 You are enjoying this summer school today is already third day the time time flies I'm soon one. I'm going to talk about measurement induced phase transitions in monitor the quantum circuits Before we start I would like to you know provide some context of you know, how these things are moving how this field is moving so That's item number zero Introduction we all know out of equilibrium quantum dynamics is extremely rich and Already, you know past two days. We talked about many concepts And one concept is idea about how to combine closed quantum Dynamics quantum many body dynamics to the statistical mechanics Which means we need to understand quantum system thermalizing the quantum thermalization Another another you know key words that we've been hearing from this school is quantum chaos. How to understand chaos, which is a one mechanism of Explaining thermalization and it's closely related to the our gaudy city So this is what we believe that it has to happen in a quantum systems without proof But we are trying to understand what's going on at the same time We are interested in when these phenomena do not happen So exceptions to the system and that's the basically what a norm was emphasizing a lot or gaudy city breaking You know particular form or entirely It is very nice and then there are many many topics to come over the next many days Which very nice however, there's a big problem the big problem is these problems extremely difficult Relative to understanding conventional condensed matter systems where we have a Well-established many advanced theoretical tools feel theoretic tools diagrammatic tools numerical tools The physics that we are talking about here. We are at the very early stage of the theoretical development So as you might have guessed For example quantum KSN or gaudy city, but I'm not only poker. We know we gave a talk many of these slides contains exact civilization of Quantum system made out of maybe 20 qubits 20 spin one half particles 20 is a large number when it comes to simulating quantum anybody dynamics But certainly this is not a macroscopic number that we deal with as a quantum many the real quantum materials And that actually demonstrates how difficult this problem is We are basically doing the brute force calculation. It's a hard to go, you know beyond that Therefore the core community is trying to develop new theoretical tool sets to understand this now to equilibrium dynamics And one of the most exciting and rapidly developing tool Is quantum information So originally quantum information Theory of quantum information science is motivated by the idea of okay. Can we control? Many degrees of a quantum system in an arbitrary fashion so that we can do interesting We can build interesting applications like building a computer's quantum computers for building sensors simulators Or maybe we can use it for secure communications At the same time we have developed the understanding of you know, what are the important ingredients to design this quantum applications? And we all heard about entanglement entropy at some point in our study. That's one of the most important concepts Tests of network methods. That's another important concepts Quantum circuits, that's another important concepts quantum channel another important concept those words and those concepts are actively in these studied in information and Therefore it's natural to adapt this quantum information theory as a tool to characterize what's going on and And one of the important concept developing quantum information field is notion of quantum circuit particularly random Unitary quantum circuit because it's a good toy model to study in the in the field of quantum information But at the same time this random unitary circuit is providing a good Intuition behind what's going on in terms of physical messages So this quantum information as a tool basically we talked about two Two branch one is what I just described the conceptual development And the other one is also Separately experimental advanced separate to these theoretical developments We do have control over quantum systems Some of them we believe to be some system some scale That's beyond what we can numerically simulate using our classical computers and Those systems just started like maybe within the past five or ten years five years They just started providing the data whether we are otherwise it's difficult to or impossible to pay and Just analyzing this data We are developing more intuition about non-equilibrium dynamics at the same time in order to try to analyze the data We are forcing ourselves to develop a better theoretical description, which it turns out to be a useful You know tool to understand is some of the many non-equilibrium dynamics and Not so surprisingly Many times whenever we introduce new tool or new experiment we happen to have discovered like new phenomena You discover it and it's all cycle like we we develop new physical phenomena that we have never imagined Sometimes just because we didn't have enough tool set or sometimes just because we have never thought about such a situation It's out of our imagination But now the things are coming into reality we think about more exotic situations or maybe not so exotic But just didn't even have a chance to think about and Then discover new phenomena in these new settings and this is going to be what we are going to talk about today this lecture The measurement induced phase transition is a particular example Where we are using quantum information tool to understand that out of equilibrium quantum antibody dynamics And at the same time we just realized that okay, there's a rich phenomena that we have never imagined to exist But seems to exist and that's why we are going to talk about today so first part of my lecture would be introducing measurement induced phase transitions our objective of these lectures is two fold the first one is to introduce This concept of measurement induced phase transition So that you guys can jump into the research field But this will remain at the conceptual level a usually high-level providing as much intuition as possible What are the difficulties in the field and how healthy our field is moving? another one that I wish I can deliver successfully is introduce Technical tool mainly analytical tool associated with random unitary circuit I hope my time management is successful But if everything is successful I'm going to talk about the first aspect in my first lecture in the second aspect in my second lecture and The goal of the second part is so that you okay you learn about these technical tools and use it for your own research It doesn't have to be measurement induced phase transition no matter what the topic is it could be just quantum information like the project itself Somehow utilize these technical tools because it's a so cool and nice tools that has been developed in quantum information community past few years any question so far Okay, so let's get started random unitary circuit. So we'll learn these random unitary circuit Provides a new class of collective phenomena Also, we'll learn this gives us a way to study the dynamics of quantum information And we'll see that some of the aspect we are going to talk about It's in some sense a little more active than passive So in the past we talked about our gaudy city breaking or the quantum chaos by studying the dynamics of unitary systems or dissipative driven systems But here we are introducing the notion of measurements and the measurement is special because you not only Disrupt the wave function of the system, but we also gain information about the system through this measurement Therefore somehow it's more like with a quote-unquote interactive dynamics, and we'll see as a special example of Phenomena that occurs in this case. We will see measurement induced phase transition But at the level at this point this be a little bit like too high level and qualitative, but hopefully it will become very solid soon Okay, measurement induced so let's jump into the settings What we are going to consider is a unitary dynamics generated by random unitary