 Let's click, let's click, let's see, the right mouse. Yeah, okay, great. Here if I'm going to speak them out, but then I also want to talk about revealing non-classicality inaccessible object. Okay, thank you, Philippe. Oh, bad move. So apparently there is, this is something. Very nice. Yeah, I can do it on the white board, on the black board actually. Hopefully we'll get back. Yeah, so it's revealing non-classicality of inaccessible objects. It's so inaccessible that we cannot even see it. Very weak measurement. It was up at some point, am I right? Yeah, I'm even, should the cable be fine? I can dance. The best projection is the sealant left touch can you switch off here? It's left touch, it's okay. I will come back, that is very weak. There we go, I don't touch anything, right? It's okay, so hopefully it will work this time. Sorry for the inconvenience. So yeah, I'm going to talk about the revelation of non-classicality in systems that you cannot access directly. And really this is say an account of the resolution of our frustration. Now resolving frustrations can help in having fun with quantum mechanics. So just as a plan for the discussion, technology is fading me today. So as the plan for this discussion, let's go through what I'm going to present. So the basic result will be basically a scheme that is inspired by quantum communication to probe quantumness of a system, so quantum features of a system that you cannot really put your hands on directly. And by proposing this scheme, we will go together through which are the implications of this indirect inference protocol for mesoscopic physics, in particular discussing some details, a scheme of quantum optomechanics, but we will also address a couple more examples in different experimental contexts. And finally just to get a little bit more exotic, a little bit more crazy, let's go to the say possible implications that what we are discussing on point one and two will have for very ambitious topics, very ambitious problems, such as for instance the investigation of the possible quantum nature of gravity or the inference of general system environment correlations in open system dynamics. It will be very unfair to say, give this presentation without acknowledging, say main contributors to this work, which are these three people. So Margarita Zupardo, she used to work in Singapore at Nanyang Technical University, but she's now in Reykjavik. Tanjom Krijnanda, who is a PhD student at Nanyang Technical University and Tomak Paterek, who is the leader of the Quantum Information Group in Nanyang, and a very good friend and all collaborator of mine. Just for the sake of credit, yeah, again, I have to say that there are other contributors to the results that I will go and show and they will appear at some point in these slides when you. So a bit of motivations, okay, so the historical context and why the frustration that I was mentioning at the beginning of this presentation. So in 2003, I was first year PhD student working on the possibility of manipulating and transmitting entanglement, transferring entanglement in quantum networks. And I was under the apparently and evidently actually very wrong impression that if you want to entangle two systems through a third party, a little bit towards what is illustrated here. You have these two guys that you want to entangle. They don't interact with each other. They are remote, they are separated, far apart. And you want to entangle them through this third party, this C guy. Well, I was under the impression that you have to entangle C with either I, A or B at some point in the protocol in a way to transfer this entanglement to B. So if C gets entangled to A, then it can transfer this entanglement to B and therefore establish a long or remote entangled network between A and B, very natural. Now I have to transfer, I have to correlate stuff to transfer this correlation to remote systems. And then I'm very unhappy myself on the archive found this paper. One morning, separable states can be used to distribute entanglement. And that really upset me a bit because it changed completely the perspective that I had on entanglement distribution. And it was, say, the abstract itself was enough to, say, turn upside down my convincement because what this gentleman showed is that no entanglement is necessary to distribute entanglement. That is, to distant particles can be entangled by sending a third particle that is never entangled with the other two. Okay, so distributing entanglement without no entanglement, with no entanglement, actually, you know, with separable states. The frustration went on for quite a long time. No, I liked the paper. I liked the paper so much that I wanted to understand it in depth. And I couldn't. I mean, the maths worked. I reproduced the results. It was great. It was good. But I couldn't understand why you could distribute entanglement without actually having an entangled particle and entangled messenger at no time in the protocol itself. And they did what mature human beings do when you have a problem. You just ignore it, right? So I just decided, okay, yeah, it works, but I don't get it, yeah. And I went on until 2011. So almost 10 years. When I had these two trips, one to Singapore and one to Canada, where these three gentlemen were at that time. No, Cavamodi and Tomek Patrek, they were at CQT in Singapore at that time. And Marco Piani was at the Institute for Quantum Computing in Waterloo at that time. Now they have moved. Well, Tomek didn't match, but anyway, it changed the university. And what