Zelo, da je zelo, kvantumetrologi se površa v seberalih subsektorov, ki je parametra estimacije, revodne detekšenje, hipotesične testov, tako, tako. Ne sem vseče, da ga ga površa vseče, ali sem pa da ga vseče genetikno vseče vseče. So the how to look of my presentation is this one. So, to begin with, I will start by briefly reviewing what a quantum measurement is. How do we formalize these processes in quantum mechanics? And then I will move, once we have set these ideas, I will move to discuss explicitly state discrimination problem in procesu, kriminacijo in parametrišnje. Ok, zelo, kvantumečnje. V kvantumečnih mečaničnih mečaničnih mečaničnih mečaničnih je basično vse proces, kaj ima tudi klasične informacije na zelo, na kvantumečnih sistem. To je basično, kaj je kvantumečnih mečaničnih.   Tako, skupne vse. Vse je odvorel, v jelberi spes. So basically a projective measurement given psi is testing whether or not this state is one of the element of such a basis. And of course it is the result of the measurement itself is not deterministic in general, but is characterized by some probability, which is given by this expression here, which basically is obtained by taking the projection between psi and j, one of the element of the basis, and you square it. This is basically the Born's rule. And of course the same notion can be generalizing to the case in which your initial state is not a pure state, but a mixed state, in that case, this same relation generalizing this form. Now as an example of a projective measurement, just consider the case in which the state of interest, the quantum state you are probing, is a single photon, which has been prepared into possible polarized orthogonal directions, and the projective measurement simply try to determine whether or not this single photon that is coming in is either vertically polarized or orthogonally polarized, then what you do, you put here a polarizing splitter and then some photo detector on the two associated parts. If you get a click here, you project this state into the vertical polarization component or otherwise in the orthogonal one, in the horizontal one. And this is a projective measurement. Now as a matter of fact, this is a very basic notion of measurement, which is very useful, but it doesn't really capture all the possible measurement you can perform on a quantum system. Let's see what we can do beyond projective measurement. So as a matter of fact, in many cases of physical interest, given as the system that you want to probe, typically you don't have direct access to s itself. In fact, what you do, you introduce an ancillary system A, which is prepared in some state sigma, then you couple the system of interest with the ancillary system through some process and at the hand of the process you measure A. So basically in this representation, you are using A as a sort of probe that allows you to test the state of the system s. And this is very common. For instance, this is the famous Ratterford experiment, which is a typical experiment of quantum mechanics. So the system of interest here is this target here, which is a gold foil. OK.  So this is indeed a measurement scheme, which correspond exactly to this representation here. And it turns out that if you look at the structure of information that this process produce on s, this is not describable as a projective measurement on s. Maybe it is a projective measurement on a, but it is not a projective measurement on the system of interest. So this kind of process is a slightly more general than the previous one, the simply direct projective measurement on the system. But it is not the full story yet. Indeed, there are other measurement process that we would like to characterize, which are not projective, which are not of the same kind of the one that we have just seen. So, for instance, we have things like noisy detection scheme. What is a noisy detection scheme?  But this time this a system is not a probing system, it's the environment that is interacting with s somehow. And after the interaction, now you perform, let's say, a projective measurement on the system s. OK? So now this time you still project the state of the system s, but however you do that after it has interact with an external environment. And this is the typical scenario of a noisy detection scheme. An example of a noisy detection scheme is given by this representation here. Once more you have, say, some photon, some field, some mode of the electromagnetic field you want to probe with the photo detector, but unfortunately, not all the photon present, which are present in this mode, reach the detector because you lost them because of the interaction with the other mode of the electromagnetic field. And in the hand what you measure is not s itself, but s after the interaction with the environment. OK? Once more, this particular measurement process does not correspond to a direct projective measurement on s. So we need to consider also these possibilities. And finally, there is a third possibility. OK. The third possibility is as follows. Again you have the system s, and now you have, let's say, an ancillary system a, again this guy is a probe. No, it's something that you can control. You let the two objects interact and at the hand of the interaction, now you measure both. You perform a projective measurement on both. OK? And once more, this process you can call it a joint detection scheme is not a projective measurement on s. It's something else. And an example of a joint detection procedure is provided here. Again, this is an optical field you want to... So this one is the optical field you want to detect. What you do, you put here a beam splitter, you mix it with the reference mode, which is the ancillary system a, which has been prepared, say, in some coherent state, you mix it into a beam splitter and then you perform photocounting on the tube. And this is exactly a joint detection scheme. As a matter of fact, this is what in optics is called a homodyne detection scheme. OK? Again, this guy cannot be represented as a direct projection on s. OK. So it turns out that all these procedures can be explicitly represented in terms of a very powerful theoretical tool, which is called the formalism of POVM. POVM is an acronym which stands for positive operators and valued measures. And basically it allows you to represent both all the value scheme we have seen so far, the joint detection scheme, the noisy detection scheme and the indirect detection scheme, as well as the direct projection measurement as follows. So basically what you do, you represent each one of those possible procedures by assigning to the detection scheme a collection of operators, e of j. These operators are positive semi-definite. OK, so they are mission with positive eigenvalues. And they have to fulfill the property that I wrote here, which is a normalization property. OK. So each one of this measurement can be described, formally described by assigning such a collection of operators. And now the probability of getting some outcome, which is labeled by this index j, is obtained by this formula here, which takes the place, which replace basically the born rule we have seen before. So if rho is the state of your system, you take the trace of the product between rho and e j. And this is it. Now it turns out that all possible scheme of measurement or detection you can think about can be casted in this form. So there is one-to-one correspondence between measurements and this way of representing them. Yeah. Yes. OK. So, OK. So let's discuss about the case in which you have s, in which you have a, this is the ancilla. Let's say this guy has been initializing rho. Let's say this guy has been initializing in some state, I call it zero zero. This is a probe, has been initializing in this kind of situation. And now let's imagine that we coupled the two through some unitary transformation u, which typically is gonna be induced by some Hamiltonian, which act interacting Hamiltonian that connect the two guys for some time t, OK. And therefore, before you perform the projective measurement on the ancilla, let's say the state which initially was this guy here, so this is the input state of the joint system, becomes this object, OK. It becomes this guy here. U dagger u, the two evolves like that. At this point, and let's say this guy is a, OK, and this guy is s, OK. Now at this point, let's say we perform a projective measurement, a pure projective measurement on the ancilla, OK, something like that. And this guy provide us some outcome j, which is labeled by this index here. And let's introduce the projectors that allows you to extract such an information. And so these projectors are associated with a collection of vectors j of a, OK. With j, which goes from one to d of a, d of a is the dimensionality of the probe. If the probe was a qubit, dh is two, otherwise is whatever. Therefore, the probability of getting j, the outcome j from our system s, can now be obtained as, so first of all, I have to compute what is the state of the ancilla after the interaction, because this is a local measurement on a, and the statistics of a projective measurement can be obtained by taking j, a, and here I put rho a prime j, a, where rho a prime is just the reduced density matrix of a after the interaction, something like trace of s u rho s tensor 00a u dagger, something like this, yeah? Well, let's replace this guy in sa, and by the way, this object, because this is a partial trace, basically correspond to taking the sum with respect to k from one to ds of ks u rho s tensor 00a u dagger ks, where ks are just element of the basis for the system s. Yeah, something like that. And now let's replace this object inside this formula here, and what you get is sum over k from one to ds, and then you get something like sum over ks tensor, and then I have j of a, this vector here, rho s tensor 00a, and then I have j of a tensor k of s, yeah? Well, so now, and here there is a u, of course, otherwise, yeah, there is a u. So now let's apply, so let's move this cat here on u, and this bra here on u dagger, and then you can rewrite the whole things like sum over k from one to ds, ds is the dimensionality of the system, of course, is something like ks, and then you have j, a, u, a, i, something like that, and then you have rho s, and then you have zero, u dagger, j, a, a, and then you have ks, and this is just simple way of representing the thing. Now, you notice that this is, so you have a sum over k here, and k is a basis, so basically this object, and this guy now is, originally u was an operator on s and a, but now I have contracted the degree of freedom away, so this guy is just an operator on the system s, okay? I call this operator m of j, because it's an, so let's call it ms of j, it's an operator which acts on the inverse space of s and depend on this index j, and this guy here is clearly, it's a joint, yeah, it's just the joint of that object, so basically, and also, so now we have an operator on s, rho s and operator on s, and then we have this kk, and this is just the trace of this operator with respect to the degree of freedom of s, so it's trace over s of ms j rho s ms j dagger, it's something like