circuit So just for the concreteness, let's consider having an array of qubit degrees of freedom So qubit is just a two-level system You can think of it as spin one half particle up and down, but I'm going to use a notation That's more familiar in the quantum information. I think it's zero and one This line you can think of it as a word line. So time is going up Sometimes I'll go time from left to the right, but now in this line, which is moving up many qubits Initial zero state. It doesn't have to be but I just chose a particular state. It's easy to write down And I consider applying a unitary rotations But this unitary rotation is acting on for example the first two qubits. So say you won on the first two qubits Unitary U2 To the third and fourth qubit. I could apply another unitary here, but let me just skip it So this defines one time step of unitary evolution, you know They are supported on different qubits. So we can apply the same, you know, like these two unitaries like simultaneously When I say U random unitary what I mean is this U is a randomly chosen element, but once I choose that's just determined to be chosen So I'm not going to consider non unitary dynamics at this point. It's just a particular unitary, but I just choose randomly From all possible unitary For example, unitary group, let's say SU for This is going to be a four by four matrix Because it's acting on four-dimensional Hilbert space, but otherwise just randomly chosen And this is independently randomly chosen and there's no correlation between those two So this defines one layer of unitary and They will apply another layer of unitary. Now, let's actually work on these two guys and these two guys So after applying this unitary All qubits are feeling each other Because they can be entangled through this unitary. Yes, Koshu It doesn't have to be I'm just drawing as an example for now and I'll stick to that cases within this lecture But it's already like it's a very good point. You have a U3 and U4 And you can build up So this is first layer and it's a second layer and this number of layer is sometimes called depth the depths of the quantum circuit Okay, so we consider the dynamics the many body dynamics of this type Whenever there's a large depth or large number of qubits, you can put into the periodic boundary condition But I'm not going to in today's lecture. We could have done that. It doesn't have to be 1d. We could have a 2d Just stack them in a reasonable way so that all qubits are interacting one another eventually By particular geometry. Okay, so this Very good point, but I'll answer that in a moment And any other questions? Yes So could you elaborate the question? Ah, ah, ah, I see I see I see Su3 is for three level system Su4 is for the full level system if I have a pair of qubit two-dimensional two-dimensional so we have a total four states Therefore the two qubit is actually four-dimensional system. Therefore Su4 is a more general one Su2 cross Su2 will be smaller than Su4 Because what what it means is you apply one unit right here one unit right here. However, not necessarily any other questions Yeah, that's right. So when I say randomness, I should have provided more technical definition Of what distribution of randomness are you talking about that's going to be my second lecture But just to answer that I'm just considering for now uniform distribution So what I mean by uniform distribution of what is group is that distribution such that if you do the change of variables by conjugating or multiplying by another group element the distribution is invariant and technically it's called a high measure Cool. So why do you care about this random? This is called a random unitary circuit like a random Unitary circuit the name is very descriptive like a random unitary and it's a circuit So it's a random unitary circuit, but why why do we why do we care about this object and then a pros and cons one of the reasons That we study this random unitary circuit. It's because it's a very generic Unitary dynamics. It's a generic dynamics. In other words, it's a feature list Feature list means it doesn't have any particular structure because it's randomly chosen And sometimes that's useful because we are now studying any particular feature coming from You know some constraint or the structures or some some special properties of these dynamics But only by the features that's commonly shared by all possible instances of random unitary circuits So what are those features? What are the features they share by this and that's locality Locality and geometry so here I intentionally to do the diagram in 1d in nearest neighbor gate and that's good because we can restrict the quantum dynamics to be over that time and then maybe we'll learn Universal or like, you know ubiquitous properties when the interaction is local like geometrically local We could have decided to do something else either two dimensions or this unitary acting on first qubit and some other qubit take far away That's still fine in quantum information community That's also called local it's so called to local because it's acting only on two qubit, but geometrically non-local Those are also part of a consideration But today we'll focus on geometrically local circuit because in some sense that's more relevant for the real-world situation And so this locality and geometry is something that can be captured by random unitary and then there was a question about Why we choose random? Because interestingly enough if you choose randomly we can solve this problem analytically So we do have analytical tool, but this analytical tool only applies to the statistical properties ensemble properties We cannot talk about the quantum dynamics for individual instance of unitary But we can analyze the properties of average overall unitary or typical behaviors of unitary So we can have an analytical tool for statistical property On the other hand, we also have a downside of this random unitary the first downside that this is generic Maybe to generate into two featureless sometimes we want to study Properties such as energy conservation and how system thermalizes to the effective temperature locally like quantum thermalization Or we want to study how chaotic behavior or gothic behavior emerges Namely how the the dynamics is described by effectively random matrix theory in this case This is already random matrix the fact that the system dynamics resembles the dynamics of random matrices It's not interesting at all because it is they are random matrices, right? So that means it's not useful to study the emergence of the random matrices because it's explicitly random Energy conservation can be they cannot be studied therefore. It's a hard to study for example energy diffusions Or any any hydrodynamics So any non-trivial quantum thermalization is difficult to study Although we do have a version of random unitary circuit where instead of Sampling you from entire su4 We only subset from the subgroup of su4 where they do respect a particular imposed symmetry So for example, you want to see me the particle number, you know conservation symmetry that is possible and that has been active That is still that active area of research However energy conservation is somewhat special because energy Hamiltonian is itself the driving the dynamics which is also considered those type of features cannot be studied in this case and Also, this does not have a no natural system where it realizes the random unitary system So we could engineer This unitary dynamics using our quantum computer. However, we don't believe this occurs naturally And finally the same point as analytic tool. We can only study statistical property artistic call and Individual instances cannot be studied so we cannot separate out what is coming from the ensemble behavior versus like individual behavior But up to this pros and cons this random unitary circuit is a very nice tools that we can utilize Under these dynamics our wave function evolves Starting from some psi zero in our case We are choosing this to be zero to the power n like n is a number of qubit And then if you