 Sixth lecture. Yes, some number. Okay, good morning everybody. Let's see, so I assume everybody's had at least an undergraduate course in quantum mechanics. Who here has heard, if anybody, about completely positive operator-valued measurements or cross-operators or quantum operations? Nobody. Good, because that's what I'll be presenting today. Sorry? Who has? Who has heard of them? Who knows about cross-operators, quantum operations, partial traces? Okay, well, you might get bored, but we'll see. Part of it will be new even for you. Okay, so recapping yesterday, what Gurjee was going over was an alternative way of actually deriving many results that are formally similar and some of them actually completely identical to the results of stochastic thermodynamics based upon infinite baths. So you have a system of interest, and one or more baths, I'll usually just be referring to one bath, one reservoir, it could be a particle reservoir, what have you. But unlike in standard statistical physics, we're going to do this radical thing and actually get a little bit realistic and say that in the real world, even if you're coupled to an infinite external universe, effectively you are only ever coupled to a finite universe because, for example, right now, I am not directly physically coupled to any degrees of freedom on Saturn or Jupiter. I'm only directly coupled effectively to a very, very small interface between me and the outside world. So one way to view that is as though I was coupled to a finite bath. And, of course, experimentally, you can always set things up so that that is your physical system, just a system of interest and a finite bath, and we're just doing experiments over this, okay? So we started, yesterday we were starting it as a product distribution, some distribution over the bath, for example, a Gibbs distribution, and there's some distribution over the system of interest initially. We start as a product distribution. The Hamiltonian is a Hamiltonian of the SOI plus a Hamiltonian of the bath. These are... This is only dependent upon the variables in the SOI. This is only dependent on the variables in the bath. And then what makes it at all interesting is there's an interaction Hamiltonian as well, okay? Then, because this is closed, it undergoes deterministic reversible dynamics. So, classically, if these were classical degrees of freedom, like in the Dzerzinski setup, it would just be a Leville's equations are being applied, it's being respected, and it's a phase-space evolution. Today I'll be talking about the case where this is actually quantum mechanical, so instead it's a density matrix that evolves according to the Leville-Bann-Neumann equation, a unitary. The important point is deterministic and invertible. As time goes on, because of this term here, the SOI and the bath start getting coupled, statistically coupled. What we are interested in though, the end of the day, we are always treating the SOI separately from the bath. At the end of the day, what we want to then say is, okay, how does the entropy of the SOI change? Subtract that from how the expected energy of the bath changes, which we are going to just identify with the heat up at our level of the thermodynamic limit, and that's going to be the dissipated work or the entropy production. Okay, so that was all of it in a nutshell yesterday. And you get things like the detail fluctuation theorem, and in some work, subsequent to what Dzerzinski did, some other papers that Gurdjay ran out of time and couldn't present them, you also get just normal integral fluctuations and so on. Today, what I'm going to do is try to present enough quantum mechanics so that we can actually derive for this exact same setup an integral fluctuation theorem. As it turns out, this is what's called, for example, a Nielsen's book, which is a great book, Nielsen and Chung, I think? Chung, yeah. This is called, they called it an open quantum system. The word open means different things to different people, and they talk about their density matrices in that case and so on, so they don't think about it in terms of quantum thermodynamics. But subsequently, many other people realize that you could use it by taking this rather than being an environment, which is what it is for Nielsen and Alia, instead making it be a Gibbs density matrix and you can then do quantum thermal, okay? So that's a preamble for today. All right, and as always, let's see, I'm getting an echo here. How do I turn my video off? I mean, I'm going to turn my audio off. Maybe that'll take care of it. Oh, okay, yes. People aren't seeing anything, are they? I need the IT guy. Why aren't you... You didn't flip back from me to the... You have to reshare it. Okay, let's see this one. And we are muted, so all is good, I think? Yes. Okay. That's not working again. Okay, well, whatever. Okay, this is all ultimately based upon quantum mechanics. We will not take that as being, in any sense, indicative of how understandable the lecture will be today. So, a standard rule for thumb in your entire future, I'm not giving you advice for how to live your life, so pay attention. Whenever you're giving a talk, whenever you're interacting with people in general, I guess, but I learned this rule of thumb when giving talks. You always start by telling the audience, it's a good idea too, it's imperative. It's