that, yeah? And now you use the cyclicity of the trace to move this guy on this other side, m of s of dagger m of s j rho s like that, and now this guy here is e of j, okay? Okay, so it's, so it's not particularly difficult, I probably lost too much time in order to derive it, but you ask it explicitly, yeah? So they are certainly positive, they are semi-definite, you should try to prove that this guy is positive, okay? Prove that this guy is positive, so it's, the claim is this guy is positive and also satisfied the normalization condition, you should try to prove it, it's not particularly difficult, but anyway, so we do have this formula, it's very handy, it can be adapted to describe all this situation. Well, and now it turns out that in many cases of physical interest, these POVM measurement are kind of more powerful than just direct projective measurement on the system itself. Depending on the problem you want to solve, say you want to extract some kind of information from the system of interest, and you ask what measurement should I choose in order to recover such an information, not always the measurement which allows you to solve the problem in an efficient way is gonna be a projective one. Maybe it's a POVM in one of those guys, okay? Okay, so this is a good kind of, this is a nice result because it means we have more tools at our disposal that possibly are more powerful than depending on the problem, than just projective, projection. Okay, of course there is also, so that means that we have more ability in extracting information from a quantum system than we originally suspect. However, you should always remind that when you try to recover information from the system, there are also the limitation. So when you try to recover information from a quantum system, there are limitation. Which kind of limitation? Of course there are technological limitation, but these are kind of trivial. We are not engineer, so for us it doesn't really matter if there are technological limitation, that's not true, but let's say. But there are more fundamental limitation for us and these have to do with uncertainty relation, for instance. So we know that certain information cannot be recovered while we are probing the system to check other property of the system. These are as to do with Eisenberg uncertainty in the hand. Not only that, but there are other limitation that quantum mechanics impose to us and these have to do, for instance, to no cloning. The fact that quantum states are not by themselves observable quantities. If you have a single copy of a system, that's it, you cannot really reconstruct it. So on one side we have POVM, which are kind of powerful, they extend our ability in order to prove the quantum system. On the other side there are fundamental limitation which are again associated with the fundamental structure of quantum mechanics. And when you try to, and the goal of optimization of a measurement has to face these two kind of problem, these two kind of, has to put together these two different competing aspect. Okay, so the fundamental problem in quantum metrology, the matter of all problems that you face in quantum metrology is this one. Which we have already seen when we were discussing quantum classical communication. So it's what you may call state discrimination. The state discrimination problem is a problem in which somebody is asking you to determine whether which state is a given one. So somebody is giving you the state row question mark. This is something that we have already seen last time and you are asked to determine if this state is one of the state row one, row two, row three that has been, that are possible candidates. And your goal is to find out which one of those possibility are. And you can try to solve this problem by performing whatever measurement you want. And in particular you can try to solve this problem by devising the most generic, the most powerful POVM you can think about. Not just projective measurement. And so this is a highly non-trivial problem which is the optimal measurement that allows you to solve efficiently this problem here. So as I mentioned, we have seen these kind of issues exactly when we were studying communication. So when you study classical communication on my quantum channel, you face this kind of problem here because Alice is preparing quantum state that encode classical messages. Bob received this object and has to check to infer what was the state Alice has to solve. And we have seen that there are some figure of merit that you can use in order to characterize this kind of efficiency. And which, and it turns out that optimal measurement in this particular context are known as pretty good measurements. They were first discovered by Olivo Schumacher in the kind of independent way. And these are other two generalization of this idea which use sequential decoding and bisection, decoding scheme that I contributed in discovering a few years ago. And basically the take-home message that I want to stress here is the fact that this optimal measurement that allows you to solve the specific problem of quantum communication, of classical quantum communication, rely on optimal strategies, optimal POVM, which are not projective. So the measurement that allows you to reach the bound that achieve the capacity in this special case are in general not projective. They are truly non-projective POVMs. Okay, so let's move on. So in the simplest example of a state discrimination problem is this one. So here you are basically given two alternative, row one and row two. Let's say with a priori probability 50%, you are given these two possibility. And now somebody is selecting either row one or row two with probability 50%, and it is in presenting to you such a state, but without telling you exactly what the state is. And the question is, can you guess? Can you determine which one of these two possibility is the state that I've given to you? You can perform measurement and you have to solve this problem. Now, you can try to solve this kind of problem, identifying the optimal measurement that solve it, but you need to introduce figure of merit in order to classify the values to answer the question. And the typical figure of merit that one can consider is what is called the error probability. So the error probability is simply the probability that you give an answer, but it doesn't correspond to the state I gave it to you. So you say one, but I gave you two, or you say two, and I gave you one. You can compute this quantity, and this is, of course, a function of the POVM that you have selected, and then you can try to optimize with all possible POVM you can think about, yeah? And it turns out that this optimization problem was solved by Hellstrom quite a few years ago, and you can show that by optimizing with respect to all possible POVM, you get this minimal error probability, which is expressed here in this very elegant and compact form. It is one minus, and then you have this quantity here, rho one minus rho two, and this is the trace norm divided by two, and everything else, then you have a factor of two to divide. Now, this object here, this same object here is indeed the trace distance between rho one and rho two. I don't know if somebody have introduced the notion of trace distance in previous lecture, but I guess so. So, this theorem by Hellstrom is telling you that the minimal error probability that you have when you try to solve the problem of this form is given basically by one minus the trace norm, the trace distance between the two. Now, you may wonder what is the optimal measurement that allows you to achieve this bound, and for this special state, it turns out that optimal measurement is indeed a projective one, okay? But this is just, no. I mean, the minimal error probability is exactly a function of the trace norm. Then if you introduce different measure of distance, they are not directly related with this one, and so there is not, I mean, the formula introduced that guy, okay? So, in some sense you can use these, if you wanna see in a different way, you can use this result to give an operational meaning to the trace norm distance, okay? So, the trace norm distance characterize the minimal error probability in the discriminating two state, if you want. That's all. Yeah? So, if row one is equal to row two, of course, this guy nullify, and the error probability, the minimal error probability is one half. So, you have 50% probability of being wrong, which is like you are just guessing. Yeah, if you want, but the point is that I'm asking you to tell me whether the result is one or two, and then I choose. Okay, it's like I put, I make a commitment, I say, this guy is one, and this guy is two. Then I'm cheating, I'm giving you exactly the same state, and then I give it to you. Guess whether I choose two or one. But, yeah, it's... Yeah, yeah, yeah. I'm just telling you, I just choose two state, and kind of cheating. These are two different state. Please tell me whether this state is one or whether this state is two. You don't have. So, I put a label on a system, on the same object, but I put it kind of randomly. So, there is no way you can guess where I put my, which label I choose. So, it's maximum error probability. Sorry? Yeah. Yeah. Okay, so let's move on. So, another result, non-result in quantum estimation theory, we can discuss later on if you want to. Another result that you... Interesting result you can... Well-established result in quantum estimation theory is this one. Now, you have the same problem. I give you row one and row two. The two possibility, row one or row two. Okay, I'm trying... You have to guess this label here. And, but now, I'm not giving you just a single copy of the system. I give you, say, n copies of it. So, you have many, many, many copies. And now, this is good, because if you have many, many, many, many, many copies, in principle, you can try to do a state tomography. You can try to construct the classical representation of the state, and once you have the classical representation of the state, you know exactly what the state is. Of course, you need, in principle, you need infinitely many copies. But let's say that, as the number of copies increased, you expect the minimal error probability to go down. And indeed, this is exactly what happens. So, of course, the minimal error probability, in this case, is given by this formula, which is exactly the formula I gave you before, but now is applied to n copies of the two objects. Sorry, this object. And it turns out that you can take an asymptotic expansion of this quantity, which goes down exponentially, which decreases exponentially, with an exponent, which is given by this quantity here. Okay, and this is called the quantum Chernov bound. Okay, the quantum Chernov bound is put an upper bound on the exponential decay of this error probability, in terms of this awful function that you see here. But it's a mathematical and analytical result that you can obtain. Of course, there are others I cannot cover. Okay, so, a related problem to quantum, to of state discrimination is process discrimination. So, in process discrimination, the issue is the following. Once more, there are two alternatives, phi