apply u1 you can apply u2. This is a unitary evolutions all the way up to some ut And This defines quantum state of time t sometimes, you know, it's not so important. How do you measure time? Here I just index them by one two three four But we could have index that this whole layer is like T equal one this whole layer could equal two So usually I'll use my intuition that my T now refers to the index, but actually the depth Because that's more intuitive This is entirely unitary dynamics So for a useful fact If this T Over n is much much larger than one. So the depth of the circuit is way deeper than the width of the circuit What we believe is this product of ut u1 it's approximately Unitary sample from the global random unitary so you can think of this whole thing as random Over as u2 to the end. So this is global random unitary So when I write this notation this is a very What I mean is okay It's not exactly, you know typical random unitary from this group But I just mean this is practically indistinguishable From global random unitary as long as depth is very large if depth gets larger and larger It'll be more and more indistinguishable from SU2 to the end Not all circuits are unitary T design But if you build up circuits, you know, whatever the geomaterial is connected And then the depth is sufficiently large and they form a unitary T design The T design is a technical term basically says it's indistinguishable from how random global unitary Up to teeth moments. Yeah, but that T is that T is different T. Yeah, just jargon. Yeah, any other questions? Okay. Yeah, yeah, that's right, but you can you can define this whole thing Okay, let's define this layer of you as like v1 and this layer of you to the v2 and let's write it as v1 and v2 Yes. Yeah, this n large n is a number of qubits. Cool. So any questions on random nature? Yeah, that's right. That's right. Actually two two main reason a I don't want any conservation law other than Unitary T and locality Okay, so that that's the only feature impose that's one reason Second reason is if this is chosen random unitary turns out we can your analytical calculations By averaging over random unitary and we know particular distribution of random unitary They uniform distributions You can utilize that for our favor to carry on the calculations for many interesting properties That's going to the main top the technical tool is going to be the main part of my lecture to know It's not like it turns out if you use a uniform distribution We can use the the well developed representation theory tool so called a sure-vile duality and then Calculation becomes simplified because you can adapt them. However, that's not the only random ensemble We can carry out the calculation Even with some other ensemble. Maybe we improve certain symmetry or certain structure Generally, whenever there's a randomness and a well-defined distribution, so we can carry out a calculation calculation becomes more difficult But in principle this idea carries out cool, so in order to understand this measurement in this entanglement Phase transition which I have not explained what it is But we do need any additional ingredients. So I'm going to talk about that ingredients now and That's measurements It's obvious it's a measurement in this phase conditions We need to talk about measurement nothing something complicated by something that we every Everyone, you know have must have learned in our Undergraduate, you know quantum mechanics or if some of you maybe have learned in your high school Maybe need meter school So the measurement is literally performing measurements on these quantum circuits And we learned if you perform measurement you the wave function collapses And then you know if you measure repeatedly measure that you get the same outcomes, etc That measurement is what I'm talking about here, but here I'm going to perform probabilistic measurement. So what I mean by that is Each time step for every qubit, I'm going to do the following So after time step one, so you just apply v1 a layer of unitaries For each qubit one qubit at a time separately I'm going to decide whether I'm going to perform measurements or not So with the probability p Or to simplify our notation, let's say probability q with the probability q I perform measurements and With the probability 1 minus q I do not perform measurements So with q I perform measurements And with probability 1 minus q no measurements If I don't do any measurements, I just don't do it. The wave function is intact Nothing changes. The wave function stays as it is However the probability q I Perform measurements and when I perform measurements, you know inevitably disrupt the wave function So here I'm going to assume that measurement is performed in the computational basis. It's zero one basis Then means the measurement outcome is going to be either zero or one with a certain probability Okay, so when perform measurements now wave functions Collapses to either of the two case When the measurement outcome m is equal to zero We collapse the wave function so that that particular qubit becomes zero after the measurements and then we obtain a Unnormalized wave function. So say m equals zero, which is defined by You project So this is a projection operator acting on that particular qubit now apply to our wave function sign and If I measure one We will get on normalized wave function m equal one, which is project to the Unnormalized wave function like side and what is the probability of obtaining this particular measurement and that's determined by norm of this Unnormalized measurement. So probability That m equals zero is the same as expectation value of the wave function before the measurement Projection to zero. So I just define this as a p zero Decide, let's say the p zero times identity Hi, so this is a measurement outcome or this is the same as I tilde m equals zero And so I tilde n equals zero the norm of the wave function because that's the probability that you get m equals zero No, we are not changing this measurement rate q So here this is we are talking about one particular measurement that could happen on the first qubit with the probability q it may not happen and Then I do the same thing here with the probability q perform measurements or minus I don't measure and Whenever we perform measurement you also collapse So you collapse potentially collapse here and also collapse here may also collapse here Maybe you don't you skip the measurements And so on and do that for every qubit and that's one layer of unit tree the measurement layer Once you do that you collapse away function you obtain the wave function like this and Condition on particular measurements suppose for now we perform the measurements and obtain a particular measurement outcome m and Then this leads to the evolution Where the psi maps to? Psi tilde m However, this is all normalized such a way that you know the norm is a probability of getting a measurement outcome So oftentimes what we are going to do we just normalize it back We just normalize and define psi of m by the literally normalizing them However, however, what the implied is whenever we normalize them We know that occurs only with a certain probability so we also need to keep track of the probability of Measuring that out and then evolve further and further and this is repeated Well, sufficiently many layers like the layer T the number of measurement we perform Also will also be the random variable because we decide on you know, they randomly Awesome totally some fraction Q of the qubits will be measured if the system size is very large and the time is Time will usually very deep the depth circuit That's a good point. So here I'm performing measurement on one qubit So the projection operator is acting on one qubit and you're acting identity to the rest of the qubits So when I say m what I should really sad is measurement outcome at a specific location Like the first qubit in this case is equal to zero If I perform measurement another time here So you perform measurement here and here and the obtain measurement outcome You know say zero and