 Tudi, taj površčen, je refinke. 1 in 2, da bomo v težkjih površčenih. Včasno, površčen. Včasno, ga so pričače, da se bo nači, da je izvrščen, da je tudi površčen, Prof. Romero Izard pojedeva. In je tudi, da smo nekaj superpozicijali na makroskopičnji ljub. Zato je tukaj zelo, da bomo tudi prišličiti. Superpozicijali makroskopičnji ljub. Tudi, da smo nekaj nekaj, Or even better what actually quantifies a system to be classical or to be quantum. So, let's start to think to categorize things. So, we start saying, OK, this is my microscopic world. Then we have our macroscopic world. And so, this will be the error of the dimension of our systems. So, for example, here we have a single atom. And on the other side, that's, for example, that's me. OK. So, clearly, whatever is on the left is considered to be quantum. Whatever is on the right instead is considered to be classical. But there is already something that we are creating some division among these two worlds. But what happens to something which is in between? For example, what happens to a cat, which is here? So, well, for example, Weinberg in some way described this situation and described the interpretation, the Copenhagen interpretation of quantum mechanics is a little bit confusing, because there is some mysterious division somewhere here that is dividing the classical and the quantum world. So, what is microscopic and what is macroscopic. But it's not really clear who is actually performing the measurement. So, who is the classical object that looks at the quantum one and who instead is measured. And usually one can think about, OK, but I am also made of atoms. So, why should I not be described in the really same terms as the single atom? So, quantum theory is linear. So, there is no evidence why I should stop describing things in a way and starting to describe it in another way as a certain point of complexity of dimension of my system. So, what I would like to have is a full, unique quantum description. OK. Then again, who is measuring what? There is the argument, we already see it in these days, of the environment. The environment is actually measuring our systems so the coherence is actually what makes our macroscopic systems classical. So, we don't have superpositions. So, OK, but where should I put the environment? Something of this sort is this environment or starting here. So, again, there is some, the division is still not really clear how this should work. And so, we still have some problems. So, at the end of the day, this is the quantum measurement problem. So, why a system is classical or why is quantum? OK, and this is actually the argument I will discuss with you. So, this is the quantum measurement problem. So, let's go back to the environment. So, is really the coherence not enough to describe this transition from quantum to classical? Well, actually it's not. Let me take an example. OK, so, we start from a state psi, which is in a superposition of 0 and 1. Let's say this is just the cut on the left, the cut on the right. The corresponding statistical operator, we can construct it just in this way, OK. And what we have is the following. We have the populations. So, these are the probability of having the state on 0 and 1. And then we have the coherences. And these are, at the end of the day, the interference fringes in an interferometric experiment. So, this is actually what is quantum there, OK. So, if we represent this state on the basis of 0 and 1, which is the most natural basis for this kind of problem, what we have is the density matrix is of this form. So, again, populations and the coherences. So, the action of the coherence is to take rho, incentive to rho prime, which is the same just without the diagonal terms. So, we have something of this sort, OK. So, now we have half of probability of having the state on 0, half of probability of having 1. And this actually corresponds to a statistical operator of the form 0, 0 plus 1, 1. Let's consider a different system, a different example. Let's suppose of having a system prepared with the one alpha probability in the state plus, and one alpha probability in the state minus. And these are just, again, superpositions of 0 and 1. Sorry, this is plus and this is minus, OK. Then I construct the statistical operator corresponding to this state. So, the rho is equal to one alpha plus, plus, plus, minus, minus, OK. But then, if we are going to represent that state there, again on the same basis, what happens is that we end up with density matrix, which is of this form. We have the same result, we have it here. We have completely different systems. So, in one case, I don't have superpositions. The coherence killed it. In the other case, I have superpositions. Still the coherence didn't act on my system. But yet my state is the same, OK. So, you see that the coherence on a particular basis is not sufficient to describe my effect. So, to solve my quantum measurement problem. And this is because my description is on the basis of the rho. And since I don't want to have superluminal signaling, in terms of rho, I need to have a linear dynamics. And this obviously doesn't solve my problem. And we will see it more in detail. OK, so, how do we solve the problem? There are various ways to do it. And one of them is given by the collapse models. Well, the statistical operator actually is also the same. If you expand that in terms of zero and one, you will get the same. Because if you take this one, which still describes a superposition, and you act with the coherence on this basis, then you are not changing anything, that you still have a superposition. So, that's the problem. So, the cut is still both alive and dead. OK, so, collapse models. So, collapse models wants to put together so this microscopic and the macroscopic world in a unique unified description. And to do that, we just go a little bit back and try to understand which is the roof of the problem. OK, so, the point is that in quantum mechanics we have actually two kind of dynamics. The first one is given by the Schrodinger equation, which is linear and deterministic. So, given state psi at time zero, then I know exactly how this is, which is the evolution of this psi. Then, obviously, there is some randomness in the outcomes. But this is not what