one or phi two. But now these phi one and phi two are not states, but processes, like black boxes. There are two processes that can act on your system, and you don't know which one is going to act, and you have to guess which one is the process that is going to act on your system. So, you have to discriminate between the two alternatives, but now we are talking about quantum evolutions that is CPT maps. Remember, all the most generic evolution of a quantum system is a CPT map. You have to discriminate between phi one and phi two. I give you this black box and ask you to say whether this is this guy phi one or is this guy phi two. Now, of course, this kind of wide state process discrimination is relevant. Well, process discrimination is relevant because it's a problem that occurs in many different contexts. So, for instance, it has to do with noise detection. So, suppose that you have a quantum computer, you have some gate, you have prepared some gates, but, and this is represented by this phi two mapping, but then maybe, at some point during the evolution, something goes wrong and instead of applying phi two, your quantum computer is operating on your inputs with phi one, which is a deteriorated version of phi two. It's phi two plus some error. And, of course, it is of interest for us to determine whether this noise has occurred or not. This is exactly what you do when you do quantum error correction. So, in the end, process discrimination is strongly related with noise detection schemes. Another instance where process discrimination occurs is this one. So, for instance, suppose that you want to probe the presence or the absence of a target. And so, these are kind of military applications, unfortunately. So, you are trying to detect the presence of an airplane, let's say. Okay? This one, which is a very realistic and concrete problem, can be casted in terms of process discrimination because in order to probe the presence of this airplane, what you do, you shine some probing signal, say, some laser beam or whatever. And then, depending whether or not you get a reflection from the object, you have two situations. If you get a reflected signal, this corresponds to a transformation of this message, which is due to some map, which is phi one. Phi one is the process that takes this signal and reflects it. Phi two, instead, is the case in which your signal is not reflected, you get no signal back. So, once more, this target discrimination can be casted in terms of process discrimination. You have to decide whether the channel is operating on your system is phi one or phi two. And this is, again, process discrimination. So, how do we address the process discrimination problem? Well, we can transform process discrimination into state discrimination, that's very simple. Just put inside the black box state and then, depending whether or not this guy is phi one or phi two, you will obtain at the output two possible states, rho one or rho two. And now the problem becomes, again, in order to answer the original question, you have to distinguish between two states and we know how to do because we can just measure them. Try to measure it, we perform it. Of course, so this is a very nice way of transforming the previous problem, the process discrimination into state discrimination. But of course there is a caveat, or there is, if you want, there is more, because now you have the possibility of choosing which one initial state of the probe you want. So depending on the choice of this row, you may have more possibility in discriminating rho one or rho two. So the idea is to find which state is the most sensible or sensitive, sorry, in order to discriminate between these two process. So here there is yet another kind of optimization stage. Not only you have to optimize with respect to the measurement that you need to perform to distinguish the two states, but also you may consider the possibility of optimizing with respect to the input state of the probe. Finding the optimal probing state. Yes, there is, yeah? And it's the next slide, okay? So, so, now we know that so one thing that you can do, instead of just preparing the state of this probe and sending through the channel, what you can do, you prepare the state of the probe and you entangled with an ancillary system, okay? So you prepare, say, here a maximum entangled state at this level, then you let this guy to be affected by the noise, by the process. And what you get on the hand is a state in which this guy has been transformed and this guy is not. It's a sort of reference state system that is still correlated with yours, hopefully. And then what you do, you measure both guys. So you perform a joint measurement on these two objects. And now the problem discriminating between phi one and phi two correspond to discriminating between these two states, where this is the transform version of the input state probe, which now was an entangled state here, according to a local noise, which is phi one, or a local noise, which is phi two. Again, it's a state discrimination procedure, which is now performed over an extended Hilbert space, not just a local one. Yeah? And now, this is very similar, is very much similar to what you do when you perform interferometric measurement in order to detect, say, a phase delay in an interferometer, and you send signals, a probing signal, you mix it with a reference input, you let the two guys interfere, and then you measure. So this kind of process here is very much reminiscent of an interferometric detection scheme, where you have a probe, which is kind of correlated with the reference signal, and then you let the two guys interact, and then you measure. Yeah? Now, it turns out that this guy here is the best you can do. And as a matter of fact, is strongly related with what is called a Choi-Jamulkovski isomorphism, which is, I didn't introduce it to you in the previous lecture, but it's a correspondence that you can establish between maps and states. So the Choi-Jamulkovski correspondence says that for each process, you can associate a quantum state that fully characterizes such process. And such Choi-Jamulkovski representation of the state is exactly the guy that you get here when you send in through this process a maximum entangled state. So in other words, if you inject here a maximum entangled state of the system and the ansinla, after the interaction of psi, you get a state which contains the full information about the map. So the full information in the sense that whatever the state does is written into the Choi-Jamulkovski state. So in this way, the state discrimination of the Choi-Jamulkovski states correspond to a full complete discrimination of the map. It gives you exactly the large, so it characterizes exactly how far away these two guys are. So in other words, this is an intrinsic way of discriminating phi. This is exactly what you were asking for. So it doesn't depend on the input. It's something that you don't have to optimize. OK, so as an application of this idea, let's go back to this target problem. And this is a protocol that we introduced a few years ago and we call it quantum illumination. This is indeed trying to detect the presence of a target or the presence of the absence of the target. And you can try to solve this problem by optimizing with respect to all possible input signal. You can shine in, OK? And here you just perform, say, photocounting. And it turns out that, of course, the best way, the best performance is also in this situation here, are obtained when you produce and entangle an input signal which is entangled with a reference. And what you do, you just take a joint measurement between the reference state and the transmitted one. And, OK, I will not enter into the details, but you can apply this idea for this problem and you find out that the error probability that you obtain in order to determine the presence of the absence of the target is improved if you use this technique. OK, so, so let's move on. And now I want to make, to discuss a problem which is slightly more complicated than the state discrimination process we have seen before. And this has to do with parameter estimation. So in parameter estimation, so remember, in state discrimination, the problem is always like that. There is a finite number of possibilities over which you have to choose, with respect to you have to choose. So I'm asking you, is this state rho one or rho two or rho three or say rho n? I give you a finite number of possible answer. And you have to detect which one of these discrete finite number of possibilities is realized by the experiment you have performed. In parameter estimation, basically what you do, you extend, you enlarge the number of possible candidates, creating a continuous set of possibilities. So now, you don't have a finite collection of possibilities, but you have a continuous collection of possibilities. And this continuous collection of possibilities is basically one parameter family of object, OK? So it's like, it's a collection of states that you can parameterize with the parameter, continuous parameter x, OK? You don't label it with a discrete index, but with a continuous one. And now you are asked to determine which one of the state I'm giving you is, where the state rho question mark is in this family, OK? At which point of this parameter? So basically, another way of looking at the problem is, this problem is to say that you have a quantum trajectory in your space of states, which is parameterized by this parameter here, and you have to determine the value of the parameter. So that's why it's called parameter estimation. So your goal is to estimate the value of that parameter. And now it's clear that this problem is much more complicated than the previous one, because the parameter is continuous. So intrinsically, there is going to be always some uncertainty in determining the specific value of this guy. OK, in order to address this problem, it is better to start from its classical version. So this problem has a classical analog use that it is worth to consider. So in classical information theory, the same problem goes as follows. Suppose that you have a random parameter x, a continuous parameter, so it's a continuous parameter, and let's say, which is encoded somewhere, OK? And now let's say that you probe these black box that encode the value of this random variable x, it's a classical random variable, and let's say that you obtain some stochastic outcome out of your measurement. And these stochastic outcomes are represented by these values here, which are also characterized by a continuous parameter. You get x1, x2, and so on, and so forth. You perform many, many measurement, and you get different value of this parameter. And let's say that these parameters, which are the outcome of your measurement, are related with the x of the black, which is written in the black box, through some conditional probability, which is given here, and which fully characterize the measurement process. So I'm just putting by hand all this fact. I don't have a POVM structure here. This is just a statistical connection between two random variables. Well, so now the question is the following. Suppose that you have performed some of those measurements, which are this one, and now you may try, from these outcomes, to infer what is the value of x. So what you do, typically you do data processing. I have a collection of data, I have new points, and I have to infer what was the value of x. So what you do, you create what is called an estimator. So maybe you take the average of this quantity, or whatever. You have to process them somehow, and from the processing of these values, you have to say, what is the value of x. OK. So this is, of course, this object is a function of the vector of outcomes that you obtain. And of course, there are many possibilities. Let's pick one. I don't care. Of course, the statistics of this vector is just given by the product of the probability that each one of the components of the vector is obtained. This is simply because the value of x is always the same, and we assume that every time we prove the black box, we get the same kind of statistics out of it. Well, and now we want to, of course, you have many different possibilities in creating the estimator. And the question is, what is the best estimator? How can I better infer the value of x from my measurement? So you need a figure of merit to answer the question. And the figure of merit we consider, you may consider, is the root mean square error of the estimation. So what is the root mean square error? Is simply this expression here.  