 OK, we can see your screen. So it's a pleasure to introduce Nicole Jünger-Halpan, who will talk about what happens to entropy production when conserved quantities fail to commute. So we'll continue further into the quantum realm, and the stage is yours. Welcome. Great. Thank you. First, I would encourage everybody to stand up and stretch, because we've been sitting for a while. And while you do so, I'd like to thank the organizers for the invitation to speak here. I'm delighted to be with you virtually. This work is based on a paper that actually came out on the archive today. Let's begin. What do we as stochastic thermodynamicists do on a daily basis? In many cases, we'll think about two systems, A and B, that have inverse temperatures. If they're classical, then they'll be prepared often in canonical ensembles. And if they're coupled together, then energy can flow between them. And as energy enters or leaves a system, it produces entropy. The entropy production that we just heard about in the last talk is stochastic. It varies from trial to trial. And the probability that the next trial will cost entropy production sigma obeys fluctuation theorems. As we know, generalizations of the second law of thermodynamics. Let's complicate this story a little bit or generalize it. Our systems could be quantum. Also, they could exchange multiple things. These things are going to be conserved globally across systems A and B. So I'll call them charges. I'll denote them with Qs. We can associate each Q with an effective inverse temperature. Common examples of these charges are, as we've heard earlier in this workshop, energy and particles and electric charge. We can couple the systems together and let the charges flow as currents. They will produce entropy. Now what happens if the charges fail to commute with each other? This question is a particularly quantum thermodynamic question because the non-commutation of operators is a hallmark of quantum theory. This question, what happens in thermodynamics when conserved charges fail to commute with each other, has engendered a small growing subfield? If you're interested in the backstory of this subfield, it's in chapter 12 of this book that I published last year, actually for the general public, but that includes scientists. And my group has written a perspective for nature reviews physics on this small growing subfield that will be out on the archive this coming week. So what does happen if charges fail to commute with each other in quantum thermodynamics? Quite a few changes have been noted already. This is just a very small sampling of the papers that have been written. The community has found that derivations of the thermal states form breakdown. The average entropy, excuse me, the average entanglement across the whole system can increase. And according to this paper, entropy production can decrease in the linear response regime. This talk is going to concern what happens to entropy production when charges fail to commute with each other arbitrarily far from equilibrium. Here's where I'd like to go. First, I'll review how in conventional thermodynamics, we have three common formulae for entropy production and these formulae equal each other. If the charges fail to commute with each other, then we need to generalize these formulae. We can generalize them in reusable ways so that they satisfy several sanity checks, but these sensible generalizations don't equal each other just if the charges fail to commute. Furthermore, different ones of these formulae reveal different physical insights about what happens to entropy production when charges fail to commute. And this work kind of opens the door of stochastic thermodynamics to non-commuting charges, particularly quantum thermodynamics. How do we deal with stochastic entropy production in ordinary quantum thermodynamics? Let's consider a very simple case in which we have just two systems with two inverse temperatures so that they exchange just energy. We attribute to system A, a Hamiltonian HA with eigenvalues eaj and eigenprojectors piaj. We attribute to system B on analogous Hamiltonian. We often reason about stochastic entropy production using the two-point measurement scheme. According to its, we prepare each system in its thermal state. Then we measure each system's energy strongly. We couple the systems via some unitary that conserves the global energy. And then again, we measure each system's energy strongly. What is the joint probability that we obtain outcomes IA and IB, then outcomes FA and FB? A joint probability is a trace. It depends centrally on the initial states. These projectors represent the first measurements. Then we have our unitary and then we have a projector associated with the final measurements. It's useful to keep this cartoon in mind. But consider the space of all possible initial energy outcomes and consider the space of all final possible energy outcomes. In each trial, the system realizes a trajectory from a point in one space to a point in the other space. This trajectory is stochastic. It's selected according to this joint probability. We're going to need to generalize this joint probability. Before we do that, let's review the three common formulae for stochastic entropy production in ordinary thermodynamics. First formula, my group likes to call the charge formula because it depends on the amount of charge, in this case, energy that enters system A and the amount of charge that enters system B. The second formula is the surprising formula. We can understand that by imagining that we prepare the initial state row, then we measure the system's energies. We expect the outcomes IA and IB, but we obtain the outcomes FA and FB. How surprised are we? How much information do we