always a good idea to start by telling the audience something they know, so they feel smart, and then you tell them something they don't know, so they think that you're smart. So, first you start off making them feel comfortable in their own skins, that their clothes fit right, and so on, and then you start taking them into a whole brand new wardrobe. Okay. So, here is stuff that I hope everybody knows, and as always, interrupt, because it is something that you're not understanding, and people are not understanding it as well. So, do at least have that working? Yes. Okay. So, this is presumably the perspective of what quantum mechanics amounts to, what quantum measurement in particular amounts to, that you've encountered in your undergraduate course. You've got something you're measuring, position, momentum, spin, what have you, and there's an operator, a single operator associated with it. In a, what's going on here, in some process, and there's been huge bloody wars about what this process is, a measurement involves the universe applies that operator to the system. There's wave packet collapse, which means, another way of saying wave packet collapse is that there's in some sense a discontinuous non-unitary jump of the system state of the quantum state of the system to a normalized eigenstate of that operator. Okay. The value of the measurement is the eigenvalue for that eigenstate. The actual jump, which eigenstate you go to, is determined randomly according to the spectral resolution, the eigen decomposition of the system state and the Born rule, and it's required that the operator be Hermitian. Everybody on board with that? Okay. So that's where you just saw something that you knew. From here on out, for most of you, these will be things that you've probably not seen. Let's first, this you may have seen as well, boy, those things do not want to work. Come on, the world. For some reason now the PDF, the Adobe is no longer working. Yeah. Today it's just not working. I'm not sure what he did, but now my arrow keys don't even work. Yeah. That's why I'm figuring out dissipated workers. So that wasn't, but how did you get the arrows to be working? It's in the fingers. It's in the magnetic field or something. Yeah. I don't know, but I mean this works. Now this works too. Okay. You should be okay. All right. So anyway, so that's the standard picture. Here is something that's equivalent. You may have come across involving what are called projection operators. We still, let's see, we still have the a single operator associated with the measurements and that operator we're still requiring that be her mission. That means that there's a set of complete normalized orthogonal eigenstates, a sub i, i indexes the eigenstates. Then you define what's called a projection operator. That's the her mission conjugate of each ai times an ai itself. It's an operator if you'll notice. It's not actually a single state. The important thing to notice is that because these are orthogonal, that the product of two projection operators, if they're identical, you just get back the projection operators. Again, if they are different, if those are projection operators for different eigenstates, you get back zero. Because since they're orthogonal, ai times aj is going to be zero. Or ai conjugate, her mission conjugate times aj, I guess I could say. And then again, just like this is, this is re-formulating what we just saw. The universe randomly chooses among these different projection operators according to this probability distribution. Because that m, if you'll notice, if you put that in there, that's going to be ai, ai. So this whole thing is going to end up just being the eigen, determining the eigen spectrum that applies for whatever your state psi is. Because of the properties of the eigenmatrix, because this is her mission, m dagger equals m, we can just replace that m with m dagger m. At this point, there's no particular reason to do that, but you should be able to see that mathematically it works. And then we say that now the discontinuous jump according to these probability distributions takes you to this state. The projection operator makes you go down to ai, and then that's to normalize the probability distribution. And then because of normalization of probability, you have what's sometimes called the completeness relation, which if you'll notice, the sum over ai of mi dagger mi, which is equivalent to just the sum over ai of ai dagger times ai, that's equal to the identity operator. Okay? Everybody comfortable with that? Groovy. Okay. Now let's look at just a little sub-part of that. So here I've written up again, and let's now consider some extensions of this. In some cases, it will be unrestrictive to say that the operators are mutually orthogonal and her mission. So how do we deal with that? So watch very, very carefully. There. That's how you deal with it. You just remove the requirement that they be normalized and her mission. Problem solved. Okay. So now we've got a more flexible formalization. Also, in some cases, we're only interested in the probabilities of the observed values, because it might not even be a state. Consider the case where you're observing, for example, what the state of a photon is with an apparatus that actually destroys that photon, which is almost always the case. The photon has no state