So this is, of course, if the estimator coincide with x, you get 0, so 0 error. Otherwise, you get some value. No, this is just a formal way of characterizing the object. And then we are going to put an upper bound, a lower bound on this quantity. So this is very well defined, and you ask yourself, how smaller can be this quantity. Now you can, and now the idea is that, how small this object can be. It can be. Now it turns out, of course, asking this question is again a kind of complicated problem. Because you have to consider all the possible estimation strategies, but nicely enough, there is a theorem by Kramer-Rau, which is called the Kramer-Rau bound, that says that no matter what you do, the root mean square error is going to be always greater or equal than this quantity here. So nu here, this nu here, is the number of measurement you have performed. So you have performed new measurement. And f of x is a function, which is called the Fisher information, which is only characterized by the probability, the conditional probability of the measurement outcome, and depend on the random variable that you are trying to compute. It doesn't depend on the estimation procedure. So it's something which is completely independent from the estimation, from the estimator that you choose. Indeed, it provides a lower bound for all possible estimation strategies. And it's just a function of the conditional probability that connect your measurement to the real outcome. So you get, so as the number of measurement increase, of course, this quantity goes down, it nullifies, and it goes down like the square root of a number of measurement, for all possible x. So you get this identity here. The nice fact about this result is that the bound itself is achievable for sufficiently large nu. So there exist optimal estimation strategy that have to do with maximum likelihood procedures that allows you to construct an estimator that indeed scale like that. So this is the best you can hope. You can determine the value of x up to that precision. No more than that. But of course, you have to find the optimal estimator. OK, so now let's move on to, let's go back to the quantum case. In the quantum case, there is no black box. Or to be more precise, the black box is provided by the state that encode the random variable we want to determine. So you start from this. And now, in order to extract this information, x is a random variable, it's a classical variable, you have to perform measurement. So let's fix a POVM. So you have to fix a POVM. For the moment, let's choose just a generic one. Fix the POVM, you measure the system, and what you get is some classical outcome. Again, c1, c2, and so on, and so forth. And now, the conditional probability that connect the value of x to the outcomes is just a functional of the POVM that you have selected. And of course, it's a functional also of the state. But that's another problem. Well, once we have that, we are in business. Because now we have again transformed the quantum, the estimation problem into a classical estimation problem in which we have to infer x out of this classical data point. Well, construct an estimator and try to minimize with respect to all possible estimator. So in the hand, what you can do, you can apply the previous result, the classical result, to determine the minimum value of the root mean square error, which is associated with this specific conditional probability. So each one choice of the POVM will give you a lower bound associated with the feature information, which is a function of the POVM. And of course, so this is exactly what I said. This is the root mean square error for a given POVM, which is lower bounded by its associated feature information, which is now a classical object. But of course, there is an issue here. And the issue is what is the mean, the best value of this quantity you can hope. So find the optimal POVM. Now, the issue is let's now choose the optimal POVM. And if you try to optimize with respect to all possible POVM, this classical bound gets replaced by this bound here, which is the quantum version of the Kramerau bound, which involve what is called the quantum feature information. The quantum feature information basically is the maximum of all possible feature information optimized with respect to all possible POVM. And so now we have this quantum result. You say that in order to determine the value of the parameter x, which is encoded in your system, the best precision you can achieve is given by this expression here, which again scales like square root of mu. And which is a function of this quantum Kramerau, which is a function of the quantum feature information. Which is a function of the quantum feature information. Well? Yeah, sure. Yeah. Yeah. So for the moment, so for me, for the moment, x, x. So let's say this is the space of quantum state of your system. So this is the set of state of your quantum system. And this is a trajectory inside this state. And this trajectory is a row of x. For the moment, I don't make any assumption. It's just some curve in that space. Let's say it's continuous. And you have to determine whether you are here, or here, or here. I don't care, OK? The full derivation still apply. No, you are looking for the best measurement that allows you to solve this problem. So the premise is the following. Your state is one of the point along this line. But we don't know which one. So I give you the state, and maybe I give you more copies of this state. You can probe it. And you have to determine whether you are where you are along this trajectory. Where is the point where the state is along? So there is this prior information, which is the state. It's not here. It's not here. It's on the line. Please find it. And I give you a mathematical representation of the trajectory. So you know exactly what this state are, but you don't know which one specifically is given to you. New copies. You have new copies. So you can perform several measurements. OK, so let's move on. No, you can even do that. I don't care. So I give you the state. You have your quantum computer. You do whatever you want. So indeed, in order to implement the most generic POVM, typically you need a quantum computer. So you have to introduce an ASILA, couple with a system, perform a joint measurement. You have to perform, say, belt measurement, whatever. So indeed, you will need very sophisticated control in order to implement the most generic POVM. But no matter what you do, you are bound to have this kind of uncertainty. You cannot go below that. That's a sort of no-go theorem. It's a. OK, so now, of course, as I told you, the quantum visual information is rather complicated. Computing it is not simple if the trajectory is whatever. But there are some cases in which this quantum visual information is easy to compute. And the most simple example of a quantum visual information is obtained when, for instance, this trajectory that we are considering, that one, is generated by some Hamiltonian. Not necessarily time-independent or parameter-independent Hamiltonian, but it can also depend explicitly on x. So this is equivalent to the evolution associated with the time-independent Hamiltonian. So let's say that you are evolving according. So your state is evolving according to some procedure. X now is the equivalent of a time, which fulfilled this dynamical equation here, where h is the generator of the dynamics. So this is a very simple kind of trajectory. So said in a different language, I'm saying that I'm considering the situation in which rho of x is given by exponent time-ordered. No, sorry, let's write it like that. Like u of x rho of 0, u of x dagger, where u of x is the formal integral of this equation, which is given by the time-ordered exponent of the associated Hamiltonian between 0 and x. And the time-ordering comes from the fact that h is explicitly time-dependent. So let's consider this special case, which is a very legitimate case. If you want an example, let's say that you try to determine the time t over which the system are evolved according to the Hamiltonian. OK, so when this is the case, then we can compute the quantum feature information very easily. So you can perform this optimization, and it turns out that, so as a matter of fact, do you get an upper bound for f of 0, which in some cases closes, and the upper bound is this one. So is the variance of the four times the variance of the time-dependent Hamiltonian of your system. OK, so you can prove this inequality f0. So if you get an upper bound for f0, you get a lower bound for the minimum uncertainty. So you get this inequality here. And in particular, if the gap here closes, if the Hamiltonian is not time-dependent. So if it is just a time-independent Hamiltonian, then f0 is exactly the variance of the Hamiltonian. Is that clear? Well, and the variance is just given. No, but you take the Hamiltonian, you subtract the expectation value of the Hamiltonian on the state at that value of the time, and then you take the square, and then you take the expectation value, it's the variance. OK, so now a nice side effect of all these is the following. So now we have that the uncertainty is upper bounded by the quantum Kramer Rao, which is lower bounded by this object here because of the inequality I've just given to you. Now if you look at this inequality, you can cast it in a very nice way. You can write that delta of x, the uncertainty in determining the value of x times the uncertainty associated with the Hamiltonian, is larger than 2 times square root of u. And this is kind of generalized energy time uncertainty relation because this guy is the uncertainty in the energy content of the system, and delta x is uncertainty in time. So this is indeed a sort of energy time inequality, the equivalent of the momentum