learn? That amount is given by this surprising formula. Finally, the trajectory formula or what we call the trajectory formula depends on this joint probability that we just saw that the system will undergo a specific trajectory in a given trial, as well as a reverse probability that if we reverse, time reverse the evolution, then the system's trajectory between these two spaces will be from the outcomes FA, FB to the outcomes IA, IB. As we know from many papers, this two-point measurement scheme has some problems because the initial state could have coherences relative to the energy eigenbasis because it's a quantum state. So these strong measurements can disturb the state. Again, many authors have written about this problem. I've singled out this paper because it heavily inspired our work. We can replace the strong measurements with weak measurements. A weak measurement doesn't disturb the quantum system very much at the price of not extracting complete information about the system. We saw that strong measurements lead naturally to this joint probability distribution which equals this trace. Similarly, weak measurements lead naturally to this object in which this projector is gone. This object is a quasi-probability. A quasi-probability is a generalization of a probability. So like a probability, for instance, quasi-probabilities are normalized to one but they can violate some axioms of probability theory. For instance, famously, some quasi-probabilities can become negative. There are many different types of quasi-probabilities you've probably encountered the Wigner function but this quasi-probability is a Kirchwood Dirac quasi-probability. Kirchwood Dirac quasi-probabilities are my favorite quasi-probabilities in all of quantum theory. I have talks just about Kirchwood Dirac quasi-probabilities and I love them in part because over the past few years, they've proven immensely useful in many sub-fields in quantum thermodynamics, quantum metrology, quantum chaos, quantum foundations and more. As much as strong measurements disturb the quantum system in that story when just energy is exchanged between the systems, strong measurements are even worse when the charges feel to commute with each other. To see that, let me make things a little bit more concrete. We're gonna consider systems A and B that exchange C charges some of which don't commute with each other. For instance, A could be a qubits and B could be a qubits. Each qubit has X, Y and Z components of its spin and we can arrange the interactions so that the two qubits exchange the components, all the components of their spin and we can associate an inverse temperature with each component. If charges, the charges in our exchange story fail to commute with each other, then we can't do the analog in the two-point measurement scheme of measuring the energies initially and measuring them finally because we can't measure all the charges initially simultaneously and we can't measure them all simultaneously at the end of the protocol because the charges fail to commute with each other. Also in a reverse sense, if we measure the system seat, then we disturb the later measurements of the charges that fail to commute with what we just measured. We propose to address this problem with a slightly different protocol. We'll prepare the two systems in a state row such that each system is in a reduced state that's thermal. Then we weekly measure each system's first charge then each system's second charge and so on. Then we couple the systems with a unitary that conserves each global charge and now we measure each system's final charge then charge C minus one and so on. This new protocol leads to an extended TIRCWID DUROC quasi-probability. I am denoting quasi-probabilities with P-TILDAs. This quasi-probability depends on the outcomes of our initial week measurements and each I index here represents a couple associated with the outcome of measuring A and the outcome of measuring B. This quasi-probability also depends on the outcomes of the final week measurements. Since I have only 20 minutes, I don't have time to present the definition of this quasi-probability, but ask if you're interested, it's in bonus slides. So this quasi-probability can become negative and also not unreal. We're gonna take advantage of that non-reality. It's going to replace the joint probability in our analysis of entropy production. Here are three generalized formulae for stochastic entropy production that accommodate really quantum charges that can fail to commute with each other. Our charge formula is kind of like the charge formula I showed before, except here we have eigenvalues of the eighth charge. So this is kind of like how much of the eighth, excuse me, the alpha-th charge flows into system A and how much of the alpha-th charge flows into system B in something like a stochastic trial. And we have a sum over all of the charges as well as the adverse temperatures. We can understand the surprising formula by imagining that we prepare the global system in the state row. Then we measure each system's alpha-th charge. Doesn't matter which one you label as alpha, our results are general, doesn't matter how you label the charges. Suppose that we expect to obtain the outcomes IA and IB, but we actually observe FA and FB. How surprised are we? How much information do we learn? The answer is this surprising formula. Finally, our trajectory formula for the stochastic entropy production depends on this quasi-probability that I introduced recently. And also a reverse quasi-probability associated with performing U dagger instead of the unitary U. These formulate why should you accept them? For example, they satisfy four sanity checks. For instance, they all equal each other if the