one here and then I need to define my m such that it contains the information for example, this will be okay the first qubit measure zero and third qubit measure one ta ta ta So over time this notion of m will be a growing array Keeping track of which qubits are measured at what time say okay So you actually these issues happen like another one, you know time index, okay t equal one first qubit zero T equal two are the t equal one third qubit one and then we should keep track of when you perform the measurements and Where you perform the measurements and what the measurement outcomes for? And then this m from now on I'm going to use as a collective index They describe all those information, you know, you know in a symbolic manner like just single m Very good question any other question That's right. There are there is finite probability Actually, that's like q to the power n where you happen to perform measurements of all qubits on one particular layer Maybe you're lucky or maybe not lucky and then the wave function will collapse to a particular product Take zero one these strings that could happen However, whenever you choose reasonable q say half Like a point one the probability of that event occurring is basically zero Because we on typical basis we only measure certain fraction, but that's theoretically possible. Yeah, any other questions? Very good. So Previously our unitary dynamics is evolved by these just literally unitary evolution and you obtain the wave function at a later time now we obtain non unitary dynamics the wave function is enumerated basically by m and also like set of Unitary gate choices it depends on the which unitary gate we choose is every layer So because notation becomes complicated, I'm not going to write down this unit right too much But we should keep that in mind that the wave function depends on a particular choice of unitary Randomly chosen and this is going to be some operator K m Acting on our initial state And it will tilt out to indicate that I'm going to talk about Unnormalized wave function and here m is this record of the all measurement outcomes And K m is whatever the linear operator that we obtain it's going to be the product of those unit rays and Then these projection operators and they multiply them and stack them over time Okay, so that's going to be my definition of K you could apply that money multiple measurement for the same time for the Different qubits for each qubit. We just decide whether it's measure or not by Q. So you could have multiple measurements. Yeah Okay, and this all normalized wave function has a nice property The one thing is if you compute the norm of the wave function Nor of this all normalized wave function squared in other words Just inner product yourself and that's actually probability of obtaining this all set of measurement outcomes This is true for single one particular Particular measurements, however, this is also true collectively if you collect many many measurements Overall norm of this all normalized wave function is actually the probability proportional to the probability of measuring that outcome set So like between two scenario, can you say that's right? So let me repeat the question So we could imagine situation where every time step We just want to choose like one of the qubits and they perform measurements versus for each qubit We perform measurement with the probability Q. Are they different or not? They are of course very different on one case on average. We have a certain fraction of qubits measured If in the limit and it's very very large on other case We do not measure the fraction but usually measure one and most one qubit and the like in terms of many body physics The effect of the measurement will be very minimum because you ever like imagine that you have a 2 to the 23 particles and measurement of them And nobody cares But if you measure like fraction of them and that's a substantial measurement so measurement plays an important role in quantum dynamics We are going to consider the letter letter case where measure fraction of qubits. Yeah. Yeah, let's write it down. This KM Yeah, I mean just defined by Unitary one You know, I say v1 the first entire layer of unitary and Then we are going to have a projection operator projection operator to say mj where the jth measurement outcome Is equal to 0 and 1. This is a projection operators. I use multiply them for j that's measured So this is a projection operator for one time step and then apply v2 second layer and Then we are going to multiply you know apply the projection operators to after the second layer that's going to the product of P and again j so maybe to layer one layer two and J in the measurement set and so on Because of this projection part this KM is on unitary E M Have you ever like superscript one that means it's a layer one What what does Sorry, I just couldn't hear you. Yeah What's in the product here a j. So you're running over qubits that's measured measured Measured yeah, it's not everyone not everything is measured, right? So we only numerate the measured one. Okay, so like one of the kind of visual picture we have in mind is a following So we start from some initial state and Then you evolve by some unit tries a v1 But at some point we perform measurement and we still you need to study like two different possibilities Measurement one and measurement two in each case. You just also evolve by unitary So this is like first measurement equals zero first measurement equal to one and then you also perform the second measurement and Then we have this bifurcating, you know these trajectories So we have a many different trajectories of wave function and each wave function is enumerated by Unnormalized wave function psi of M or psi of M prime So psi of double prime many different trajectories and also of course a set of choice of unitaries And we are going to consider this whole thing as an ensemble of pure state That was set of ensemble pure state either consider as an unnormalized wave function ensemble or Normalized wave function ensemble Associated with the probability P of M Which is the same thing because like, you know, this is a norm of the wave function This is just a normalized version of psi tilde So let me make a one comment This situation is slightly and actually drastically different from the conventional Algebra equilibrium dynamics and open system dynamics. We usually consider In physics. So in the usual physics, we usually talk about Average over all these trajectories or ensembles. So we don't talk about ensemble of pure state We usually talk about or identify ensemble pure state as if it's the same as a density matrix For example, many cases we define the density matrix as Summing over all this measurement trajectory the probability of M S I M S I M Which is the same as sum over M K M So I 0 So I 0 and K M But generally we could have talked about some generic mixed initial state sum over M operator K M row zero In KM dagger Row zero is a initial density matrix and this particular form of map from the density matrix to the final density matrix It's generally called a quantum channel. He has to satisfy a certain properties Namely, the if it rolls zero is normalized say the trace is equal to one The final one has to be trust equal to one We know the eigen values of row zero has to be all non negative because it's associated with the probability distribution The same has to true that means that it's a positivity that you preserve the positivity And also we could consider this row zero as some mixed state Which is actually secretually entangled with some you know some other degrees of freedom And even in that case after applying this map the final state also has to be physical state And if you satisfy all the relevant, you know physical constraints is called a completely positive trace preserving map See so called a CP TPM app and this expression is also called the operator sum representation And then this individual operator KM is has a name. It's called the crowds operator Okay, so I'm introducing those terminologies so that we can connect to the conventional