in these two different research visits, these three guys did was welcoming me with a cup of coffee and this paper. So distributing, separable state can be used to distribute entanglement. So apparently the same frustration was present at different locations for quite some time. So we decided to work on that problem trying to understand where the crux was and what made the scheme by Qubit et al possible. And so it took a little bit of time, but in the end, say in 2012, we came up with at least something that was instrumental to resolve my frustration and the take home message of that and there here is the following. Entanglement game along the lines of what these guys in this paper by Qubit et al had in mind. So the possibility of distributing entanglement in a quantum communication protocol such as the one that they had in mind where C interacts with A and B to distribute entanglement is bound by the degree of pre-available Quantum Discord. So you must have something, a resource of a quantum nature like Discord not as demanding as entanglement, a little bit loser, so it's Discord, but that is key if you want a scheme such as that to work. So we have particle A, we have particle B and as I said, you want these two guys to be quantum correlated, to be entangled at some point. You have this messenger boy, see, that interacts first with A and this is what I will call the encoding operation then stops interacting with A, it's transferred to B where there is a decoding operation and your hope is that at the end of this decoding operation you get entanglement between A and B but C on its own remains separable, remains separated from A and B all together. Now, a bit of say natural expectations, no the natural expectations is that if you take a measure of information, no say for instance, mutual information, measure of general correlations between A and B, the mutual information between A and B at the end of the protocol would differ from what they shared at the beginning, right? I'm talking about gain, so I'm not excluding the possibility that at the beginning of my protocol, A and B shared already some correlation. I just want to, for this correlation to increase, so not necessarily establishing them say from scratch, from zero, but maybe just an increase, right? So the difference between initial and final correlations will be basically dependent on how much correlations I'll be able to transfer from A to B. This goes along the lines of my naive expectation back in 2023, right? So you have to establish some entanglement somewhere to transfer it, to be able to transfer it to the remote station. What you would expect is that say the final entanglement minus the initial entanglement, okay, so the gain that you have in this communication protocol will be dependent on the amount of entanglement that you're able to transfer. Again, along the lines of what at least myself was expecting back in 2023. The actual truth is that in such a scheme, and this is an actual theorem, no? For any tripartite system, and there is no limitation the dimension of the systems themselves, of the birth space of the systems themselves, the gain that you have in entanglement does not depend on the amount of entanglement that you communicate, but depends on the amount of pre-avail of discord. So a quantum correlation figure of mate, but of a different nature. And again, just for the sake of completeness and credit, I have to say that, well, come on, go back, yeah, good. I have to say that again, the same idea condensed at different places, because almost simultaneously when writing up the paper we learned that the group in Neusendorf, by the group led by Dagmar Bruce, Alex Trelsov should be around, Alex, Alex is there, so they had, sorry, I didn't see you before, so they had more or less the same, they reached more or less the same conclusions, the same results, and then Alistair Kay in the UK he had a very related, a very related set of results. So if you are interested in, yeah, more on a personal level and the resolution of that sort of frustration, more generally and scientifically on how you can distribute entanglement through separable carriers, then every that these three papers you should be able to get a semi-complete picture of the problem. And now say, I mean, this is not the topic of what of this presentation, right? So this is the foundations, right? That result is the foundation for what we are going to see what I would like to present at this point, okay? And therefore, so how the resolution of frustration helps you going forward, no? The results of what I'm going to, so basically an account of what I'm going to discuss is in this paper on the archive last year and this is going to appear in PRL recently, we got the proof two days ago. So given that somehow the logic towards inferring this, say, quantumness of inaccessible systems is a little bit convoluted, let's go through it together step by step, okay? So let's go through the logic of the result that I would like to show. So what is the goal? The goal is to determine if the state of a system that you cannot access directly, so you cannot put your finger on it, you cannot poke it. Whether or not the state is classical. So you want to determine whether or not it's classical without touching it, okay? So without measuring, without doing anything on it because you simply can't. And this matches most of the situations in the lab that we have, not in our quantum labs, not always you have the possibility of accessing directly the system that you want to probe, manipulate