position Heisenberg uncertainty, which now has been obtained in the context of quantum state discrimination or parameter estimation theory. Are you happy? OK. So in other words, the precision in determining the value of the time x depend on the spread of the energy of the system. And by the way, this inequality here, as I told you, becomes an identity. So these guys are the same if the Hamiltonian is explicitly time independent. And because the quantum Kramer Rao boundary is achievable, then also this inequality is achievable when the Hamiltonian is time independent. Yes, yes, in that sense, yes. Fortunately, yes. But still, no, still. So, for a final number of measurements, it's correct. OK, so I think how much time do we have? Five minutes. That's fine. So I want to discuss a final topic, which is associated with this parameter estimation problem. And this has to do with the difference between shot noise and Heisenberg scaling in parameter estimation. And the subtitle of this section is entanglement as a resource. And we know that entanglement is a resource. And we see in which sense, in this context, entanglement is also a resource. OK, so let's consider the following problem. We have a probe, and we have a black box that is affecting the probe through some unitary process represented by u of phi. And basically, the unitary transformation depend on this parameter phi that we want to determine. And let's assume that u is given simply by the exponent of some Hamiltonian time, the value of the parameter phi. So phi, again, can be seen as a time, as a temporal evolution. And your goal is exactly to determine the value of phi. And you can choose whatever state you want here. You can choose whatever measurement you want there. Let's try to see what's the result of this very simple estimation problem, which is very common in quantum optics. You typically have to perform this kind of measurement. It's equivalent to determine the delay of a signal or of a phase shift, or a phase shift. OK, so, yeah, there are several examples. Now, but of course, in order to address this problem, we allow you to probe your black box several time under the promise that the black box is the same. So every time you interrogate the black box, it responds exactly in the same way. Always add the phase phi. So now you can perform several tests and perform several measurements. But of course, you can even try to send entangled resources in through these multiple copies of the black box and then perform joint measurement. OK? So if you wonder how you can do that, because maybe you have a single black box and you don't have, say, n black box, but just one, how can I do that? How do you do that? If you have a physics, just one copy of the black box, which always respond to you in the same way. So how can you implement such a scheme? You cannot copy the box. The box is just one. So basically, you can just send one pulse of the time. Can you entangle pulses that live at different times or not? Entanglement is something that applied to the present state of reality. Sorry, I'm entangled. These two guys are entangled now. But what does it mean these guys entangled with this guy two days ago? Doesn't work very well. Yep. Yeah, you can do that. But it doesn't really explain how can you send, so the point is that, OK, use the same box many times, but how do you send an entangled state there? Again, there is a pulse here. And then there is a pulse I'm going to send in 10 minutes. How can I entangle the two? It lives in different spacetime point. So the answer is require the use of quantum memories, of course. So the idea is the following. So you have, say, two quantum memories. These are just, say, qubits in your computer. You create an entangled state of these two quantum memories now. And now you let this guy, you move the state of this quantum state of this qubit into some traveling mode, say, some photo that moves away, but remaining entangled with that guy. You send it to the box. When you arrive at this stage, instead of measuring it, you charge it into another quantum memory. So this is quantum memory 1. This is quantum memory 2. This is quantum memory 3. And this is quantum memory 4. So you create this guy. You send it through it, and you load it here. And now you have still some kind of entanglement between the two. Now you move this guy into a photon, send it through here, and then you charge here. And at the end of the day, you can realize this kind of configuration. So when you see this kind of picture, as a matter of fact, they kind of implicitly assume your ability of having memories that can be entangled, over which you can kind of process the theme. So in other words, I wanted to stress this point, because I'm not cheating here. If this is possible, it's difficult, but it's possible. Of course, if you have many copies of the black box, no problem whatsoever. But if it's just one, yeah. OK, so let's move on. And now you can perform the same experiment many, many, many times. Let's say you, every time you prepare n photons, and you send it through hit, and so on and so forth, you reprate the same experiment, new time. And the question is, what is the scaling that you get? And it turns out that, of course, you have different possibility. So for instance, you can decide not to put entanglement between the various boxes to begin with, or you can put entanglement. And also you can perform local measurement on the black boxes, or you can perform joint measurement, meaning you have to use this machinery here. Now, it turns out that you can prove that if, as long as you don't put entanglement between the various boxes, the scaling with respect to n, the number of probes that you say, using a single run of the experiment, is going to be like square root of n, which is, as a shot noise, kind of scaling. While on the other hand, if you entangle the probes all together, the square root of n becomes n, 1 over n. And this is called the Heisenberg regime of the probing procedure. Is the number of pulses that you entangled together. So you pass from the shot noise to the Heisenberg regime limit. So unless there are questions, I think I finish my presentation here, and I thank you all for the attention.


And today we will be talking about an important aspect of corrosion that is called as galvanic corrosion or dissimilar metal corrosion. And the topic of the discussion is galvanic or dissimilar metal corrosion. Now, why you should be concerned with the galvanic corrosion? Why we should be worried about the dissimilar metal corrosion? If you look at the engineering components, in many cases consists of multiple metallurgy, multiple metals. They may use in combination like steel and stainless steel or they may apply as a brass the steel, titanium may be along with the stainless steels. And there are several reasons for that. For example, if you consider a pipeline carrying water and you want to have a tap where you want to take the water out and you want to regulate it. We have seen when we talked about the galvanic corrosion that the materials requirement for a given application depending upon how critical it is, how critical the corrosion resistance is. In this case you take a pipeline, if there is a small amount of corrosion there is a loss in thickness and you are not going to see an apparent leak in the pipeline. But if there is a small amount of corrosion in the tap or in the valve, then what happens? The valve or the tap starts leaking. You may not able to control this. So, while you select a steel for the pipeline, you cannot for obvious reasons choose steel for the tap and valve. Many of you know, you will have seen various fittings right. They are made up of stainless steel or made up of brasses. Because stainless steels and brasses they corrode at a lower rate. And so, the leak is avoided. Now they bring in similar problems at other places as well. I give an industrial example ok. You know of this right. What are this unit called? Heat exchanger right. If you look at the heat exchanger, the shell what you see here is in this case made up of a carbon steel. Actually it is a this is called an HB heater, high pressure heater in a thermal power plant. The water is heated to high temperatures. And so, there is high pressure. So, this is made up of carbon steel. But you make the tubes with the stainless steel because the water in the tubes flow at very high rates. And the carbon steel cannot withstand erosion corrosion. So, people go for a stainless steel. Now the tubes are stainless steels. Look at the tube sheets. This is called a tube sheet ok. This is called a tube sheet right.  Similarly look at the header for example, this is header here right header. This inlet and this outlet here.  And these tubes what you see here they are impregnated into the into the sheet. You can see this here. So, these are all tubes which are you know will are welded onto the tube sheet. So, you have a combination of carbon steel and a stainless steel being used. I give another illustration. You sometimes weld the structures. It is a stainless steel here. You weld this. The weldment the weld fusion zone we call it could have a different chemical composition as compared to the tube itself. So, there are different chemistry. Basically you can call this is metal 1. This is your weld fusion zone. And the tube we can call it as they are another metal. They are coming in contact. Please look at they are coming in electrical contact. They are coming in electrical contact with each other. So, one on the top you see here slightly a different reason where the corrosion occur here. This is a carbon steel. Another carbon steel is alloy steel tube. It is a chrome moly steel tube used in the boilers. Now, what has happened here was copper deposition in these locations. The copper deposition has happened because the water carried corroded copper metals, corroded copper metal ions. And in fact, that copper ions came some of you might know that when you when you look at the thermal power plant that is also condenser. You have a steam. The steam goes to steam turbine, it does work. The steam is getting converted into water again by a condenser. And condenser in this case was made up of cupronical. And copper dissolved and that copper carried all through and got deposited over here. They are copper metal and this is steel they led to what is called as galvanic corrosion. There are several reasons several places you see. The other obvious example that can happen is you sometimes coat steel with a nickel. You have seen in car bumpers you know you have seen in in washroom fittings to make it looking very good for aesthetic reasons. If the coating is not proper then you will see a galvanic corrosion between the nickel coating and the steel substrate. So, whenever you have two different metals coming in contact with each other. When I say contact I mean the electrical contact with each other then it leads to what is called as the galvanic corrosion. So, two so two different metals, two dissimilar metals shorted electrically can lead to galvanic corrosion. This is the origin. The origin of galvanic corrosion comes like this. You know a very famous device most of us use in a very constructive manner