Schrodinger equation describes. This is a linear differential equation. So, it's linear and deterministic. We have, on the other hand, the reduction. This is the mass. So, here, this dynamics instead is nonlinear and this is also stochastic. So, I do a measurement. I get a single state, which is chosen randomly. So, this is the stochastic part. It's chosen not in a superposition among, so it's not a superposition of the eigenstates of the operator I'm measuring, but it's a single of that hypothesis. And that's in the linearity part. OK, so, the idea of collapse models is to bring together these pieces. OK, so, we want to construct, to modify the Schrodinger equation, adding some terms, which are both nonlinear and stochastic. Yes, I work. So, this is the measurement process in quantum mechanics. OK, we have a state psi and some operator O and we do a measurement of the operator O on the state psi. So, what we can do always is to describe the state as a linear superposition with some probabilities. OK, so, sorry. Let's put it in this, OK, this way. OK, so, we are describing on the basis of the operator O, our state. OK, so, you can do the same with the operator. So, it's J. This will be the eigenvalues and then we have the projectors. On that state. Then we do the measurement. So, the operator O is actually not psi. Obviously, this is not an eigenstate. It's a superposition. So, I will not get or something times psi. I will get something different. But the question is, OK, after I apply this, which will be the state instead of psi, I will change it in my state. And that's the collapse of the wave function. So, the wave packet reduction. OK, so, I will get the state OI with a probability Pi. And I will get as outcome of the measurement the eigenvalue OI. Yes? OK, nice. So, now, a question is more or less natural, no? Should I really mess with both these properties, both the nonlinearity and the stochastic features of this kind of dynamics? Cannot just make some linear and stochastic modifications or nonlinear but deterministic? OK, so, let's try. Let's see what happens. We start with linear and stochastic. OK? If I take something of this sort, then I end up of having the same kind of dynamics I have for the coherence. I have due to the environment. OK? So, I can describe everything in terms, for example, of Hamiltonian, Schrodinger, Hamiltonian, plus a potential which is stochastic. OK. But in this way, still I don't have the collapse of my function. So, the effect of this kind of dynamics is still that of leading me this state here in this. And so still I have the same problems in the coherence problems. So, this kind of dynamics doesn't actually solve a problem. Let's try the other one. So, nonlinear but deterministic. In this case, I have a big problem. I do superluminal signaling. How do I show you this? So, consider entengle pair. OK? Our system, which is described by... It's a system of two photons which are entangled. And a photon is sent to Bob. A photon is sent to Alice. And the full state rho A B is described in these terms. i, j. And then I have... I choose these states here as eigenstates of an operator that Alice can perform a measurement. And this is the operator X. OK? Then Alice have also another operator, and this is Y. This can be, for example, Pauli matrices. And in terms of this Y, I can just rewrite this state in that basis. And I have something completely similar. Just with a tilde here. And then, instead of X, I have Ys. So, this is the part of Alice and this is Bob. And that's it. OK? Let's assume that Alice makes a measurement of X or of the Y operator. So, in that case, the Alice part of the system will collapse as well as that of Bob in one of the eigenstates of X or of Y. So, for example, Alice measures X. Then Bob's data will be Bob's data will be of this form. And then this state here evolves due to this nonlinear and deterministic evolution. And so, after some time, this will be just X i of t, X i bar of t. What happens if Alice instead measures Y? I have something similar. Let's put a prime here. And this will evolve again with this nonlinear deterministic evolution to Bob prime of t. What's the trick? This state here is not can be different from this one. Because my evolution is nonlinear. So, depends on actually which is the initial state. Which is a particular initial state. So, this row here is not equivalent to this one. And so, the trick is, okay, encode Alice encodes the information 0 in X encodes the information 1 in Y. Okay, I want to send this information, 0, 0, 1. I do two measurement of X and one of Y on three different pairs of entangled photos. Then, Bob measures an expectation value with his data. But the so, the let's say the expectation value of this will be different if Felix measured X or measured Y. So, in this way, one can actually perform superluminal signaling. Clearly, we don't want this. So, again, we have problems with this kind of transformation of dynamics. Okay. So, we end up with sorry, sorry, deterministic. You don't say that it's deterministic. So, is described still by the Schrodinger question. So, for example, okay, an example of that dynamics could be of this form. So, we have the Hamiltonian. Okay. And this is, let's put here the team. So, this is the Schrodinger part of the equation. Okay. Do you recognize it as a Schrodinger equation? Yes, nice. And then here, I put, for example, an operator, generic. But then this is nonlinear in the psi. But still, there is nothing that is stochastic here. Okay. So, that is the case of that kind of dynamics. What we end up is that modifications of Schrodinger equation that can actually solve the measurement problem need to have two characteristics. And these are the nonlinearity and the stochasticity of the equation. An example kind of equation that solves the problem is of this form. So, here this kind of equation is the Schrodinger equation. Then this is the Hamiltonian. Then we have another operator and a part which depends on the expectation value of that operator. So, this actually leads to in a coherent way and this kind of equation actually solves the measurement problem. Okay. Do we have questions at this stage? No? So, maybe we take five minutes and then we go on with the rest of the lecture. Okay. Thank you.