and say what property you want to infer. You need some form of the indirect access. How much can you learn about the state of this guy if you cannot access it directly? The tool that we are going to use is precisely the sort of entanglement gain that was at the basis of the distribution of entanglement through separable state that we went through before, right? So I'm going to use the entanglement gain between remote systems, remote probes as a tool to infer the non-classicality of the system that I cannot access. What is the result that I want to report? I want to show that one can actually say, one can actually infer this non-classicality through this gain, through this entanglement gain, only if during the evolution, during the interaction between your probes and the system that you cannot access, there is some discourse that gets built up, that gets established. Okay, so between the system that you cannot approach and the probes themselves. Why should we put the emphasis on discord and why is that a good quantity to analyze, to study when one wants to infer the non-classicality of the state of such a non-accessible system? Well, by its own definition, it's very definition, this code is non-zero if there is no projective measurement that leaves the state unchanged, right? So I perform a projective measurement on a party of a discordant state, of a multi-partite state in general. If there is not such measurement, then this code is non-zero. So this is precisely the key because such a possibility is there only if the state of the party that you are measuring contains needs non-orthogonal states to be described. So this statement, which is at the basis of the definition of discord, is posse is true only if I have say non-orthogonal state and it's precisely this non-orthogonality that we are going to, this lack of distinguishability that we are going to use as the non-classicality that we want to infer, okay? So let's see what is the context which attempts and conditions for this protocol to be effective and for the protocol that I want to illustrate to work. So as I said, I have a guy in the middle, I have C and I cannot access C directly. I have A and B and these are my two probes and what I am allowing for is just bi-local interactions, so I am allowing for A to be interacting with C through an Hamiltonian HAC, which I am not going to specify. I allow for the same thing between B and C so in this subsystem, so there is an Hamiltonian model that describes the interaction between B and C but there is no term that connects A and B, no Hamiltonian, no interaction that connects A to B directly, okay? Do I want to put any constraint on the dimension of C? No, I don't want to do that, I want to set it completely to leave it entirely free. Therefore, the scheme that I am going to show and the results that I will discuss, okay? They are valid, whether you have an elementary system or a very complex, a very complicated one as C, okay? So you can have a single particle, like one of the systems that we have seen this morning or a spin chain, like say, a many-body system in general, like the scenarios addressed in these morning's talks. It doesn't matter. What I can also allow for is the interaction between each of these subsystems with their own environment. So my scheme doesn't work only for unitary closed system dynamics but actually it works in general for any Markovian-like dynamics of probes and inaccessible system. And yes, I want C to be completely inaccessible, so it's in a cage, I don't touch it. I'm not going to do anything on it while I allow myself the luxury and the freedom to be able to do anything I want on A and B, including the possibility of performing measurement, arbitrary measurements on that, okay? And reconstruct the state of A and B fully. Does it make sense? Okay, so picture is clear so far. If you lose me here, you're not going to get what I want to illustrate later on. Okay, so shout now or shut up forever. Yeah, good. Okay, so this is the scenario. And just to make things easy and approachable at the beginning, I'm going to consider for now the case where there is no environment and this is just for the sake of the argument. The paper illustrates the full open system dynamics case, so this is not cheating, it's simply because I have 35 minutes plus questions and I want to go straight to the point, okay? So let's address the closed system case and also allow me to make a simplification. Again, this is our first step, we are going to the full case in a bit. So my first assumption is that the Hamiltonian between A and C commutes with the Hamiltonian between C and B, okay? So if you now consider the time evolution operator, right? The total time evolution operator of the system, this will be something like this. If you can read my handwriting, okay? So in general, that's what we have but under this assumption, under the assumption of commutation between these two Hamiltonians, then I can exactly break it down into that, okay? So into, can you read from that corner, yeah? Into two time evolution operators determined by HAC and HBC only, makes sense. Moreover, I can decide which is the order of the two, so it's completely in material because they commute, so it's completely in material if C interacts with A first and B later or vice versa. A bit of notation, whatever is primed in these slides is what results from the interaction between A and C. So I'm assuming that it's actually disordering, okay? So A first and B later, okay? So C interacts with A first and then with B, okay? So rodash