 Go ahead. OK. Can you hear me OK? Yeah? Fine? OK. So thank you for being back. This is the second installment of this crash course on non-equilibrium quantum, say, thermodynamics of quantum processes. And what we did yesterday was basically focusing on these two objects, right? So on the work probability distribution, and towards the end of yesterday's discussion, we had a look at the characteristic function, the associated characteristic function. So the take-home message from yesterday's discussion was that in this non-equilibrium framework, work becomes a stochastic variable. Distributed according to this probability distribution, p of w, w is the actual amount of work that you are doing on the system or that the system is performing. And this is actually equal to the difference in energy between the difference between the two outcomes of the measurements that we have performed yesterday in the so-called two-measurement process or protocol. And on the blackboard yesterday, we went through the formal derivation of this closed expression for the characteristic function of work distribution. So just to somehow refresh what the ingredients of this expression are, rho g is the initial thermal state of my system. And I'm going to comment on this in a second. While u is the time-evolution operator induced by my Hamiltonian process, I remind you that in our framework, the environment has been detached from the system right before starting the protocol itself, right? So right before performing the first measurement of my two-step protocol, U dagger is the Hermitian conjugate of this time-evolution operator. H is the Hamiltonian of my process. And lambda is the, in general, time-dependent parameter that I change when I want to perform work on the system itself. And the way I'm changing lambda is completely arbitrary. I'm not setting any constriction on that. I'm simply changing from a value lambda naught to a value lambda tau in an arbitrary fashion. U is the conjugate variable to work that enters into my characteristic function. Now, yesterday afternoon, after the question and answers session, a couple of guys asked me about the assumption of initial thermal state for my system. And this is, I mean, a perfectly meaningful point that the guys raised. Do I need, so the question is, do I actually need to start from an equilibrium state of my system? And the answer is absolutely no. I can start from any state, and that will be absolutely legitimate and absolutely fine. The apparatus of building up, designed to build up the work probability distribution, would work with any, so it would work the same, the same fashion in the same way, regardless of the initial state of my system. The only need for starting from an initial thermal state is that, to some extent, it simplifies the calculation that I did on the blackboard yesterday. And on the second hand, on the other hand, this assumption we are going to see is very important for fluctuation theorems. And this is what we are going to see in a second. So this is somehow recapping what we have seen yesterday together, and we now move forward. So the plan for the next 45 minutes, at what time do we stop? Fabrizio, at what time am I supposed to end up? 10 past? OK, so 10 past, 10 past, 12, OK. So what we are going to do in the next 45 minutes, 40 minutes, is to go together through what are called fluctuation theorems, very briefly, and then assessing a way, say, a protocol that allows you to gather information about these two quantities that we have, these two objects that we have in this slide, the probability distribution for work and the characteristic function. And then towards the end of today's lecture, we dig into what Martin has anticipated this morning. So the implications are a brief discussion of Landau principle and the implication of these framework for non-equilibrium thermodynamics for the exchange of heat. So we are going to open the box into which we have placed our system. OK, let's start. So again, these two objects you know, and you now meet two of the most celebrated fluctuation theorems in this framework of non-equilibrium thermodynamics. These are called Yajinsky identity and the Tassaki-Krux identity, OK, or the Tassaki-Krux relation. Let's try now to. Yeah, so historically, Yajinsky found a relation for the classical stochastic case, stochastic thermodynamics case, and Krux derived this relation, again, for a classical process. Later on, and I think this paper is the generalization of Yajinsky to the quantum case, later on, Yajinsky extended the argument to quantum dynamics. And Tassaki proved Krux, the validity of Krux relation, also in the quantum domain. Now, this is my take of it. Now, I've met recently Al Tassaki in Granada, probably at the end of June. And it told me you shouldn't call it Tassaki-Krux. What I did was simply to put together two papers. And I found the relation, right? So I didn't do anything. He said the merit is all on Krux. I still think that the contribution was interesting and relevant to me, as well as for the community, John. So yeah, it's typically portion, yes. In fact, that's what Al Tassaki says. So there is this paper by Krux and another one, and Krux one, that he put them together and he got the relation. So historically, these two relations were found for classical processes and later on extended to the quantum domain. Now, see, anecdotes apart, let's try and see what