charges commute with each other. And also they satisfy fluctuation theorems in some cases with corrections that depend on commentators of the charges. Each formula gives its own physical insight about how non-commuting charges change stochastic entropy production. The charge formula tells us that we can think of stochastic trajectories as actually violating charge conservation, even though we observe charge conservation on the whole physically if the charges fail to commute with each other. The surprising entropy production has an average that can become negative if the charges fail to commute. And we know that negativity of the average entropy production is related to the flow of time. So we have the non-commutation of charges kind of enabling a resource for effectively reversing the arrow of time. Finally, this trajectory entropy production can become non-real, which might sound very strange. Entropy production is usually supposed to be real. But this is actually a good thing for us because this non-real stochastic entropy production signals a rigorous form of non-classicality in an experiment that involves currents of charges that fail to commute with each other. So let me give a little bit more detail about that result. We can express this trajectory formula in terms of weak values. What are weak values? They're kind of like conditioned expectation values. Suppose that we perform a procedure extended in time. We initially prepare a state omega. Then sometime later we measure an observable big X and we obtain an outcome little X. What is the value most reasonably attributed to a different observable Y in between these two events? Suppose that Y does not commute with X and omega does not commute with Y. What we kind of wanna do is take the information about the state preparation and propagate it forward and take the information about this outcome and propagate it backward and get something like a conditioned expectation value of Y. And since I have only 20 minutes, I don't have time to show you the expression for the weak value, but that too is in bonus slide. So feel free to ask for it. A weak value can be anomalous. It can lie outside the spectrum of Y. For instance, the weak value can be non-real. And if the weak value is non-real, then it signals contextuality in this protocol that I just described more or less with a weak measurement of Y in the middle here. Contextuality is a rigorous form of non-classicality. It enables quantum computers to achieve their speedups. What is contextuality? We can model quantum systems as being in microstates, similarly to classical statistical mechanical systems, microstates that we don't know. And you might expect that in this model, we would model operationally indistinguishable procedures by the same pieces of math. But if you want for this model to reproduce the predictions of quantum theory, then you have to model operationally indistinguishable operations by different pieces of math. So some extra context in the background that you thought would be irrelevant to your mathematical model actually turns out to be irrelevant. And the fact that this extra context matters is the contextuality of quantum theory. It's true non-classicality because classical theory is non-connectional. Though as I said, we can express our stochastic entropy production in terms of weak values. Namely, the entropy production equals the log of our ratio of the magnitudes of two weak values plus i times the complex phases of these weak values difference plus a log of some probabilities. I suppose that this stochastic entropy production becomes non-real. That means that this difference of phases is non-real. So at least one weak value phase is non-real. The weak value is non-real. So that weak value is anomalous. Therefore, there is contextuality. And we show that this contextuality is in a protocol in which charges that fail to commit with each other flow. So this stochastic entropy production emerges as one of the very, very few thermodynamic signatures of contextuality, rigorous non-classicality. I think that this work opens up some really rich results for really rich opportunities for future work. We can take any phenomenon in stochastic thermodynamics and ask what happens to it if the dynamics can serve charges that fail to commute with each other. For example, thermodynamic uncertainty relations feature currents. According to such a relation, the relative variance of the current is bounded with an entropy production. And non-commuting charges can decrease entropy production. So maybe they lead to particularly uncertain currents. Also engines have efficiencies that benefit from low entropy production. So non-commuting charges can decrease entropy production. Maybe they can provide advantages to the efficiency. And you can also take more or less any topic in this workshop and ask could the dynamics conserve charges that fail to commute with each other? So does any of the physics change? We have experimental opportunities to complement the theoretical ones. One can test this, our predictions, particularly our fluctuation theorems. The first experiments into the thermodynamics of non-commuting charges was recently performed with trapped ions in the lab of Christian Russe and Reiner Blotz in Innsbruck, Austria. So that experiment featured trapped ions, but amenable platforms include also superconducting qubits, quantum dots, and more. Our work also features sequential and weak measurements. Those have been performed with superconducting qubits and photonic systems. In conclusion, we saw that charges failing to commute with each other qualitatively alters stochastic entropy production. Three common formulae for the stochastic entropy production stop you equally in each other