 Part of the two, I'm just going to introduce Nina, Meghie from essentially Milano, right? No, I was in Milano, and now I'm in Gdansk. Okay, so, anyhow. Yes, I will tell you when you are four minutes, okay? Yeah, thank you very much. So at the beginning, thanks to the organizers for inviting me. So the work was done, so I'm now in Gdansk, the work was done mostly in Milano, but with collaboration with Roy Dan and Ronny Kosloff from University in Jerusalem. So I will be talking about the same setup as Joerky was talking about, so we have investigate quantum open system, so our reduced system is coupled to some environmental degrees of freedom, and because there are so many degrees of freedom in the environment, we only focus on this evolution on the reduced level. So here I will be more concerned with how the evolution equations looks like rather than what are the definitions of non-Marcovianity in quantum regime. So in general, to derive the mass equations of the evolution equations for the reduced density operators, so the proper way to do it is to start with total density operators, write the von Neumann equation for it, and then trace out environmental degrees of freedom, but of course, in general, it's a difficult problem. So nonetheless, one can say something about the structure of the mass equation if one assume some properties of the dynamical map, which takes the initial reduced density operator, so from time t equals zero to time t, some later time. So and as Joerky was mentioned, if we assume that we have the map as complete positive trace preserving, and we also have the semi-group, so the property that the dynamical map can be divided for all s, for all t and s larger than zero, it can be divided dynamical map at time t plus s can be divided in dynamical map at time s, and then later on, we have to simply apply the same map at time t. So with this assumption, we have this well-known form of the master equation, so GKSL master equation also called Lin-Blatform. So we have this coherent part where this Hs, it's commuted with the, so it's Hamiltonian of the reduced system but it can be shifted by some lamp shift and this non-coherent part, which is defined by the dispositive rates which are time independent and the Lin-Blat operators which can be chosen in such a way that they are traceless and orthogonal to each other. And people like this equation because firstly, of course, it's easy to solve, but also from the structure, you see you can see that it's trace preserving. It's also complete positive, so this goes in both directions. So not only if the map is complete positive trace preserving and we have semi-group, we get this form but this goes also all the way around. So if we encounter such a master equation, we can be sure we have a complete positive map and it can be also generalized to time dependent version. So generalized in the sense that we make this gammas and this Lin-Blat operator time dependent, however, we still assume that this gammas are positive and this would, at least for final dimensional setup to which I restrict here, this would correspond to the CP divisibility of the dynamical map. So we know that this map which stays the state from time S to some later time T plus S is also complete positive. Yes, and this can be easily seen but considering lambdas which take, which acts at the time T till the time T plus delta T or delta T is very small. So this can be arbitrarily well approximated but this time, so we conclude from this, does this generator, time locator generator for fixed time has also satisfied this Lin-Blat form. So all of the gammas have to be positive. Okay, and, sorry, and so this is one result. So if we assume that we have a dynamical map which is, so it's complete positive, three is preserving but we also have semi-group, we get this Lin-Blat form. But the other thing is to derive this Lin-Blat form from the microscopy description. So and mostly the derivation comes with following assumptions. For example, quantum optics, so we make a born assumption, so we assume that the total density operator can be approximated but factorized states for all times when the environmental state does not change. This of course does not mean that we think that the total density operator really looks like this but on the reduced level, on the dynamic, on the reduced level, it simply does not make the difference and it's also compatible with the semi-group property. The second assumption is Markov assumption which simply says, or it's, which is valid if the environmental time scales or the time scales at which environment evolves it's much smaller than the time scale of the reduced system and the third approximation is secular approximation. And this is done actually if we look on the interaction, Hamiltonian interaction picture and we use the spectral decomposition of the system operators. So this pi are the projections on the energy eigenstates and we can always decompose this in this way and then this time dependency in interaction pictures comes on this reduced system side as this exponent when the omegas are born frequencies so the difference in energy of the corresponding energy eigenstates. And if we do by the derivation this born approximation, Markov approximation then we obtain such a mass equation so which where the operational part is almost the form of the Leibniz equation so we have still the sum over all omegas and different omegas and omegas prime but we also have this time dependent prefactors e to the i omega prime minus omega times t. And in the secular approximation we simply neglect all the terms where this omega prime is unequal omega and we say then that we can do this as the time scale connected given by one over omega prime minus omega is much smaller than the time scale of the reduced dynamics. So if we do then the integral of omega then these terms were very were oscillated rapidly so at the end that they will cancel out and they simply do not contribute to the reduced dynamic. And with