is the physics that these two relations entail. And then very briefly, I'll go through a derivation, a very quick derivation of Yajinsky's identity while I leave the derivation to the proof of Tassaki-Krux to you as an exercise. It takes two lines, two lines of calculations to do it. So, Yajinsky, so let's have a look at this relation, yeah? So it's pretty, not pretty elegant. An identity is nice. It involves the expectation value of an object. This object involves work, the work that you do on the system, this stochastic variable that we have introduced yesterday. However, it also encompasses an element of, say, exoticness, if you want. You have to take the average, not of the work itself, but the average of the exponentiated work, right? So the left-hand side deals with this e to minus beta w. Beta, I remind you, is the inverse temperature set by the bath with which the system was in contact at the beginning of our process. Well, the expectation value of this quantity is equal to e to minus beta delta f. And delta f is the change in free energy of the system, right? So a couple of points of notice. First of all, this object, the change in free energy. You open, again, not the usual standard books on thermodynamics. Change in free energy is defined between two equilibrium points, right, to equilibrium states of my system. So these entails that my system asks us that my system started from an equilibrium state, should end into an equilibrium state, right? In order for me to calculate this quantity in a meaningful, in a meaningful fashion. However, as I said yesterday, and I repeated it briefly, five minutes ago, I'm not prescribing anything about the way I'm kicking the system, right? So I'm not putting any constraint, any requirements, on how gently or hardly we are pushing the system that we want to bring out, that we want to let evolve. So more generally speaking, my unitary evolution, so the unitary evolution that we subjected the system to, can occur in finite time. So it's definitely not necessarily a quasi-stactic transformation, which will take an infinite amount of time, not a very slow process that takes your system from an initial to a final configuration, leaving it at thermodynamic equilibrium every time, right? It's not necessarily like that, which means that if I now borrow a standard cartoon that you would see in all of these papers on non-equilibrium, non-equilibrium stochastic thermodynamics, both classical and quantum, well, say if this dot represents my initial equilibrium state, right? So this is the thermal state at time t equal to 0, not the initial state that we have prepared in our protocol. And if this meaningless line means not the evolution induced by my unitary process, right? This is the kick that I'm doing to the system that I'm, say, subjecting the system to. In principle, and this observation was also somehow hinted up by Fabrizio yesterday, I end up into a state rho at time tau, right? So the arrow of time is here. This is the final time of my experiment. This is the initial one. Can you see it? Yeah, guys? Down there? Yeah, or up there? Yeah, fine. In principle, there is no guarantee that this state is the equilibrium state at time tau, right? So there is no guarantee that this state is an e to minus beta h at time tau over the partition function at time tau. Yeah? Why should it be like that? I'm kicking the system, and I'm doing it in an arbitrary, in general, an arbitrary way. So I could be doing it very, very abruptly, or extremely gently. It's still allowed, right? So it's both cases, both sides of the spectrum of kicking, so to say, and compassed by the framework that we have illustrated yesterday. So this means that there is an equilibrium state somewhere in this abstract and stupid space of states that differs, in general, from the final state of my evolution. Well, this change in free energy, this object, delta f, this change of free energy is defined between these two points. So it's the change in free energy between the initial equilibrium state of my system and the hypothetical equilibrium state of my system that would be reached by a perfectly adiabatic, so quasi-static transformation. Makes sense? Yeah? So I'm dealing with three enemies here. The initial state, the final state, the true final state of my evolution, and the hypothetical equilibrium state of the system at time tau. The change in free energy is defined between these two guys. It doesn't involve rho tau because I cannot use rho tau. Rho tau is a non-equilibrium state. It doesn't make sense to use it to calculate a free energy change. So the right-hand side of Yajinsky, it's a fully equilibrium statement. It involves quantity well-defined at equilibrium. That's it, full stop. The left-hand side, on the other hand, well, I mean, I'm taking the average of a quantity, right? So I don't need to do that. But I'm just writing explicitly what this quantity is, right? So this is e to minus beta w, the forward probability distribution for work, dw, right? So I'm taking the average. Please. I apologize. I'm deaf. I didn't hear you. What average? What average? What temperature? What temperature? Sorry. Yes. Very, very relevant question. Is the same temperature of the initial bath, of the bath that I've used at the beginning of my protocol? So somehow, Yajinsky is hiding under the carpet the assumption of an isothermal transformation. OK? I pre-empt the question, can