after the observation. It's gone. Now you can deal with this using quantum electrodynamics and so on and so forth, but at this level here, very, very clearly, you're not going to be interested in requiring anything about the state of the system being measured after the measurement because it doesn't exist after the measurement. It's a destructive measurement. So how can you deal with that? Well, you do this again. Very easy. Erasers are great things. So whereas before we were saying that what the measurement does is it takes you down to this projected state like that, now we're saying, well, maybe it doesn't do anything. Okay. This is a partial equivalence, no longer fully equivalent formally to what you're used to. Okay. So then there's these, what gives us actually is what are called positive operator-valued measurements. This is an alternative way of considering, let me actually go back a second, to this. Yeah. So if we go to this, these particular measurement operators, M, that's actually the way that you will see the process of measurement considered in quantum computation. In quantum information processing for various reasons is much more convenient to use these kinds of operators which are consistent with projection operator formalism but can actually apply for other scenarios as well to be working in terms of those rather than there's no stipulation of some kind of operator A with an eigen-spectrum that determines the probabilities and so on. You just write down a set of M's directly. Okay. And in many cases it's sufficient to only be interested to only be considering this subset where you don't, the subset of conditions where you don't even specify the state after the measurement and that's what are called positive operator-valued measurements. Basically, let's see, do I have it in terms of GIs? Yes, those are the GIs. So let me just go back one. So rather than in terms of these MIs we can abstract it even more one more level. A GI is a positive operator-valued measurement. It's equivalent to what we were just seeing as M dagger M. And so all that we're stipulating is let's see, this should be, I'm sorry, that should be GI rather than EI. All that we're stipulating is the probability of which measurement value we get. So for example, when there's no photon after you actually make the measurements you can be using positive operator-valued measurements. Everything from now on is going to be in terms of these measurement operators. Are people comfortable with that? Just think of them as being projection operators to guide your intuition. Okay, all well and good. That was a reformulation of quantum mechanics in another way that are actually formally equivalent. It's not clear at this point that I'm going to continue, but now I'm going to present some results which are actually easier to prove and derive using this new formalism. Okay, so great. We knew Schrodinger's equation before we came to class. We've just been told all about these measurement operators. If we want to get exotic we're not really sure we understand what it means to destroy a system when you measure it, but nonetheless the big dude up in the lecture he's just introduced us to positive value-valued measurements. Okay, but in the real world, especially in statistical physics real world you're not going to be certain about what the state of your system is. Everything that he just presented was all about what happens when you know exactly what your wave function is, what your wave vector is. The real world, remember the lecture that I presented, all about Diffiniti and all that other kind of stuff, you're not going to know the actual state. You're going to have a probability distribution over states. So in the real world, so that means that in the real world you're going to have what is sometimes called in the literature an ensemble which is going to be a set of multiple quantum states and associated probabilities. You will see this referred to as an ensemble of quantum states. So, okay, if that's what I've actually gotten in front of me, how do I deal with this beast mathematically? Well, it turns out that something a very convenient shorthand is to express that ensemble in terms of this thing right here which is called a density operator or density matrix. I want to emphasize, this is a sort of philosophical point that tripped me up until I learned to relax and love the bomb as the saying goes. And that's in which that somehow is real. It's a tool. It's a mathematical tool that's going to allow us to start by saying we've got this particular ensemble and calculate things like what will the ensemble look like 10 seconds from now, what would it look like if we were to do some measurements and so on and so forth. This is just a tool that's going to facilitate our doing those calculations, the density matrix. Don't necessarily think of it as being real. It's kind of if you want to, but you don't need to. Okay? So, a density matrix or a density operator, it's called a density matrix when you actually specify the set of states there. In terms of some terminology, this is called a pure state if the PI are a delta function, otherwise it's called a mixed state. Very, very easy to confirm that because these PIs must sum up to one that the trace of a density matrix squared, a density operator squared is an operator, so squaring it is