where the galvanic corrosion works and works for the benefit. What is that dry cell batteries? You know dry cell battery right. In the dry cell battery take it dry cell battery what is this made up of in a few it is made up of a zinc sheet casing right zinc casing. In the center you see graphite of course, you will see a cap here right you see a cap and there is a copper cap. The inside you may have something like ammonium chloride and you know and you also have manganese dioxide. How does the current get degenerated here? It is getting generated because it is shorted now ok. When these two are shorted when you short when you short these two when you short the when you short graphite and zinc happens. It delivers a current delivers a current and and voltage. Now what is happening to zinc? Just zinc remind intact. How does the current come? The current comes because zinc dissolves and gives rise to electrons. The two electrons are used to reduce manganese dioxide to manganese oxide. So, that is the voltage between difference between the zinc and this zinc and this this graphite electrode, but graphite electrode is not an electrochemical system by itself. It is manganese dioxide and manganese mn 4 plus and mn 2 plus equilibrium here. So, between these two the current flows and zinc is rendered as anode and this is rendered as a cathode and and so, the corrosion occurs. This is a well known example of the so called galvanic corrosion used very constructively. But this happens in the industries and that is not going to be good for us. You have a carbon steel, they have stainless steel. If one of these two metals corrode then there is going to be what? There is going to be a damage to the component. So, there can be leak or there can be you know it can happen. So, so dissimilar metals can lead to the corrosion actually. Now why? Why it occurs? One is say that there are differing corrosion rates and the differing potentials. So, if I have two metals you know you take any any environment ok. This is m 1 I just put environment and then this shot and one is you know. So, since they have two different potentials you know ok and and so, that can lead to the things. But we even before a shot is possible that one will have a higher corrosion rate other will have lower corrosion rate. But if you look at closely the corrosion rate differences are not the reason. No this is not not the major reason. One is not the major reason. Sometimes it may not be reason at all. The second is the reason. Now so, if you know these galvanic corrosion can occur when you have two different metals they come in contact with this. Then we need to know two things. How do we predict which of the two will undergo corrosion at a higher rate and if it undergoes at accelerated corrosion due to galvanic you know interaction what is the corrosion rate of that ok. So, we need to know this case ok ok. Can we galvanic corrosion? First question. Can we calculate can we calculate galvanic corrosion? So, what is the basis for that? What is the basis for that? So, that we will see. This the third question that comes is what are the factors right. If you know what are the factors the parameters that control galvanic corrosion then it is possible for you to methods you know to devise methods of controlling corrosion control galvanic corrosion. Of course, next is how do we evaluate galvanic corrosion? How do you evaluate them? So, let us now look into one after another and see that we can have a clarity in all this. So, that if you are doing a research or if you are developing a material or developing a processes or you are developing a monitoring technique for example, it is easy for you to do all this if you can understand all the five aspects of the galvanic corrosion. Let us take the first one. Can we predict? We all know the galvanic corrosion occurs primarily because of the potential difference. So, potential difference is a reason. We are aware of calculating the equilibrium potential right. You buy some about the potentials which is the thermodynamic property. In the standard state what you call that potential? So, the equilibrium is called as standard potentials ok. So, the one readily available data in the literature is the standard potentials and from the standard potentials you can also calculate the equilibrium potentials provided you are given the concentration or activity of the species involved there. For simplicity if you take the standard potentials and put them in some order either in the descending order of the standard potentials or in the ascending order of the standard potentials. You can list them like this right and for different equilibrium and such a list such a series is called as electromotive force series ok. This is called as electromotive force is not it? Potentially is a force a driving force for the reaction to occur ok. This is called as electromotive force. You can have you know a list I mean you can have hundreds of equilibrium not necessarily the equilibrium related to metal and metal ions you can have between the you know zone species you know. For example, there are some organic molecules there are even there are inorganic molecules Fe2 plus Fe3 plus in equilibrium can happen not necessarily metal is involved there. So, you can have an exhaustive list, but for as if you can look at you know the metals which are somewhat related to engineering applications just have a look at them it is very somewhat interesting. Look at some of them for example, let us say let us take the system like gold and the old and look at the standard potential of that it has got their olds right. When you say standard potential it is implicit it is in relation to the standard hydrogen electrode. So, I do not really have to make mention about it, but all other cases you should make mention in whose reference the potential is measured. You can also have just given few examples here ok and you can also get it from the lots of published literature ok. So, this gives you a driving force. Now, the electrode potential as a standard electrode potential is given in the decreasing order from the highest we have here is the gold and here lowest is magnesium here ok and somewhere you see here the hydrogen and you know how to interpret this data right. Suppose any metal that is lying you know below this hydrogen you immerse it in the assay solution of standard state and the ions are standard state then hydrogen is going to evolve and all this above this metal you will not get hydrogen evolution taking place. So, corrosion will not take place. So, you normally call this as noble right and you call this as active generally that is how you talk about it and the nobility increases as you move from the top to the bottom and you can compare any two the the one just lies above is noble and the one below is considered as relatively active and relatively noble ok. So, when we talk about all this we talk about the relative values not the absolute values right. If you take let us say gold in gold chloride and platinum and platinum chloride you just couple them together the platinum will be acting as a anode and gold will act as a cathode. So, you can predict which of this will undergo corrosion what is the basis normally we use a basis earlier called as E cell is equal to E cathode minus E anode it has to be positive right you recollect our discussion earlier right ok. So, it is possible to use you know this ah EMF series and and and find out which of these metals which are shorter electrically can become anode and become a cathode ok based on the equilibrium potentials ok. But can this be true can this happen in the real life situations? In the real situation do you have a an equilibrium potential or no standard potentials in the state where the metal is corroding the steel is immersed in water zinc is immersed in water it will establish a potential right what is that potential called when zinc immersed in in water and it is what is the what is the potential called that is simply called as a corrosion potential. So, it is it is going to establish a potential which is called as a corrosion potential, but what is the what is the criteria for the metal to establish corrosion potential how does that value is arrived there how does the metal arrives at value what is the basis for that. When you when you take a steel I mean you just take iron and put it in acid and at some time it will attain a potential which is steady state slowly it becomes steady state and you measure the potential and that is called corrosion potential. Why does the metal reach that particular potential? Yeah yeah see that is the criteria for that it has to satisfy the mixed potential theory the rate of anodic reaction must be equal to that of cathodic reaction that is the place where the metal will stabilize and the corrosion potential will stabilize all are the cases the potential will drift drift to us that actually that is the criteria for that ok. So, in actual real-life situations the aim of series or the equilibrium potentials they have very less relevance to predict the galvanic corrosion. So, what is the potential that is relevant actually? So, we look at what is called as corrosion potentials is relevant and equilibrium potential is irrelevant. What drives them? What drives the current between let us say zinc and iron immersed in any environment you know maybe water maybe acid? What drives actually? What potential is that? It is the corrosion potential that drives the current between these two metals. Now, when you take corrosion potential there is a problem. What is the problem? The corrosion potentially is it unique to unique to a metal actually high-tech steel? Do you think the corrosion potential of steel will be same in in all corrosive medium yeah? No. No. So, the corrosion potential of the metal will very much depend upon the environment, the temperature, the pressure, the concentration of the species, nature of the species, pH. The so many variations that you will get and you probably must have a million of million data of equal values to compare. So, that is simply not possible in practice. So, they developed a potential based on metals and alloys in pure seawater. You can use seawater also questionable right with the composition of seawater in Arabian sea maybe we have Bengal maybe different ok. It was measured in US at Ann Arbor ok. It is deep in the sea so that the pollutants and all are not going to influence the corrosion potentials. Again I do not know how many of you know even in a given environment the corrosion potential is not going to be unique like you see in the case of equilibrium potentials it can vary by few millivolts 5 millivolts 10 millivolts can really change. The way you prepare the sample sometime you prepare a very nice smooth samples sometime mirror finish. So, you are not going to have a unique equal value even a given environment even for a given metal and alloy. So, you can only look at the relative performance, relative values of the corrosion potentials. That is why we see the Fadrana books many times they give you a relative positions and not the red values. You also see the ASM handbook the corrosion values are given as a band of values you are not going