is this form of intermediate state that I get when I apply the interaction between A and C only. And this is the evolved state. Now, if I go back to a couple of slides before, so if I go back to the result that Modi, myself, Patrek and Piani were able to prove to prove to understand how it is possible to distribute entanglement without any entanglement. Well, I can make use of that bound to the entanglement game based on this code. And what I do is that I apply it, I use it twice. I use it at the beginning, so this must be true at the beginning. That relation was true at any time. And this must also be true after the application of HAC. So these two conditions are simply the statement of my theorem back in 2012, and also say the reformulation of Alex's results. Not back in 2012, nothing else. So at this point I did nothing but just using what we knew in 2012. Now, the entanglement between B and AC is invariant if I do something on AC only, right? In that bipartition, a joint unitary on A and C, it's a local operation in this bipartition, so this entanglement cannot increase. Therefore, if I apply the Hamiltonian HAC to get something primed, that entanglement cannot change. So the entanglement between B and AC at the beginning is the same entanglement that I get between B and AC after the application of this unitary. Similarly, the entanglement between A and BC after I applied this Hamiltonian remains unchanged at the end of the protocol. So when B interacts with C, because with respect to this bipartition, the unitary between B and C is a local operation. So I have these two relations again, okay, these two relations that I can now sum, I can add mutually. There are two terms that cancel, so this guy gets canceled by this guy, and what I get is that the entanglement gain in this bipartition, so between A and BC, right? I'll begin, no, between initial and final time of my protocol, depends on these two quantities. Depends on the initial discord between C and AB, and depends on the amount of discord that I'm able to generate in this bipartition due to the interaction, AC, makes sense. Now, let's simplify things. Let's assume that at the beginning, I had no discord whatsoever between C and AB, so I had a completely classically correlated state between C and AB, and this tells us that in order to observe, again, in the entanglement between A and BC, I need to have some discord that gets generated by the interaction established between A and C. Does it make sense, guys? So what we have at this stage is the suggestion that entanglement gain in this bipartition is strongly determined by the mediated discord, not the quantum correlations of the discord form mediated by C. And the non-classicality of C along the lines of what I've introduced a couple of slides ago entails the entanglement gain. Of course, see that discord just upper-bounds this entanglement gain, so the gain might be zero. I might have discord in this bipartition after the interaction without no entanglement gain. Let's see if we can state something a little bit more precise, but before doing that, I just want to kill, pre-empt the question, yeah, but how about C interacts continuously with A and B? Now I'm doing this sort of shuttling. C interacts with A, then with B. But most of the time, C is always interacting with A and B. It's tough to gate, to gate the interaction between the particle that I have to probe and the two probes themselves, right? So, well, if I have a continuous interaction, if I have a continuous interaction, what I can do, and if I have non-commuting Hamiltonians, what I can do is to use precisely the trick that is typically used in order to understand the physical picture behind continuous interaction, which is Suzuki-Trotter. I can trotterize my evolution. I can chop it in small slices of length delta T with delta T, a very short time interval, and therefore having, again, a discretization of my dynamics, which is only approximately true, but if my time intervals are very small and are many, it's a very good approximation of what happens in the continuous limit, and therefore replace my statement, the statement that I showed you in the previous slide, with something that instead of depending on the final time T, depends on the time interval delta T that every time I'm using. Makes sense? So, my take-home message at this point is that the result holds unaffected whether you have a discrete interaction between probes and system, or a continuous one. And now the statement, so the two theorems that are proven in this paper. So, and the answer, the question, can I increase, no, can I get entanglement gain if C is always classical? So, if I don't have any gain, and any creation of discord, and the answer is no. So, there are two theorems, one for the closed system dynamics and one for the open one that I'm going to illustrate. So, the first one states that for a three-particle system with this sort of bilocal interaction, whether the commuter or not is immaterial, the gain in entanglement is zero. Actually, okay, the entanglement in this by partition is constant, so the gain is zero. If this discord is zero at all times, okay? So, unless I get some discord in the dynamics, I cannot gain entanglement. And this puts us in a perfect position to be able to assess the non-classicality of C by using, see, the gain in entanglement between A and B. Is there a question down there? No, okay. How about the open system picture? Well, we can generalize the protocol, we can generalize the result, and address just the case of open system