so like I think here's here's maybe what I'm getting at what I was discussing was the problem where um the system size is actually taken to be Infinity right uh sure so in that case you know there's no there's not really fluctuations right right so so I think that's why I didn't want to get into what happens at finite size when you have finite size yes you have the domain

Channel: BSS Physics School

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Id: BMLD9ogMXSc

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Published: Thu Jul 06 2023