situations Whenever we have a quantum channel We can always rewrite in these operator some representations for some choice of cross operators This cross operators have this new this index M which tells you what trajectory we are considering While conventional case considers averaging over This different M or summing over different M In our case, we are considering individual trajectory separately and we will consider the properties of them Not only especially the non-linear properties of them You know averaged over probability M that the probability of Pm, but that's going to be the overall you know the Situation that we are going to talk about so any questions so far good We are already halfway through the lecture. So let's actually explain what this measurement is phenomena. The entanglement phenomena is Entanglement phase change No, so we are going to talk about entanglement entropy of the individual wave functions so we consider random unitary circuits and probabilistic measurements and obtain an ensemble of wave functions enumerated by probability of M and Associated normalize wave function Now take a particular instance psi of M and Then there is going to be some kind of many-body wave functions So I'm implicitly using tensonative diagrams So this individual X represents a different cubic degrees of freedom and this is some wave function psi of M I I want to divide them into the two parts Not exactly half and half, but let's say roughly half and half So the left half is what I call a subsystem a The other half is a bar is a complement of a bar and then we'll Quantify amount of quantum correlation between the left and right for this particular pure state And this quantum correlation can be characterized by so-called entanglement entropy so entanglement entropy The entanglement S of a of the subsystem a with the rest of the system for the wave functions psi of M Is defined as Well, we first define a density matrix of reduced density matrix for this wave function by tracing out a bar from Wave function so we get the wave function and trace out All the degrees of freedom they bar and get the reduced density matrix for a and then talk about entropy minus trace of row a log row a And of course this row depends on Particular measurements that outcome M and also choice of unitary, but I just suppressed the notation This entanglement entropy so this essay is a function of This measurement outcome and also all set of unitary is that we applied what we are going to do is we will average this entanglement over all possible violations So we will compute average entanglement essay bar Define by we start from as a that depends on the measurement outcomes and the choice of unitary first and Then multiply the probability of that measurement are you measuring that particular outcome and sum over all M? So this is average entanglement Average over different measurement outcomes and then we are going to average over all Ensemble of unitary gate that we could have applied Stick to the particular geometry so average over this guy average of this guy because it's the independent random variables so average over these unitaries and Then that's how we define the average entanglement of a Is this clear that's right, that's right so we are given this diagram with a specific trace of use here and there and Then let's say your specific realization of measurement. Okay, you're here. I perform measurement here I perform measurement here I perform measurement and then we have a measurement outcome Okay, I'm one equals to one and two equals to two zero You know I'm three equals to zero and measurement outcome and then we can talk about wave function That will take you obtain as a consequence of this non-unitary dynamics. That's a psi of M Just normalize Okay, so normalize your normalize wave function because you normalize and Then do this humidity composition or do we just as entry matrix and then compute entanglement entropy is PM This I am Here. Yeah, this PM is a probability of obtaining psi M Yeah, probably obtaining a psi M for a particular measurement outcome This PM also depends on the choice of unitary which I did not specify yet but here we're averaging over all measurement outcomes and Then we are also averaging over different choice of unitary that you could have applied They doubly average like two different average. Why it depends on the measurements S is an entanglement entropy between the left half and the right half So it depends on the wave function and the wave function depends on the measurement outcome M is a measurement outcome. Yeah I mean, sorry, maybe it wasn't clear like M is a measurement out collection of measurement outcomes at every space time Whenever we perform measurements not by material It's not a product. It's just you you sample different M. So you this is performed measurement You could either zero or one So you're following this trajectory Yes, it's not like that. So this wave function is some very complicated Many body wave function. It's not a single qubit wave function and qubit wave function Unnormalized it's obtained from this formula But just one wave function. This is a different wave function And you know that they are enumerated by the records like an M. Let's go that talk about it there. Yes That's right for each wave function. It's pure state wave function with computer entanglement entropy For the pure state wave function and then average over these trajectories that's right It's different trajectory corresponds to different measurement outcomes Exactly, exactly Exactly All clear Okay, this is rather unconventional because as I said the conventional picture is we obtain the ensemble wave function the average the wave function first and Then obtain the mixed date and then do evaluated different quantities Okay, neutral informations or some different types of entanglement, but here we are not doing that It's that we are taking a rather unconventional route where we compute the entanglement of pure state first and then average over different trajectories a different measurement outcomes Measurement out when I say trajectory what I really mean is a trajectory of the measurement And surprising finding is that there exists a phase transition in the dynamics of SA bar So if you plot Is SA bar was sufficiently large systems and a is some large fraction slightly less than one half for example I just divide into the half like say like you know like one-third and two-thirds or like 50 50 That's right. Yeah That's right. So we obtain the psi m for the global wave function and then divide the system into a and a bar and Then consider, you know, psi a bar as a function of These depth t we could have done these calculations for the different depths of time It's more and more complicated wave function and time t And the wave function SA bar So what's gonna happen and time equals to zero when you don't apply any unitary and Taking money zero. There's only one trajectory because we have not performed any measurements Right, so it's going to be starting here Now if you run the time illusion you apply some layers of unitary and Then the systems get entangled one another, you know from one part same thing from the other part So entanglement will eventually somewhat develop However, we also perform measurements So the measurement actually destroyed entanglement because once it measures this qubit collapsed to either zero and one That means it's not entangled with any other degrees of freedom. So there's a little bit of competition So if the measurement rate Q these Q the probability of performing measurement these Q is sufficiently high higher than a particular threshold Qc The entanglement entropy may be developed a little bit But the century to a certain value and then stay there in the steady state at least on average when you perform Average over measurement particular trajectory will behave in a funky way. However, once on average you behave in this way Especially the typical behavior on the other hand when this measurement rate is less than a certain threshold Qc and Entanglement actually grows and Grows almost