I go beyond this assumption? The answer is yes. And I can look at transformations that are not isothermal as well and come up with modified different alternative versions of fluctuation theorems. And tomorrow, we would see also different quantifiers of irreversibility based on this fluctuation theorem. OK? Very good. Very good. The answer is, to some extent, yes. But the framework becomes extremely more involved. And in general, what happens is that you have a fluctuation theorem with a correction term. So these two terms stay there, are they, in principle? But you have to add another term here, to this right hand side, that accounts somehow for the exotic nature of your bath. OK? So the answer is a partial answer. And I cannot go more quantitatively than what I said. But the answer is that, to some extent, you can't treat non-equilibrium environments. I think an explicit case was a recent one. It's a paper by Parondo Zambrini and someone else, which I don't remember. They're looking at a non-equilibrium process of an harmonic oscillator in a couple to a squeezed bath. OK? And they derive the explicit form for the corresponding fluctuation theorem. OK, so if I can go back to that, this is the left-hand side of my fluctuation theorem, which means that I have to account for the probability distribution for work. And this probability distribution for work encodes the, in principle, non-equilibrium nature of my transformation. So in general, yes, Martin, the average is, of course, encompassing all the possible trajectory connecting the initial two. The probability distribution accounts for all the dynamical trajectories, because it puts together all the possible processes, say, all the possible transition probabilities connecting your initial equilibrium state to the state that you reach at the end of your dynamics. Let's look into that later on. I think is that clear so far? Yeah, OK. So I have something that is explicitly encompassing the possibility for non-equilibrium and something that is defined as equilibrium. So what this relation is telling you is that on average, so that quantity to minus beta w would be entirely determined by equilibrium properties of my system. The non-equilibrium nature of my process gets washed out somehow by the average. And what matters are the equilibrium properties defined between this initial state and the hypothetical one that I will reach by a quasi-static transformation. Second relation, so the Tassaki-Krux one, on the other hand, is a little less intuitive now and introduces explicitly the backward process. The process that we, yesterday I intered at, but we didn't go into detail. So we know already what Pfw is. What is this Pb of minus w? This is the analogous probability distribution, but for the backward process. For the process where instead of starting from this initial state and evolving, so to say, according to this evolution and ending up on my final state tau, the backward process is the one where I start from the equilibrium state of my final Hamiltonian, evolve according to the time-reversed evolution operator towards a new final state. Does it make sense? Yeah? So this time, so this is forward, so to say, and the backward one, the backward one would entail the preparation of an equilibrium state of the final Hamiltonian, precisely this object, time-reversed, with the application of the time-reversed operation to it. The evolution according to, well, I shouldn't do it. I shouldn't make it to similar to the previous one. OK, the evolution, well, why not? The evolution according to the time-reversed process and not which will bring you, which will lead you to another final state, right? So associated to this process here, there is a probability distribution Pb of w, which you can define in an analogous way to the Pf that you have there. No, you have to account for the transition probabilities from a given eigenstate of the final Hamiltonian, which now becomes the initial one, to eigenstates of the formally initial Hamiltonian, which becomes now the final Hamiltonian for your backward process. So what the second crooks are telling us is that, say, the probability of performing work in the backward process is somehow exponentially suppressed by a quantity that depends on w, the stochastic value, the value that the variable takes for that specific trajectory and the change in free energy. For those of you that are interested in irreversibility, these are very nice implications for the concept of time reversal. So it doesn't seem to be symmetric. It doesn't look like a symmetric statement when you reverse time. You pay in a different way. The forward and backward process allow you to pay in different ways. OK, these two relations are very elegant, extremely nice. By the way, you can rephrase them in terms of the characteristic function for work distribution in a pretty easy way. And we are going to see it in a second. So given that pf and chif have the same predictive power, so to say, they contain the same information on the process itself, you can restate these two relations, these two identities in terms of chi. So now, in a minute, let's try and have a look at how we can indeed make sense of the Yajinsky identity in a formal way. So what I need is that, why don't I use it here? So what I need is this quantity. Now, if I want to evaluate this left-hand side of the Yajinsky identity, what I need is this quantity