to get the unique values why it is kinetics ok. Because of kinetics you are not going to have unique equal value and so there is going to be a brand of value you are going to have it. So, what I have done is the pure seawater and they list the corrosion potentials of that and that is called as the galvanic series. I can just give some illustrations here ok and and if you see in the in the published literature you see lots of data, but let us give some important you know metals and alloys platinum tops here then you get gold then you get the graphite titanium ok. You have silver you have say 188 moly stainless steel passive nickel passive, cupronickel you have copper zinc and you have tin stainless steels which are active cast iron steel ok, aluminum alloys have zinc before that I think you will get cadmium your aluminum have zinc combination. We are just moving from here like that. So, you see some interesting behavior in the in the metals and alloys in galvanic series we saw this the AMF series there are some of them very obvious you can see that ok. I hope you will you could have recollect some of them ok. I just put back here you can have a look at this galvanic series I am sorry the AMF series you see how they are looking like this here ok and titanium comes somewhere here titanium comes titanium will come in fact, the titanium will come even somewhere here actually ok. Titanium will come even below magnesium is not it? How do you extract titanium? Use magnesium to extract titanium actually the growth process ok yeah. So, titanium come even below actually one of the most reactive material is is titanium. Compare this compare this and what are the obvious differences do you see right what are the obvious differences you see in this no no differences yeah what do you see yeah. Yeah the first obvious difference you find is in the AMF series you do not see any alloys you see only pure metals right. In the galvanic series you do see alloys and metals that is one major difference what is the other difference? Of course, this is a sea water ok of course, there is that is of course, a standard state or it has to be its own ions ok that is of course, there is another implicit difference that you have anything more you see? Yes. Yes yeah we see you know some of the which are reactive in the automotive series as I said titanium will be figuring somewhere here ok. The titanium is going go up even between platinum and gold the platinum acts relatively noble as compared to the gold actually see that is a reversal taking place and even more important is the stainless steel exhibits the galvanic potentials at two locations the passive state and active state you know what the passive state is you know what active state is right. So, metal when it is passive the potential is noble when it is active the potential is it will let you be more negative potentials active potentials you know the reason why it is right why does it happen? When the metal spontaneously pass away what happens in the car does it go up or come down what happens anybody can recollect that it goes up right you if you look at the active passive trans passive polarization curve you best recollect that the car they will lie at a very you know close to equilibrium potential f if the alloy is not spontaneously pass waiting if it is pass waiting the potential will will shift up. So, obviously, that potential is noble compared to the metal exhibiting active potentials. So, stainless steel itself can form galvanic corrosion between themselves in the active state in the passive state you will see later that you know in a pit for example, it will act as a anode surrounding the pit you will act as a cathode we will see that later ok. I hope you are getting this point right. So, you please do understand this if you do not understand this you cannot predict galvanic corrosion in reality where that happens in the field ok. I hope you are able to see the differences very clearly between these things ok. So, practically we use the galvanic potentials ideally if I am choosing a alloy somebody somebody comes to you ok and they ask you to select two different materials for application. What data do you seek? I need to use steel and stainless steel or maybe somebody says oh I want to use titanium and I want to use stainless steel I need to do together you know. So, what data do you seek from that?  Yeah. So, what is the environment you are looking at you know and that put and that environment you should get the corrosion potential that is what the right thing to do that. But if you do not have that data now I have two tables I have a table one which is a galvanic series and the table two which is a EMF series we are available readily for you which one do you use? I use galvanic series because galvanic series is more reliable than the EMF series. But do understand that there is no galvanic series which is applicable all through any environment. I give an example suppose in the rooftop you know you are going to use you know two different metals. The galvanic series for that atmosphere corrosion is different from the galvanic series that you find in seawater because some of them may pass away at something can happen ok. So, it is it is it is important that you keep in mind that the galvanic series what you will see here is seawater that should be used very cautiously only when you do not have the data corresponding to given application otherwise you need to generate the data and that data can only be employed in selecting the materials material selection. Now, let us take the case of the galvanic corrosion. What are the factors that affects the galvanic corrosion? We saw the potentials right. Now, what is that that governs the galvanic corrosion of the metals? Let us take two distinct examples a simple one and a complicated one. So, that you can understand the endochemical parameters that affect the galvanic corrosion rate of a metal. We have seen that when the galvanic potentials are different the one with the noble potential will act as a cathode the one with the active potential will act as a anode that you know that, but you do not know what rate at which the active metal will undergo corrosion. So, how do you electro chemically you understand that? So, let us use the so, called the Evans diagrams to understand this. Let us take the case of say iron and say platinum in let us say one molar hydrochloric acid. How do I get this? Can you do now I think you guys have mastered the kinetics and we discussed this for about 10 10 hours of lectures we discussed right. How do you do this? Anybody has a clue? You go apply mixed potential theory right. So, first of all you apply mixed potential theory for platinum separately mixed potential theory for steels I mean iron separately you get the E car and I car for iron and platinum in that water without coupling them galvanically you know the corrosion rate. You put them together now now you rearrange all of them actually. Again mixed potential theory is applied what do you what do you say the total anodic reaction is equal to total cathodic reaction in overall system ok. You solve that equation then you will get which undergoes what or what rate it is happening at all taking place. So, simple galvanic theory you can it can be evolved by applying the Evans diagrams ok. Suppose I take platinum let us take platinum I just take platinum here and you know the case of let us say take the platinum here one case. Now, what happens? You go for standard states for the for simplicity let us go standard state what is the equilibrium potential of platinum? This is 1.2 what is this platinum 2 plus plus 2 electron gives you platinum am I right. And what about the other equilibrium? So, what happens in this case? Will platinum corrode? Does platinum corrode? Platinum does not corrode at all. So, practically we do not use the dissolution or the platinum equilibrium when you talk about iron right because you see here now when you put platinum and a in a in a in an acid and you shrouded with a with a hydrogen you will always have the hydrogen gas dissociating into H plus and platinum will not dissolve ok. So, we do not normally use this if you see in the pattern of book you do not see that this is there at all ok. And for iron it is easier right what is this cathodic reaction here? This is your exchange condensity for what? Adrenal iron please understand that ok.  So, iron when iron corrodes you have iodine evolution taking place. Now, let us put this together and see how the galvanic reaction occur between platinum and these things right. So, this is a platinum separately immersed in the acid iron immersed in the acid separately. Now, if I am going to be together what happens now? Let us look at this this is iron and platinum shorted is iron separate platinum separate ok. What are the equilibria you will have to be considered there are only two equilibria to be considered one is H plus is in equilibrium with hydrogen iron is in equilibrium with Fe 2 Fe 2 plus am I right or not? Platinum 2 plus platinum does not come into picture at all will it come does not come into picture right. So, let us take the case of iron here let us take the case of iron right this is your Fe going as Fe 2 plus plus 2 electrons. Now, what is this called? Exchange condensity for H plus and H on iron please look at that ok and this if you want to put iron here you can draw this diagram for iron. This is this is your exchange condensity for Fe 2 plus and this is your E car. Now, I want to represent now this is clear right this is same thing I have written here only. Now, I want to now represent platinum in the solution how to represent now? What will happen on the platinum surface you put it in acid when I have hydrogen what equilibrium will be there in the surface of platinum H plus is right. So, what is the how do you represent here? What will be the potential there will the potential of H plus H on platinum and iron will be different or same? The equilibrium potential for H plus H on iron and platinum it will be same or different why it will be different? How do you how do you find out the equilibrium potential for H plus H here come on quick how do you find out the equilibrium potential can you find out or not? How do you find out? So, it is it is nurse equation there is not it whereas, it come into picture where is platinum or iron. The equilibrium potential does not depend upon iron or nickel whatever it is talks about the partial pressure of hydrogen and the concentration of H plus ions that is not going to change. So, the equilibrium potential of that will remain the same not going to change this is the equilibrium potential of H plus and H right, but what is going to change on platinum yeah. Come on I want to see if people talk you guys know come on what what what is going to happen can you look at your your hand out anybody has the hand out now what is the exchange condensate approximately for for H plus H on platinum 10 power minus 3 watt amperes centimeter square for iron how much minus 6 ok. So, it is over 3 orders of magnitude higher for that. So, the exchange condensate for that is going to come come somewhere here am I right? The tuple slope remain the same. So, I