this we obtain a mass equation of this form so when we have simple so when you have only a single sum of over omegas. And here the point I wanted to make here is that, okay now we can show that this mass equation can be bring in the Linn platform GKSL master form. However, here we have only this sum single sum of the omegas so the different and the contributions with different omegas they do not mix. So they do not interact like to say. So that's mean that from to obtain this Linn blood equation from the microscopic description we actually obtain an equation which has somehow constrained form. So this is not most general master Linn blood master equation. However, we have to do the secular approximation without it, we have red fill equation and this is even not positive. So we get somehow from microscopic description a restricted form of the Linn blood equation. And okay, if we additionally assume that the born frequencies are non degenerate and the Hamiltonian is also non degenerate we get such a spatial form where this non unitary invariant eigen operators then do not mix under each other and they also do not mix with this operators which are built from the projections. So they decouples from each other. Okay, but this we don't want to do. We don't want to do non-Makovian dynamics. So that our starting point was actually to make some thermodynamically motivated postulate and then derive a master corresponding master equation. And the post the assumption we made is that we have a system where we have strict energy conservation. So the sum of the local system and environmental Hamiltonian commutes with the interaction. So it does not mean that the system energy is conserved but the sum of the system and environment energy is conserved. So no energy is somehow captured in the interaction part. So this one called this operation, thermal operation. And we also assume that initially the environmental state is stationary which is also very standard assumption. And with this two assumptions without making any Markov assumptions one can show that the dynamical that we get such a time translation symmetry also called in qubit case often phase covariant dynamics. So that the dynamical map, it's commutes with the free local dynamics. So when this view is defined in this way. And with this fact as they commute with each other then we know that the eigen operators which are eigen operators of you if they have non-degenerate eigen values then they are also eigen operators of this dynamical map. So this F if we assume that the born frequencies are not degenerate this F. So this non-unitary invariant part of this eigen operators. It's also eigen operator of the dynamical map and the projection of the energy eigen states not an eigen operators but they like to say they stay only under each other. So here it's illustrated in this basis of F and pi. And however this mixing this it's also can be time dependent. And with this we can derive the master equation of such a form. So we actually get exactly the same or from the only looking on this operational part the same equation as the one I saw before. So when we have done this secular approximation however we get here some time dependent rates. So if we have an underlying symmetry in our system then like to say that the symmetry fix the operational part of the master equation and the only thing we have to guess if we have to describe the dynamics are this kinematic or are this time functions which makes the think of course easier but not trivial. For example we can using this time translation as symmetry we can use that the interaction Hamiltonian it's not time dependent and interaction picture and then approximate or write it make some Taylor expansion and time around zero and this can help us by truncated the expansion starting from some order we can derive then some approximated time dependent rates. What is more what we also can to use it once more use this symmetry underlying symmetry of the total system and use the fact that the asymmetry cannot increase under symmetric dynamics and this gives the specific constraints on this rate how they can behave in this particular case one can follow that such a time integrals of this race can be smaller than zero. That of course does not mean that we can like to say derive the same the result which is also so strong as this concern GKSL master equation. So these are only necessary conditions not sufficient however it can help us to guess or which master equation are complete positive for example. Okay and now some simple example to illustrate the thing so we have a James Cummings model so we are qubit which interacts with a single harmonic oscillator and interacts with the sigma plus B and sigma minus B dagger type of interaction so what we call the rotating wave approximation and if we then assume make our assumption that the initial environmental state commutes with the environmental Hamiltonian we get a master equation exactly of the form I have derived so we have the first two terms they correspond to this unitary non invariant part of the spectrum and this terms with sigma Z corresponds to this projection which mix under each other and then of course one could ask if under this constraints the dynamics is somehow maybe not very trivial or in some sense Markovian however firstly okay here I depict simply what the dynamics means or does so we have a block sphere which is then mapped to the ellipsoid which can be then moved around along the Z axis but is symmetric and the rotation around Z axis but even in this case one can show that the rates can be negative and we can for example we can calculate the eigen operators of this time local generators and we indeed see that the unitary invariant part they are eigen operators and in this unitary non invariant part are eigen operators and this unitary invariant part that they mix under each other and this mixing can be time dependent and because of this time dependency this dynamics is even non commutative so exactly