here. And now what I want to do is to notice the similarity between the quantity e to minus beta w average and the characteristic function for work distribution. So if I look at the characteristic function for work distribution, I have that. If I replace u with an iw, right? So if I calculate chif, sorry, no, iw, i beta, if I look at chif of i beta, chif of i beta is precisely integral of minus beta w pf w dw. Yeah, trivially. So chi of i beta is precisely the expectation value of e to minus beta w. Let's dig a little bit more into what chi of i beta is. Because I have not in this slide, but what we did yesterday and what was in the previous slide is that chi of u can in general be written as the trace of u e to minus i u h naught times my equilibrium state at time t equal to 0, which I'm calling rho eq naught, times u dagger, times c to i u h final. Guys, can you still read it? Yeah, are you OK? Yeah, OK. Now I want to replace u with i beta in order to work out another expression for this quantity, right? So chi of i beta, according to this expression, will be trace of u e to beta h naught, equilibrium 0, u dagger. And then I have e to minus beta h final. And now what is the equilibrium state of my initial Hamiltonian? So I write it here is e to minus beta h naught over the initial partition function. So I can plug it into this expression. Again, I do something that I shouldn't do. No, I'm doing it on this expression. So I'm deleting this guy and replace it with e to minus beta h naught over zeta naught. And of course, z naught is a number, so I can take it out of my trace. And then what I do is that I can notice that these objects and these objects simplify with each other. There is h naught in both. So this expression just gives me a 1 over zeta naught trace of u dagger e to minus beta hf. My process is unitary. So these two guys cancel. It should give me the identity. So I get the trace e to minus beta h final, which is the definition of the partition function for the final Hamiltonian. Is it the wrong one? With the, what do you mean with the wrong one? What is z? This is precisely the zf, no? So zf over z naught is what I get. And now group picture, shall I leave it? Yeah? And now, by definition, the free energy is defined as 1 over beta, the log of the corresponding partition function. So what I have is that the free energy of time tau is this object, the free energy of time 0, given that I'm still at the same temperature. Is this object so I can get, no? I take the exponential left and right-hand side. And what I get from these two conditions, from these two equations, is that the ratio between zf and z naught is precisely the exponential of minus beta times f tau minus f naught, which is my delta f. So on one hand, the characteristic function, and this is still kf, so I forgot to put the label there, the characteristic function at i beta is given by this e to minus beta delta f. On the other hand, kf of i beta is equal to the expectation value of e to minus beta w. So you put the two things together, you get Jajinsky. From this non-equilibrium starting, from this non-equilibrium framework, and the two measurement protocol that we have used. And when I said, for Jajinsky, it is actually very important to start from an initial equilibrium state. Well, you can appreciate it from here, from my new passage, right? So the passage that allowed me to replace this guy with my e to my Gibbs distribution, no? OK. So to say, these are not the only, the only statements, the only factation theorems that you can draw. There are other versions, but historically and also somehow, operatively these are the two that are most celebrated and somehow very useful for considerations that we are going to make later on, later on tomorrow. OK. I would like to raise one point, no? And the point is the following. So it's actually not my own question, but it is a question that was raised by some of the people that came up with the paper that we commented upon yesterday. And also, fluctuation theorem work is not unobservable. So the basin group, the Osberg group, is people, say, pointed at something that was right before your eyes. So the fact that it is, in general, difficult to infer the probability distribution for work, the characteristic function, so to get the statistics of this non-equilibrium quantity. This is, in general, a tough goal. And in this sentence, in this paper, in this review paper, they claim that, say, the major obstacle for the experimental observation or verification of the work fluctuation relations is posed by the necessity of performing the projective measurements of energy. And also, in the two-measurement process, you have to measure on eigenstates of two different Hamiltonians, non-commuting Hamiltonians at the beginning, in general, at the beginning and at the end of your process. And if you think about, say, somehow, the paradigm for a working medium, which is a complex system, in general, for instance, a many-body system, right? And if you want to not think of a gas, and if you want to export that into the quantum domain, you might have to project onto many-body, in general, entangled states of this complex working medium. So it might be, on its own, a very difficult task to accomplish. And truth was that, truth is that up until a few years ago, there was a substantial body of experimental work on the classical verification, so the verification of