write the tuple slope here this is the exchange condensate I naught H plus H on platinum agreed or not agreed. So, this is on platinum. Now, what is the mixed potential theory? You sum up all the cathodic reactions and sum up all the iron reaction am I right or not? So, sum up this when sum up this becomes so small this is going to be your place where the mixed potential theory is applicable and this is going to be this what this is your i car of what of iron combined with platinum here. What happened to this one? This is my equilibrium potential sorry this is your e car of iron when you shot with platinum agreed ok agreed or not agreed ok. Now, if you agreed on it now I would ask the question now to you. Let us make the observations here now we will make let us look at the observations ok. They are an acid let us say 1 molar HCl please notice they are not shorter now just put it here. If you can closely observe the surface of iron surface of platinum what do you what will you see there? Will you able to see anything or not? Yeah there will be. Bubble. What bubble what what bubbles here will be where will there be bubble? Platinum where there will be bubble an iron right is iron there will be gas bubbles what do you have what will be the gas the gas will be. Hydrogen. Hydrogen gas I think you should have given simply one answer hydrogen gas that talks about you know if people are clear about you know ok. So, you have hydrogen evolution taking place on the on on on this will there be hydrogen evolution on platinum? No because platinum cannot get oxidized ok. So, there will be hydrogen evolution taking place. Now what happens I am going to take it out and I am going to shot this you observe now what happens where where will the hydrogen evolve? Platinum. Platinum right now the hydrogen will not evolve there will be less hydrogen evolution on iron then more hydrogen evolve on the platinum right. So, the hydrogen evolution will start shifting from here to this what happens? So, there will be more hydrogen evolution on iron upon the on the platinum the amount of hydrogen evolution on iron is less, but then the amount of hydrogen evolution on iron and platinum put together will be equal to the amount of iron corroding on iron surface can I can I say that actually? Am I making sense? The amount of hydrogen and evolved on platinum and iron if you look at it ok iron take this quantified this which will be equal to the amount of iron that is getting corroded on the iron surfaces that is how the mixed person theory is what is mixed person theory? The total amount of reduction is equal to total amount of oxidation takes place. Now in this case if I measure the potential here using a voltmeter what will happen to potential of when you shorten this the potential of this this will be same only am I right assuming that the conductivity is very very very high conductivity here when I put then this together the potential will change. Now what will happen to the potential of this system when you short it? When I do not short it the platinum will exhibit a potential is equal to what potential equal to hydrogen equilibrium whereas, iron will exhibit a potential equal to corrosion potential right. Now when I short this what will happen to potential of these two? Go up please look at your diagram only ok what about this potentials? It will increase right the potential starts moving from here to this I want you to look at the diagram if you do not look at the diagram analyze you cannot you cannot guess it I think what it guess it you have finished you have to look at the diagram see what happens here is is the potential start moving from here to this that is what happens mixed potential ok it moves up here ok the potential moves from here to this because this is the mixed potential is now because moves from here to this the hydrogen evolution on iron what happens now look at this the hydrogen evolution on iron is decreasing the amount of hydrogen evolution on platinum is more because now it is moved from here this potential it is moved to this potential. So, the hydrogen evolution on platinum is increasing because at when you when it just immerse platinum in the solution this is the potential you are going to have it and shorting the potential moves from here to this, but for iron the potential moves from here to this. So, that means, the hydrogen evolution on iron is decreasing the hydrogen evolution on platinum is increasing from here to this. So, that is what will happen I hope I think there will be an experiment in the lab I say I do not know how many of you have already done it. If you have not done it you please do observe this this particular test. I want you to observe it and see does it really happen or not happen at all ok. So, this is something you should be looking at actually ok. You clear about yeah what means you are talking about this right yeah ok. Yes it is not correct which is yeah it is I it is interesting this point that is right. Yeah. So, how do you explain this now? Suppose I take a platinum right I immerse dip it in a sulphuric acid and I shroud the sulphuric sulphuric acid completely with hydrogen let us say one atmospheric hydrogen. So, what do you think will happen there? What what do you think will happen? What you can happen is of course, you will establish an equilibrium actually ok. He will establish an equilibrium. Will this will ok will this will this go to this one? It may not go here also. In fact, this will not go here ok. Suppose assume that this guy is getting oxidized suppose you are getting oxidized right that is must be a reducing species to take. If I have platinum ions in the solution what will happen to platinum ions? The platinum ions it will get reduced. So, you visualize other situation I have platinum in the solution as platinum 2 plus and I have one more or hydrochloric acid and I am also putting hydrogen in the system. I keep putting hydrogen in the system what will happen now? Slowly platinum ions will get deposited as platinum metal will happen ok. If on the other hand there is no platinum ions in the solution what will happen? Nothing will happen the potential remain here only because if iodine dissociates as to as electron there must be some species to take electrons at all right. There is no species to take electrons at all ok. That means, nothing will happen there that is why platinum is considered as a reference electrode in this case. What is the platinum actually? Platinum immersed in acid and shrouded with hydrogen what is the equilibrium there? The equilibrium is between H plus and H there is no oxidation no reduction takes place it goes both ways right that is why platinum in acid is considered as a reference electrode where there is no net oxidation no net reduction taking place. If you add some platinum ions obviously, all the platinum ions will slowly get reduced to platinum at all actually. So, question now you are right you will not go here it will remain here only ok. But, but, but then the moment I am going to put platinum chloride this line is applicable is not it?  Yeah you are right if you are going to use if you are going to read this title iron platinum in in 1 m s e l probably this fellow does not even figure here ok ok. From that point of view it is correct I just drawn is only to illustrate to you how the equilibrium will start shifting if you do not have platinum ions there will be no way you will get hydrogen dissociation as H plus ok. It will dissociate again will will will reduce there will be equilibrium establishing between platinum between the H plus and and H on the platinum surface that becomes your reference electrode ok that is what happens at all actually. Yes. Dot line the percent according to the exponential j is the final outcome right. The yeah what is this dot line correspond to yeah that is good what is this dot line what is what is this dot line means? The sum of this the hydrogen reduction reaction for this hydrogen reaction this hydrogen reaction occurs in the ion surface given by this kinetics hydrogen reduction reaction occurs in a platinum surface followed by this solid solid lines. So, what does it mean? If I hold if I hold iron at that particular potential the amount of hydrogen on iron is only this much but on platinum it will be this much but if I have both platinum and iron together the total will be this much isn't it you just I know assume that I have a platinum and platinum platinum and iron together I short it and apply a common voltage what will be the amount of hydrogen will takes place this is on the iron this is on the platinum the total will be for that particular potential that is the amount of thing that happens. Instead of platinum I am going to put gold you look at the gold what is the value of gold actually there exchange intensity is given not given there 10 to the power minus 6 right. So, 10 to the power minus 6 is coming very close to this right 10 to the power minus 6 probably you will not get a huge difference in the hydrogen evolution reaction if it is a gold ok. So, the line will shift towards this. So, the effect in fact, we will see later the effect of galvanic interaction of gold and steel will be low as compared to the galvanic interaction between platinum and steel ok because the exchange current density on gold is lower for hydrogen equilibria as compared to that of platinum ok. So, so that is what we are now looking at what the kinetic parameters what are the kinetic parameters that they will not affect this now. Before we close I want to look at this here look at what are the ok and look at this this is the the anodic tuple slope of the active metal and the cathodic tuple slope of the other one is ok is ok I think we will talk about this when we when you deal with two active metals here it is noble metal here it is an active metal. So, we will see what happens when you have two active metals how they will change ok and I think today I mean we can stop here provided you have any questions or for clarification. Now, is that meant that in general we see that iron on iron there will be no hydrogen evolution when they are. Virtually no hydrogen, but you cannot say no hydrogen. Virtually no hydrogen. Virtually no hydrogen. Will it suppress that much steel? Yeah that is because of that it is again we go back to same thing the Evans diagram only talks about the relation between the potential and the current you have a platinum here that is all. But what is the role of platinum the role of platinum means to lift the potential from here to this otherwise the platinum is not talking to the steel at all anyway right because it is lifting the potential here it is happening otherwise any other metal will lift to that where where it will happen.  So, it is platinum or something else something else does not matter at all because as long as you you know you follow the same Tafel line this is the Tafel line at all ok. On the other hand you move down the hydrogen evolution on iron will increase right ok. So, that we will see later when will you move down that we will see actually that could happen if you are going to take about a more active metal than iron things will be different we will talk about it in the next class.