indefinitely until he saturated the later point where the saturation value He's actually limited by the system size So some fraction alpha times n is alpha is a number from zero to one So basically it's limited by the system size So alpha could be one half and that would be actually the case when you do not perform any measurement If Q equals zero Entanglement will develop almost indefinitely and it will be almost maximally entangled So the size of entanglement will be basically size of the a so you could even put and a here But I'm assuming and a is kind of some fraction of total system size. So I'm going to define according me So I'll find some number and Saturate this okay, and this is very distinct here and here Here we have an entanglement that scales with the volume of the subsystem So in that case we say it follows a volume law Scaling of entanglement entropy. So those are so called the volume of states and Here the entanglement is kind of you know developed a little bit but stay small and this is a auto one quantity like some constant number Why because Every time step there's a some probability of applying this unit time even or odd time step and If you happen to not apply any measurements There will be some entanglement across this boundary Right, so it may be zero, but you may not be zero and basically that's the picture You have some local entanglement around the boundary, but not made large scale of boundary. So this amount of entanglement here Skills with the boundary area of the subsystem. So we could say boundary row instead of the volume law But more conventional jargon in quantum information is so-called area law This is very confusing because in one dimensional system boundary is not area But it's very confusing, but they did just jargon so I'm sorry about that We can just call it boundary law versus volume law or bulk law But just convention in the communities. We talk of volume law Let's say the size of the cubits and boundary area law to say the boundary area of the system and then idea is This transition from one case to the other case occurs as a critical phenomena as a quantum phase transition So at the critical point q equals qc We have an intermediate behavior Logarithmically scaling entanglement trophy Scaling with the subsystem size and then we have a diverging correlation links for examples and the critical behavior And this phenomena is discovered and that goes my that's by name measurement measurement induced entanglement phase transition this is interesting because this is literally new critical phenomena a critical physics and occurs in The space of random unitary and measurement this interplay is very interesting This is not coming from the equilibrium systems this is occurring in the intrinsically dynamical systems and quote-unquote order parameter is not obvious because it's not a symmetry breaking phase transition It's as like you know professor Norma now talked about There's no symmetry that's broken because it's a generic random unitary. There's no structure However, there's an entanglement entanglement entropy, which is an information theory quantity But that undergoes a phase transition similar to the spontaneous symmetry breaking. So that's very exotic and very interesting But at the same time at the current moment, it's not exactly clear. What's the physics behind this? What does it mean that entanglement entropy undergoes a phase transition? So I'm very sympathetic like and I kind of agree with you, but as a person a personal perspective So maybe the intuition is this okay the wave function initially is all there is a product state We know that the small corner of the many body Hilbert space Majority actually, you know almost entire wave function in the large two-to-the-end dimension Hilbert space is very highly entangled Right, so what it means is if you perform measurement very frequently You're not exploring entire wave function, but we'll need to see this scratching the surfaces You know just staying in the corner of the Hilbert space from that sense roughly speaking. Okay, it seems like some organic is broken. I Agree with that picture. However, I have not seen kind of existing research along that line So I think it'll be very interesting research perspective But I will say maybe we need to handle what it means by you know Orgotic in this exotic sense because that notion as far as understand has not been studied very carefully in these particular settings Right in some this tricky because it's a random unitary So randomly kind of right in something randomness emergence of kind of ergodicity that people could talk about typically, you know, this you know random magic theory That that conventional definition hard to apply or maybe we should modify some picture, but that's a very good question Yeah Yeah, yeah, okay. Okay, so I could just put some alpha NA But we know this NA is some fraction of N Right, so I just redefine alpha by alpha times F. It's just some number from zero to one That's what I meant. So I I could just say this is like alpha tilde n where alpha tilde is this alpha times F Oh, yeah, that's right. That's right. That's right. So let's let's talk about these alpha So I saw sorry that this is alpha tilde. Okay In the ratio of n if the measurement probability is zero, you don't measure anything and Then you will generate the wave function of this form in the limit. He's very large At some point when he's very large in saturation value, you will behave as if You're applying some global random unitary to the system And then you get some typical wave function from the many body he was space and you can actually prove that for typical wave functions Antenna my entropy of any white partition is basically maximum. That means it's an over two So in that limit this value this kind of you can you can perceive as if an entanglement density Which is incorrect intuition, but because I'm going to explain it in the next five minutes But let's still call it entanglement density and then that's it will be one half Just because we are cutting into the half like half and half If this is not half and half like one-third and two-thirds and that alpha will be one-third just that means that at maximum So this alpha will be one because it's an entanglement to people with basically the size of the subsystem Q equal want zero case, but Q equal non zero case. It'll be lower At some point, you know, not just developed volume novel say area. Yes, excellent question Yeah, but can I default the question because I want to have a proper discussion about that point Okay, so this has a dynamical notion because it's a phase transition about how entanglement developed over time However, it doesn't really need to have to define the characteristics in time because we can take the T Going to very large and they consider steady state for the finite size system Right. So one way to characterize this is just getting the swap. Okay here the slope is zero here The slope is not zero. So in the thermodynamic limit and goes to infinity first and then you will develop Indefinitely, that's one approach or choose n to be finite and then take the time goes to infinity first and then talk about the Saturation value. I'm taking the second route Go to the time infinity first before n goes to infinity and then talk about saturation value and Compare how what ratio to the total system size because we take time infinity first We don't have to worry about, you know, characteristic timescale. It's not universal. It's just some number I don't even remember the number because it's not universal. It varies over like details For example, like what ensemble unit are you use? Or what geometry you use? See if you can introduce like next nearest neighbor unitary or like long range interaction unitary The QC depends on a lot of different factors. No worry to make scaling. Yeah Absolutely, like so it looks like so the question was when Q equal QC It has a signature of a continuous phase transition like typical to other universality class So the natural question is what is a universality class of This problem we have signature that