fluctuation relations at the classical level, but very little done at the quantum level, experimentally. Theoretically, there were proposals. There were a few, say, most noticeably, one by the group of Eric Lutz and Schmidt Kalle dealing with trapezoid ions, trapezoid ion technology. So precisely, the context that was illustrated yesterday by Christoph, the proposal brought about some difficulties was rather involved, a bit far from what they could do at that time in the lab. And it did see light in a way or another, not in a, say, changed version more recently by a Chinese group. But until a few years ago, there was no, basically, no verification of fluctuation theorems at the quantum level. And the difficulty was precisely, put your hands on the probability distribution for work or the characteristic function. So the question I want to raise is, how can we reconstruct for a given process, how can we reconstruct the work probability distribution for a general process, for a general process undergone by a quantum system? And the answer, a possible answer, a possible answer comes from this scheme, from this diagram. Now, I don't know how many of you are familiar with quantum circuits, but we are going to, I'm trying to illustrate what the ingredients of these circuits are in a second. So let's have a look. I should stress this is not the only way of putting your hands on the characteristic function of work distribution or on the probability distribution for work. In fact, I say another case that you might be interested in is provided by the group of Juan Pablo Paz in Buenos Aires, where they made a very nice observation, which is that the two measurement protocol that I have illustrated yesterday can be actually understood in terms of a single generalized measurement performed on the system. So not a projective measurement, but a generalized measurement. And they come up with a protocol for the inference of the probability distribution by starting from that view point, from that standing point. So we are sticking with the two measurement protocol as far as this slide is concerned. And this is the way, the proposal for the reconstruction of the characteristic function of work distribution. So for those of you that are used to quantum optics, you can understand this guy easily in terms of a very standard, I mean, very basic interferometer, nothing else. This is actually a phase inference protocol or circuit, nothing else, phase estimation protocol. So what do we have here? S stands for my system. So the system upon which I want to perform work. So it's the guy that I've kicked so far, so to say. A is someone, it's a secondary system, which we call an ancilla, so an helper, someone that helps us performing a given task. So from this point on, I'm not just focusing on my system, I'm looking at the system and the second subsystem that will help me reconstructing the probability distribution, actually, the characteristic function for work distribution. And the general underlying assumption is that I can do anything on the ancilla. I have the freedom, the luxury, so to say, to prepare it in any state I want and to measure it, to measure it on any basis. So I can do what I want on the system. I also have, say, allowing myself the capability of engineering these weird object g. So besides the exotic drawing that you have there, g is nothing else but a joint unitary evolution. So it's a unitary operator that combines the degrees of freedom of my system and the ancilla. So this is a unitary operation that lets the two guys interact, makes the two guys interact and evolve in time. On the other hand, h here, now these two guys is not for the Hamiltonian, now stands for what is called a Hadamad gate. A Hadamad gate is, again, is a transformation that allows us to create superpositions of logical states of a qubit. So here I'm assuming that my ancilla for convenience, that my ancilla is a simple two-level system. So A lives in a Hilbert space whose dimension is two. So I only have two logical states, and I'm labeling them as 0 and 1. Precisely as Martin did. H, what H does is that it acts on 0 and transforms it into a superposition of 0 and 1. And what it does on 1 is that it prepares the orthogonal superposition. And this is called Hadamad transform. So what I'm doing here is that I'm starting from a fiducial state, right, from a state that I know I can arrange easily in the lab. And I create one of such superposition. I create 0 plus 1 for convenience. Of course, you can replace this all first block here with the preparation of this state only. And that will be perfectly fine. And then what I do is the following. I prepare my system S in the initial thermal state that I was interested in, or in the initial state that I'm interested in, in case I don't need a thermal state, right? So S is prepared independently on its own initial state. And after I created coherence in the state of the ancilla, I let these two guys evolve together in a specifically arranged manner. So let's have a look at what G is. And I'm writing explicitly the fact that G depends in general on the time tau, my time of my evolution, and on the variable u, the tension in my characteristic function. So this is the conjugate variable to work. So let's have a look at what G does. Suppose that the ancilla is prepared in 0. Suppose that the ancilla is in the 0 state. Then if the ancilla is in 0, then the system