when it is commutative then this gammas are proportional to each other and we then get no time dependency in this eigen operators and then actually one can show that also this eigen operators are also the eigen operators of the dynamical map itself okay and for unitary dynamics this term will even disappear okay and however if you look closely actually I here even did not assume that we have the strict energy conservation because the strict energy conservation for strict energy conservation for this model I do not only have to assume rotating wave approximation but also this omegas and omega e have to be the same so that this have to be satisfied this unresonance condition however here the other symmetry is satisfied so the total number of excitation is conserved and this more or less all consideration I do I have done for time translation or symmetry can be actually also done for other symmetries so in this case we have this symmetry with total number of excitation when k are this eigen eigen state and k is counting the level the eigen state so we can define then corresponding symmetry by this unitary and then where we do not have here the system Hamiltonian but this particle number and then we can show that indeed the dynamical map then commute with this unitary and one can derive the general form of the corresponding master equation that satisfies this symmetry however in this case of qubits it actually is so or in this very simple case it's actually so that this symmetry coincides so the master equation is exactly the same for this time translation or symmetry and where the total number of excitation is conserved okay so I have found you that the symmetries fix the operational part of the dynamics but also they put some constraints on the kinetic coefficients and this can help us to modulate some non Markovian master equation or to guess how the evolution equations looks like for some for some realistic models thank you okay so thank you very much for being perfectly in time actually so are there questions here in the room yes hi thank you very much for a very interesting talk so my question is related to the thing that you said earlier about not imposing strict energy conservation so if the commutation of the interaction term with the free ones is not imposing strict energy conservation then my question is do you think it might be possible to use this as a as an average energy conservation so my question is let me rephrase it like this so in thermal operations sometimes like one of the critics is that strict energy conservation is a very strict assumption and so it would be desirable to have also on the resource theory approach like something that is a little more relaxed that is conservation of total energy on average but of course from the resource theory approach it would be difficult to impose this could this be a way to maybe have I don't know a dynamical viewpoint on something that is a bit more relaxed than strict energy conservation and from this maybe infer back something from like an extended thermal operations that do not obey like strict energy conservation so my question is sorry as you said for thermal operations this is usually the assumption the exact commutation of the interaction part and the free parts and I got the impression that now you said in this case you can derive a master equation with this assumption but this is not actually imposing strict energy conservation so here I did not impose strict energy conservation but the number of excitations is conserved and do you have any intuition if it's possible to use this master equation and to tweak it a bit and insert some small perturbations to it and small terms that maybe modify this and insert some fluctuations of average energy but under control in the sense that because it's a posterior approach you either impose strict energy conservation or energy is completely not conserved but then you can violate all sorts of bounds but in this case I'm wondering if it's possible to actually have a perturbative so then my first idea would be because if we so in general if we restrict the strict energy conservation form of the master equation okay we have strict energy conservation so then would be an idea to add some additional term where I control that the rate is for example small in comparison to other rates so that the difference would be not large and this would be my first guess how okay there are other questions well I have one myself so but it's a bit so essentially what you were saying is that I mean as far as if you say okay now if I impose this translational symmetry then essentially I'm forced to have for the density matrix and evolution which is of the limbalat form it's almost so the form is operational form is somehow fixed yes but the questions of course what does time dependence function does because I mean my problem is the following with the limbalat a personal problem with the limbalat form but because you know if you take the limbalat form of the evolution and you say now I would like to describe a system which is in contact with a bath in such a way that my limbalat form will give me an equilibrium dynamics in such a way that the fluctuation distribution theorem is satisfied actually you cannot do that with the limbalat form so in a sense limbalat is somewhat bound to give you something which even if seems to be an equilibrium state it actually does not satisfy fluctuation distribution theorem so then my point cannot understand how this goes together with the theorem that you proved so I think the thing is so in general so that the whole like to say the problem is what it is time dependent rates makes because one could actually show that if the dynamics is invertible then one can bring it in such a generalize limbalat form however where the rates can get negative on the list of questions if this at all is complete positive okay so there is room with the okay no no I see your point okay are there other questions then if not let's thank Nina again