another operator. Its trace is always between 0 and 1 and it equals 1 if and only if rho is a pure state. Basically, that's just reflecting the fact that if you take any probability distribution and view it as a vector and take its dot product with itself, you're never going to get anything greater than 1 and you will get value 1 if and only if that distribution is a delta function. Okay? Everybody good with this? Let me get rid of this meeting control. It's blocking my view of things. I'm scared though if I try to do this I might let's hope this doesn't lose everything. Oh, that killed it. That is very interesting. Okay, so we now know what the trick is. Let me just get it down then. Okay, very important point. In general, this gets yes, question. Oh, in general it need not be. Okay, I want to know why you consider only the case the density operator is a diagonal matrix. Yeah, why it's diagonal? Because of how I'm motivating it. I'm saying you got a bunch of states and we don't know their probabilities. So in that particular basis, those SIIs where I happen to know these are the states whose probability distribution I know but I don't know which one of them that gets you this density matrix but you're looking forward you're exactly correct in general this will evolve to being something that's non-diagonal. But in terms of this motivation we can view it as being diagonal. And in this basis it would not be diagonal. So I can always just take rather than expanding it in the SII I can expand it in some other space as an operator so it's just the normal similarity transform. Does that answer the question? Okay, that means PI is an eigenvector PI. Is it an eigenvector? Identical? PI is a probability or the SII. I'm not here assuming that the eigenstates of anything. This is a set of states and I don't necessarily know which one you're in but I know a distribution over them. They don't need to be orthogonal I'm not making any of those requirements. Okay? Very, very general. Yeah. Now an important thing is that in general two different ensembles might actually result in the same density operator. This is getting to actually exactly the point that you were just making. If I've got a different basis then this will be a different. So in fact you can also have two different ensembles in both of which it's diagonal but they're different ensembles they have different PI's and they're both diagonal but nonetheless they're actually the same density operator. What that's telling you is even if you diagonalize a density matrix don't describe too much meaning to the actual basis vectors of that diagonalized density matrix. It doesn't necessarily mean anything. Okay? And there are actually conditions you can solve for saying when do two different ensembles represent the exact same density matrix in general and things like that. Okay? Okay good. And now this thing's not working anymore. Oh come on. This has lost its amusement factor. Okay. So. Okay so the first question is that we're not defining what the density matrix is. There's a set of probability distributions and let's just say we're not doing any measurement or something. So we just evolve it forward in time. How does the density matrix, the operator evolve in time? Well, it's very, very simple to figure that out. Let you be the unitary operator that's evolving you forward in time. So we're assuming here notice that actually the system is closed. Then we know that every single state goes to u of psi. The the Hermitian conjugate goes to u dagger of psi. So by linearity at the end of this some dynamics according to the unitary operator u what we're going to be ending up in rho will actually go to u rho u dagger. Okay? Is this too slow for people too fast? About right? Kind of? Okay cool. Okay then we can also similarly look at rather than unitary is the dynamics when you're actually evolving across a non infinitesimal amount of time, the dynamics when it's an infinitesimal amount of time of course that's a derivative instead and we can use Schrodinger's equation to evaluate what the time derivative of the density matrix is. So the plugging it in, plugging it in is always a very good idea. The less math you can do the better it is. Always go after the little hanging fruit. The sum over p i, so p i again we're not changing that in time we're not doing any kind of measurement or anything. You take the time derivative of this psi i dagger times psi i but you don't apply just the chain rule it breaks up into two different terms you apply Schrodinger's equation twice, one to each of those and recall that the Schrodinger's equation for the Hermitian conjugate state you're going to actually be conjugating the i so it becomes a minus i. So what you're going to end up with right here is going to be the commutator of h and of the psi i. So in other words this is the dynamics of the density matrix infinitesimally and the reason I wrote it as h over 2 pi is I couldn't find a stupid h bar on my microsoft. So powerpoint. So there you go. This is called the Lieuval-Vernonman equation. So it's consistent of course with that one there and that you can play the normal games that you can write down what u is by exponentiating this beast right there in the case that the Hamiltonian is time invariant and so on and so forth. Okay, now I've been very very careful up to this point let me