dynamics. So, if I have A, B, and C, each couple to its own local environment, then the gain in entanglement is zero, if this discord is zero at all times, okay? So, this is the formal result that we use then to analyze some examples of, say, some experimental interest, okay? So, I'm going to use the picture where A, B, and C are open to their own environment, and what we did was focusing on an optomechanical device, on an optomechanical system. Now, as far as I understand the way, I wasn't here, unfortunately, Monday and Tuesday, but there were talks on optomechanics. Am I right, Fabrizio? Yeah? So, by now you are familiar with the system. I have a slightly more convoluted optomechanical system than what was probably presented to you earlier in this workshop. I have what is called a membrane in the middle configuration. Thank you so much. So, I have a double, say, a double cavity, a double Fabripero cavity, with a vibrating membrane in the middle. And as I hope the speakers, before me, illustrated, well, this system is typically open, right? So, there is electromagnetic noise surrounding the cavity, affecting each mode of this double cavity, and there is thermal noise affecting Brownian-like noise, actually, affecting the movable mirror, okay? So, this is an explicitly open system, and if you ask my experimentalist colleague, they are working in optomechanics, they tell you, well, it's impossible to know exactly what is happening to the mirror without the field. So, it's the perfect scenario. A doesn't interact with B. They both interact with C, which is my mechanical oscillator, and they want to infer the non-classical nature of this mechanical oscillator. This is the interaction in Newtonian, so it's radiation pressure. You have a pressure, you have seen it in previous talks, and these are the results, okay? So, this is the entanglement that you establish between the two fields in time for some parameters which fit the experimental reality at increasing power of the pumps that you are using from outside. And this is the disco that you establish between the mirror and the two cavity fields in time. So, at the beginning and at the end, you have no discord whatsoever between your mechanical oscillator and the two cavity fields. At some point in the interaction, you build up discord, and this establishes entanglement between A and B. This result can be applied used to other configuration, other physical configurations. Actually, I'm very important to experiments that have been conducted recently. These guys published this paper in Nature Physics two years ago where they reported the entanglement between these two spins mediated by a spin chain, an anti-ferromagnetic spin chain, a many-body system, of which they didn't know absolutely anything. So, this scheme can be used to understand the nature of the state of the spin chain itself. So, instead of focusing only on these two guys, one can understand what is happening to the many-body system that passes, that transfers, establishes the entanglement between the remote spins. And the similar result was reported in the group of van der Seeven two... Yeah, van der Seeven, two years... No, this year, yes. So, the paper appeared on the archive a bit ago. Where they have two remote quantum dots mediated by a third one in the middle, and they were able to entangle them by using this mediator. Okay, so I'm done. I just have two very speculative, extremely qualitative slides that show that one can understand a little bit more than just entanglement distribution and entanglement gain through this scheme. Because one can learn something about system environment correlations. I think I have, say, one of my major accomplishments is that I now have two papers on APS with sex on it. And, I mean, you can't have, you can't have shedding the cuts, but you can have this acronym on NPRL, and they are fine with this. So, you can understand system environment correlations by pretending that C is some form of environment, right? So, if it's something that you cannot access, if only AMB you can access, then C is akin to an environment, is not different to an environment. And, if you establish gain in entanglement between AMB, this means that at some point in the evolution, you must have had some correlations of some nature, yeah, these could so, of some quantum nature between your environment and the system, AMB. And, you don't need any tomography. You can understand, you can infer this non-classicality, so the system environment correlations, through a simple entanglement witness that witnesses the entanglement between AMB. And, finally, and this is definitely my last slide, well, you can reverse the picture, say, how about I cannot access AMB, how about I have them completely isolated from the environment in such a way that I must be sure, I can't be sure that the only interaction between AMB, if any, is of a gravitational nature. Then, if you probe the entanglement gain between AMB, if you see a gain in entanglement between AMB, then you must necessarily conclude that this can come only from the quantum nature, the non-classical nature of the gravitational field that mediates the interaction between AMB. And, this was the basis of another paper that we have on the archive. I'm done, so this is the group in Belfast plus edition, so Bina is around, Bina is there, and it was not present in this picture, so these are the guys that say suffer my tantrums, these are the guys that put bread on our table, I thank you for your patience, and I'll take your question if you have any.