that this universality class is a particular and not a particular It's a it's a non-unitary CFT However, we cannot pinpoint which universality class it is. It's open problem and we want to understand that And my second lecture will be basically on efforts to the figure that out even though it doesn't give us the answer But but it develops a pretty good intuitions. Yes. Yeah, the questions okay This is perfect and I want to open up one discussion about These things let me just leave it here. It's not important. The randomness is not important. Yes Yes, exactly. So, okay, is this phenomena does this phenomenon require randomness? No, if you consider particular chaotic spin chain Hamiltonian dynamics and do basically just even continuous time evolution But you randomly perform measurement spread over space time I'm not sure whether the particular numerical simulation is done But there's something similar has been done and this phenomenology can be recovered So this phenomenology does not depends on the randomness in the random-unitary circuit However, random-unitary circuit will be useful because we can carry out a calculation But I want to get into this, you know number two The non-local effect in some sense if you think about this it's a very weird and let me explain why it's weird So intuitively this random-unitary circuit is Antangling systems one-on-one so this is roughly speaking our entanglement production on the other hand this measurement Is basically what destroys the entertainment? So I think I'm in destruction and it's kind of natural that we have a competition Okay, so entanglement developed and they you just kill the entanglement by just performing measurements and there's a competition Whenever there's a competition you could have a second or the phase transition a continuous phase transitions and critical behavior So it may be not so surprising, but let's think a little bit further Here we are talking about the steady-state behavior So at the steady-state on average over time the coarse-grained picture the entanglement will not change So average change in the entanglement across certain cut will be zero I'm building up the arguments based on this intuition and then I'll lead it to the contradiction So the entanglement change will be zero and this entanglement change is coming from the two factor Which is an entanglement increase by random-unitary circuit And there's entanglement decrease by measurement and They have to be equal because this is a balanced in the steady-stage okay Roughly speaking Renting an increase by random-unitary circuit goes by following if you cut the system into half say say this is a and this is a bar We are talking about how entanglement entropy is developing Due to the unitary evolution if I apply any unitary like this guy Which is entirely supported with any bar or within a Entanglement entropy does not develop because we are talking about correlation between across this cut But not correlation among each subsystems so the entanglement Entropy between a and a but grows only through the unitary Applied across the cut in 1d. There's only one boundary area is order one therefore per time These entanglement growth is an order one constant number in the d-dimensional system It will be d you know d minus one dimensional object only on the perimeter So it is so called a very a lot scaling behavior. However, if you fulfill measurements We are completely disentangling those degrees of freedom from the circuit So every time step in the flow mode and I'm living and it's very very large, but fine I you measure some fraction of qubit like two times and that's like substantial number So and after the measurements those entanglement will complete get completely You know this entanglement from the rest of the system Okay, so if we use for example this entanglement density of a tilde as a Intuitive notion Entanglement reduction will be entanglement density basically entanglement per particle times Q times n which is the or na which is a number of particles that's measured at each time step on average Okay, and because they are completely disentangled from the rest of the systems We can imagine that entanglement reduction will be proportional to the density per particles q times na number of particles Okay I'm explaining the wrong thing just to illustrate what goes wrong So if what we want is this difference to be zero What it means is so each order want some constant. So let's say the constant What it means is alpha tilde Scale roughly some constant over q times na Or in other words like alpha star alpha tilde times na scales as constant over q So what it means on this q is zero These entanglement has to be the order one independent system size So you can always have this area law entangling phase unless q equals zero So it could be large value, but strictly speaking any finite q you should have finite or Venetian entanglement density or finite entanglement of the subsystem These implies No volume of face. I want to put a pressure mark because in fact this is not true No, we do see very strong signature from numerical simulations that this indeed happens and Also, now we have a better Theoretical descriptions of how to understand the volume of entanglement state Okay, and that's associated with the subtle nature of no local aspect of entanglemental p And I'll spend the rest of the 10 minutes explaining what that means So any questions here? The the let me tell you the conclusion We should not treat entanglemental p as if it's a local observer It's not like energy density. It's not like charge density. It's not local observable It's an information theoretic content. It's an information theoretic quantity that characterize a amount of amount of correlations And also measurement is not a unitary operation. It's a non-local in operations That that our terms of wave function in a very subtle way and To illustrate that I'll give you a specific example Let's consider a quantum circuit diagram. So we are going to have three qubit The first qubit is in a so-called a plus state, which is a superposition of zero and one Second qubit is also in the plus state and the third qubit is also in the plus state. Yes question Of course, it's not observable. It's not a Hermitian observable. So it's not an observable in In the in the in the technical sense You can see you can but you can ask Okay, so that is to be in the realm of research, right? So who cares about the existing definition, right? Just modify the definition observable like can we say entanglement is kind of generalized observable And can we can we consider it? Is it like it as a plain language English? Is it observable? You can ask the question, right? I have personal opinion and my opinion is no However, you know, I'm not 100% sure maybe 99% sure but that's my personal opinion. That's a very good research In fact, that question is partially studied in computer science So they ask about complexity Computational complexity, but another thing is so-called a sample complexity So what is the number of repetition number of measurements you need to do in order to estimate a month of entanglement in The worst-case scenario that's exponentially difficult in the value of entanglement and that's a provable statement So now we are kind of entering the volume. We are you know, we need to talk about complexity You know, so it's observable in principle, but if you only repeat exponentially many measurements Maybe in some cases you have a smart choice. So it's not the worst case So we physics is deals usually about typical case better than worst case The typical case might be actually better than the worst case under certain structured case But that's I think the excellent research project. Yeah, okay So we are going to consider a particular quantum circuit By applying so-called control Z gate. So it is a CZ gate So what CZ gate does is the following If this qubit is zero, I don't do anything here, but if this gate is one, I'll play Z gate here And it turns out that's the same as the other way around if it's zero I don't do anything or apply identity if it's one I apply Z