evolves according to this part of G. Suppose that in the other end, the ancilla is prepared in 1, then the system will evolve according to this other bit of my joint operation G. So in an even more cartoonish fashion, what I'm doing is that I'm splitting. So this initial point is put together system and ancilla, each prepared in its own state. After the Adamard gate is as if I'm performing two different evolutions of the system depending on which logical state the ancilla is in. So if the ancilla is in 0, then the system will evolve according to u e to minus i u h i. And if the system, sorry, if the ancilla is in 1, the system will evolve. Again, this is the system only. The system will evolve according to e to minus i u h final times u dagger. No, times u itself. Yes. So here I need the initial Hamiltonian. Here I need the final Hamiltonian. And then what I do is that I do, again, an Adamard transform. I redo my Adamard transform. What the second Adamard transform does, I mean, it's completely immaterial, but it's there simply because it's nice somehow, is to recombine, not let it interfere these two path, these two evolution path at the end. So it's just like some form of interferometer that people, for instance, in quantum optics labs would implement, in optical labs will implement, in an optics lab will implement. So the system evolves into different madness depending on the state of the ancilla. At the end of the protocol, and this is the nice part, I don't care about the system. I discard the system. I don't even look at it. So here you can put your bin and throw the system into the bin. So mathematically speaking, I trace out all the degrees of freedom of my system. And what matters is only the ancilla. So what I do is that I measure, I try to reconstruct the state of the ancilla. And if the ancilla is just a qubit, that is a pretty standard operation to perform into the lab. So I can reconstruct with experimental uncertainties, but I can reconstruct with some confidence the state of the ancilla. So I'm bypassing somehow the problem of inferring the properties of the system into the reconstruction of the state of the ancilla. In some cases, this might be an advantage. In particular, if the dimension of my ancilla is low, right? OK, so we have five minutes before the lunch break. Is the lunch break from now? No, there's another. Oh, there's Christoph Lecture. Yes. So in these five minutes, I would like to go through the calculation that allows me to show you that at the end of the protocol, or maybe I'll just bypass it, and you believe me, and you trust me, and if you are interested, we can go into the details together. So at the end of the protocol, so when I discard the state of the system, when I don't look at the state of the system and I reconstruct the state of the ancilla, I notice that the state of the ancilla, in the basis 0 of 0 and 1, in the basis of 0 and 1, can be written in this manner. So let's write it explicitly as a matrix. I don't need this diagram. So if I want to write down this density matrix, rho a, what I find is that it takes a 2 by 2 matrix because I have assumed that my ancilla is just a qubit. What I have is 1 half here on the diagonals, and this is the identity, right? Then I have this quantity, not this Pauli matrix, sigma z, times alpha. And alpha is the real part of my characteristic function. So what I have here is a real part of chi over 2 and minus real part of chi over 2. While on the off diagonal elements of the density matrix, I have a beta, unfortunately, sorry guys, this might be confusing. Here beta is not the inverse temperature, but is the imaginary part of the characteristic function. Sorry, I'm noticing it here after three years. Sorry. So here I have minus i over 2 the imaginary part of chi and i over 2 the imaginary part of chi. OK, so the state of the ancilla, the state of the ancilla encodes information on the characteristic function. If I reconstruct what is called, say, the longitudinal magnetization, which is the expectation value, basically of sigma z, and the transverse magnetization, which is the expectation value of sigma y, I have full information on the characteristic function for work distribution. So by simply acting on the ancilla, by simply reconstructing the state of the ancilla, I would be able to get full information on a protocol that is arbitrarily complicated. So u is given there. When I implement this object, I don't have to worry about what u is. You gave it to me. Make sense? So no matter how abrupt my process is, how violent my process is, or how gentle the kicking is to the system, so no matter how stronger the non-equilibrium features of this guy, I will be able to reconstruct them through this procedure. And this has been used in a couple of experiments that were in the quantum domain, inferring the statistical properties, the non-equilibrium properties, of a quantum system. So I think I'm on time. Fabrizio, shall I stop here? Yes, it's 10 past 12. I'm done. I think I can stop here, so it's a good time for getting. Yes, I think they are. It's a physical state, say. It's a legitimate state. I mean, no, no, no, no. Kai enters basically as a phase in your state. Yes, in fact, say, OK, just if I can add, if you avoid the second Adam R, then you write everything on the basis of z of plus and minus. It's a lot more clear, so that Kai appears as a phase. Yeah, I'll stop here. And questions?