if charges fail to commute. And different formulae reveal different physical effects on the stochastic entropy production of non-commuting charges. So this work opens up for business, the topic of non-commuting charges in stochastic thermodynamics. This work is based on this paper that again came out on the archive today that I wrote with my students, Tuashu Pagaya, my postdoc, Billy Brush and our wonderful collaborator, Gabrielle Lundy. So this work is one in a series that my group has been working on in the field of non-commuting charges in quantum thermodynamics. This is our other work. And again, a perspective on this little subfields will be on the archive next week. Thanks very much for your time. Thanks a lot, Nicole, for this great talk. So I'm sure we will have many questions. If you have questions, please, okay, there are some clapping hands first. Let's see whether we have questions. Okay, not for now. Then I will start with asking questions. If I recall correctly, you said that you consider for your initial states in this framework, not just product states of sort of generalized gifts or some of those states whose margin notes are correct, is that right? Right, we just want for the marginals to be in thermal states. But a lot of this work or in a fair amount of this work, most of it, we assume that the initial state is just a product of thermal states so that there are no initial correlations that are providing the advantages that we talked about. Okay, yeah, this was something I was wondering about. Okay, there's another question by Naruo Oga. Please, just unmute yourself. Thank you for your interesting talk. Do you have any comments on the case where one of the system is very big, very large so that this basically describes the relaxation process of one of the system toward the other value of the other effective temperatures? Like, for example, in the case that some of the definitions becomes equivalent or something. We haven't considered that situation in this paper. It is an interesting situation, but there were a few of these other papers that concerned that situation. For instance, in these two and this one and this one. Here we were thinking of the global system as a closed quantum antibody system of the sort whose internal thermalization has been studied in theory and experiment a lot over the past several years. And we would think of one small subsystem as the system of interest and the rest of the global system as the effective environment. So there we were considering more of the setup that you had in mind, but we were not thinking of stochastic entropy production in those cases. Thank you. Okay, we have a question in the chat by Steven Leibar. Do you want to unmute yourself? Yeah, sure, I can quickly ask the question. I'm just wondering if it, since it's entry production, usually you think of that being positive and real. Are there contributions that are kind of missing that would make like the global entropy production positive and real? We were thinking about the entropy production of the system AB. And we were assuming that nothing is entering the system or leaving the system. At least not any of these can serve charges. So that is the global entropy production. I agree that it's weird when an entropy production becomes negative or non-real. So we basically follow the philosophy of our collaborator, Gabriel Landy, who wrote a giant review on entropy production. And this philosophy is basically, we generally expect the entropy production to be real and non-negative. And if it deviates, you'd better have a good reason for it or it would better be for a good reason. So the, for instance, I mentioned that according to the suprisal formula, this or rather the suprisal stagastic entropy production its average can become negative. And we associated that negativity specifically with the non-commutation of the charges. Namely, if you have just a product of thermal states as your initial state then you really wouldn't expect anything weird to happen. But if the charges fail to commute with each other then each of these thermal reduce states has coherence relative to some of the charges only if the charges fail to commute. And it is that coherence that allows the average entropy production to become negative. And that's why I said that we're thinking of here non-commuting charges or charges non-commutation as emerging as a kind of something that underlies kind of our resource kind of similar to work how work allows entropy production to go negative or as allowing entropy production to go negative in this case. Okay, thank you. Okay, thanks. So can you associate this also somewhat of the resource theory of asymmetry then is there a connection that you're brought about already? Oh, this perspective has a fair amount about what has been done with resource theories to model thermodynamics of charges that fail to commute with each other. We in this work were not thinking about resource theories but this shows that people have been thinking about for instance, the forms of batteries and reference frames and earlier work on essentially the math that went into the resource theories of or the resource theory of asymmetry that did end up helpful in a work reviewed here on non-commuting charges in thermodynamics generally. Okay, thanks. So it seems that currently you don't have specific questions for this talk anymore but anyway, we have our discussion session now. So I would now like to open up the discussion session for questions about all the three talks we have seen in the last session. I remember there were quite a few questions on the first talk left. So if the people that have these questions to remember them, please feel free to ask them now also. I have one. Okay, perfect.