see how what would be good for how long to go for this first part of today for the first lecture go to quarter two do you think you are asking about time? Yeah, so actually this morning we don't have because there is no lecture after yours so this morning it's just your lecture so we are rather flexible. Okay, well we still need to go to coffee we still need to go to coffee at quarter two. So there is no coffee break today so there is coffee there. Because there is no break. Okay. Alright, so let me okay, so I'll keep going with this a little bit and then we'll take a break. So so far I've been very very careful to say oh the PIs are not changing because we're not doing any kind of a measurement and that allows us to use luminarity to see how the density matrix evolves with time now let's actually look at what happens when we do a measurement. And we're going to be using as I mentioned before these measurement operators are a convenient way to do that and again to reiterate to guide your intuition you can just think of them as projection operators where you're just projecting on to a particular eigenstate. So let's think about it carefully if the state of the system is psi i that's one of the states in your ensemble then the probability of getting a measurement value m if you are in states I recall the rules that we wrote down before this is the probability for we can I can go backwards will this work yes so remember back here yeah this equation the probability of measurement i coming out with the measurement operators given that you're in some particular state is given by this so that we just apply that to each individual state I and we get the probability of a measurement m if we are in state I is just given by this expression here which by the cyclic property of trace we can rewrite it like that okay also recall Bayes rule from many many lectures ago which says that the probability of a measurement is just the sum over all possible states you're in right now times their prior sum over all possible states you're in right now the prior times the conditional of getting that measurement if you're in that state we know what the p i are this is how the measurement operator gives us the p of m given i so you just plug it in and this is your final answer by linearity if you make a measurement the probability of the value coming out being m is the trace of m sum m dagger m m times row okay question in general it need not be it is went to projection operator but in general it not be remember we were weakening the condition of her mission by this great little expedient of just erasing it so in general it doesn't need to be okay okay any other questions so if we then going on what happens to the actual density matrix itself we'll also recall from what we presented in terms of measurement operators that if there is a state after the measurement in that particular situation then each psi i if you measure its state and you get the value lowercase m psi i goes to m sub m times psi i over the normalize over the normalization factor plugging that in again using linearity what that tells us is that if we actually measure the state we get the value small and then the density matrix becomes this right here okay so the probability of getting the value m is given by this equation and the density matrix if we do so is given by that equation so we've gone through dynamics that's the unitary stuff and the level of our normal equation and here we've also gone through measurements okay I'm thinking it might be just about a good place to start well let me go a little bit further okay so now let's go a little bit beyond that one thing that's very very often either due to savages axioms any quadratic loss function or maybe just because you like first moments is the expectation value of an operator so let's say that I've got some function f which is a function of the measurement see m itself is just an index that lowercase m it's just saying which measurement operator said yes it's me it doesn't have associated with it any particular value that's associated a value with every possible measurement what then is the expected value of that of that measurements now that we've got a value associated with it and then again you just use linearity what you're going to be getting is that the expected value under a density matrix of any particular function if you do the measurement is trace of this beast right here times the density matrix okay all right okay yeah I like this one this is to just give you a little bit of warnings about strange properties actually not just of density matrices density operators but of the measurement process in general basically if a tree falls in a quantum forest and nobody sees it it's still its density matrix changes so what do I mean by this you can see a little phrase right here let's say as always we have a set of density measurement we have a set of measurement operators we perform a measurement so that means that row M after if we get the measurement value M is given by this right here as always so if we measure the state of the system but do not know the result of that measurement so something measures it over there it gets measured it is physically you could take an Everett from many worlds perspective which I would adhere to or if you're somebody who prefers like the boars view of measurement