gate the Pauli Z gate So that's the CZ gate. I do the CZ gate here and then obtain the wave function and this wave function is Entangled between these two qubits due to this and this gate and also it's entangled between these two qubits because of this gate so if you cut the system into Part a and the complement a bar before measurements As a is going to be Log two Or if we choose a log base two, it's just equal to one That means there's a one qubit and that qubit is entangled with the rest So it's a maximum value for this minimum example before measure We will talk about what happens we perform measurement And specifically we are going to measure this guy in the middle in the X basis so what I mean by X basis is you can understand in two different way apply the Cubic rotation so X becomes Z and then measuring Z basis Okay, but to simplify my argument. I just measured X basis that means measurement outcome could be either plus or minus Even though this startup plus state because it's a being entangled Locally these wave function will look like maximally mixed state Therefore, it's not obvious that you will get plus stage because of the entanglement with the other one In fact, we can easily show that probability of measuring plus and probably measuring minus equal and that's a 50 50 For this particular circuit Let's analyze what happens to the wave function to these systems. Okay, so that's do that Before measurement our wave function can be understood in this way I'm going to divide into two case whether this is zero or whether this is a one Okay, if it's zero you don't apply here and you don't apply here Therefore the wave function is going to be plus on the first qubit and zero in the middle qubit and Plus in the third qubit If the middle one is one and Then you apply Z gate and if you apply Pali Z you flip the phase here So this whole state becomes a minus state And then you have a minus state for the first qubit and one for the second qubit a minus state for the third qubit But we are starting from the superposition of those two branches Therefore your wave function will be superposition of these two branches This is a wave function before measurement, but let's see what happens when you perform measurements when you perform measurements 50-50 random, so let's just choose one and then analyze that particular case. Let's choose a plus one the plus state what we do you get is wave function on normalize is Identity on the first qubit and project to the plus for the second qubit and identity for the third qubit acting on This wave function, okay So let's compute that this plus state is a superposition of zero and one right One way to analyze this is you have this psi, which is a superposition of zero and one So one is in the zero one basis for the second qubit. He's a plus minus basis for the second qubit So let's just choose one. So I decided to work on the zero one basis. Okay, so This branch of the wave function if you act on this projector What happens is these two parts first qubit and third qubit will not be affected because it's identity However, which project to the plus state the overlap between the plus state and zero is one over square root two Okay, so what we get is this will give you One over square root two plus plus plus This plus state because we have a plus here these plus because we have a plus here This plus because the plus here and one of our screw to because we have an overlap between this plus and the zero Which is one of the screw to okay in superposition of this branch But here the overlap between plus and one is also one of our screw to so basically we have the same thing one of a screw to minus plus minus and Then we have an overall factor one of our screw to coming from here Okay These can be rewritten in the following way just pulling out these plus in the middle This is the same as one over screw to one over screw to Plus for the first qubit plus for the third qubit in superposition of minus for the first qubit and mine for the third qubit And plus for the second qubit Okay, so this is not normalized because of the extra one of a screw to two and Then makes the norm of the wave functions norm squared to be one over two That's makes sense because of probability of 5050 But condition on that particular way function. Let's normalize by eliminating this factor What we find is actually Even though the second qubit is completely disentangled it's in the product state Actually the first qubit and third qubit are maximally entangled it's in a bell state and one of the bell state right, so interestingly even after the measurement Entanglement of subsystem It's still okay. So you perform measurements. You completely disentangle the middle one However, you do not disentangle subsystem a from the subsystem a prime eight a bar What's going on? So what's going on is a following? Entanglement is a concept of correlations. We are talking about how a and a bar are correlated here It's correlated through the middle qubit When you perform measurements The environment or the whatever the person who performs a measurement is trying to destroy the correlations However in quantum mechanics if you perform measurement in a particular basis You do not reveal any information about the correlations and then such a information can be protected and This is precisely the idea of quantum error correction So I need to wrap them in two minutes. So I'll just sketch on it at a high level So what is quantum error correction? The quantum error correction is roughly this Suppose I have some all-known quantum state that I want to protect How should I do? I? bring extra physical ingredient the degrees of freedom and Then apply some unitary you encoding Such a way that Even if I throw away some fraction of qubits on the output in other words I lose access to them because I lost them. I Can still recover information of a side from the remaining ones Okay, so pictorially. Okay. I just Throw you know throw away. So there's a trash box here and trash box here Nevertheless, I can apply some kind of decoding unitary so that we have a we can recover this side out if you can do this for the large fraction of Disposal of qubit. It's a good error correcting code Because we can protect the information against many, you know a large amount of error Right. So designing good error correcting code is a big deal in quantum information science and quantum information theory Here we are having a particular instance of phenomena where instead of throwing it away We are performing measurements and measurement is different from throwing it away Because if you measure you get the measurement outcome and you know what the measurement outcome is But throwing it away You just threw you don't have any access still Measurement can be understood as some type of error Because by performing measurements you collapse away function and this quantum state essentially becomes classical either zero or one What it means is any coherence any superposition Associated with this particular qubit is gone and we only have access to classical information So some degrees of quantum information is eliminated But not all information is eliminated because the classical bit zero or one is retained and you can read it off So measurement is a special type of error that we can talk about in these specific settings And what happens is in random unitary circuit? It's not a competition between entanglement production and entanglement destruction it is competition between Entanglement hiding like encoding like how you how fast can you do this encoding in the random unitary circuit? Versus how fast you reveal the informations by performing the measurements? So that took some time to realize in this community Where we we figure that in the general unitary case such as random unitary circuit in fact this error correcting property emerges Naturally and therefore this error correcting property is what allows us to have this volume length entanglement scaling Yeah, I think the time is up like maybe any last questions Okay, so let me continue on the lecture to on a more technical side. Thank you