then you've got a real hard problem but let's say there's a measurement over there that nobody actually knows the value whatever that measurement process is well since we don't know the value that means that the density matrix by the standard probability theory that's going to be an average it's just going to be the average over the possible individual measurements according to this probability distribution so row changes into this beast in general they differ and so that's why the glib phrase that if a tree falls in a quantum forest but nobody sees it still actually it's density matrix changes it's in taking Everett's perspective it could still be that nobody sees it but you do get entangled yes yep exactly that's one of the so what Gilger is getting at is this the very very good point okay wait just repeat for the zoom people because I'm not in the zoom okay so I just ask David if we take the Everett many worlds perspective if we violate what he writes in the red text or not so it's a very good question so it's a little bit of a side and this might be a good way to actually end this first part of today so people I assume have encountered this huge wars about what measurement actually means in quantum mechanics because the measurement process is a non-unitary and that violates all the axioms of quantum mechanics not all of them but it violates in any axiomatization it violates one of the axioms Bohr and others fellow travelers famously said shut up don't think just calculate and that's pretty much the mantra that's what you'll read in your textbooks there's this discontinuous jump nobody's saying what's really going on people like Wigner said maybe it's some kind of a collapse of the wave packet in the human mind nebulous weird everybody getting mushy and squishy and basically dropping too much LSD and that's kind of what the standard traditional view is in textbooks Hugh Everett and he's got actually an amazing story about how he was abused by the physics establishment but he in his PhD thesis which actually is a 20 page article in physical review that anybody here can read it's a beautiful piece of work he did this under John Wheeler he simply pointed out that if you say that what a measurement is is that you have a physical system over here and a physical measurement apparatus over there and there's an interaction in Hamiltonian which has certain properties then after that if you look at the state over here of the measurement apparatus you will learn something about the state of the system that it was that it got interaction with and it actually all the rules of measurement come out if the entire thing goes according to Schrodinger's equation so Everett actually never used the word many worlds that was I forget who that was that was somebody several decades later but the idea is that each of these possible joint pairs of state of the system and the associated state of the measurement apparatus each of them some people say is a different world but really it's just Schrodinger's equation and so whatever it was saying was X and A undo all this crap about what mysterious thing happened under measurement just do Schrodinger's equation and the reason that we right here see one particular result of a measurement apparatus is not because in some sense that's what reality is reality is also another set of us so to speak that sees another value of a measurement apparatus you don't like that the universe doesn't give a petunia about whether you are comfortable with something or not that Schrodinger's equation explains everything you can see experimentally you physicists have to change your brain to match reality not the other way around don't get too full of yourselves human beings that's one of the that's actually to me the primary lesson of all quantum mechanics that human intuition ain't worth crap and stop following it so anyway that's all on the side I'm getting at that that actually answers Gilda's question because if a tree falls in the forest if you make a measurement to Everett that actually means something concretely it means there is this interaction between the measurement apparatus and the system being measured so people like Bohr and Wigner they don't can't say if a measurement is being done or not it's this nebulous undetermined thing so in Everett this is actually slightly modified for the many worlds view where you would say if a tree falls in the forest and if it actually there's a measurement apparatus that we don't get to see but nonetheless that does actually interact with the system in that case it's density matrix changes okay one thing to notice though in general whether you take the many worlds perspective or the Bohr perspective if rho is diagonal in a particular basis and if the measurement operators are projection operators in that basis under those circumstances the right-hand side of this is equal to the left-hand side so if a tree falls in the forest and it gets measured by something that's exactly aligned with the state of the tree so to speak then nothing happens to the tree so in that case you don't have a Zeng twist okay partial traces let me just see what I want to do now okay let's take a break then and I'll start to do quantum information theory when we come back okay any questions or if not let's just take a five minute break okay five minutes break and it's stopped