 Error in the assignment, right, it was, should have been a probably amplitude and it wasn't a normal, I think it was a possibility, so it's a good way out. Made the expectation very nice and clean. I thought it was a good idea. There is another student coming. As I mentioned, I feel on travel, there's afternoon. And so there won't be a problem session this week up. Gopi, are you available? Yeah. There's no. No, not this time. Okay, so maybe during those hours you can hold the office hours. I'll discuss the homework with you. Yeah, sure. You can meet with folks in 190 this time. Or anything you're offered. Okay, we'll discuss that. I'll send a message to everyone. But as we saw, unfortunately there's a room conflict at 11 o'clock in Room 190. So here's, and there's just no other room available. So here's what I propose. The chair seminar finishes at 15-ish. So we'll get started at 10-15 in 190. And I'll go over the problem set with you. Make sure everyone understands what's going on. And address any questions. And then we'll, you know, get to work on it. And we'll continue to come around. We'll use the front of Gopi here for a continued discussion from 11 to noon. All right. And I'll start next week. Then I'll start a week in Friday. But I'll send the marching orders for this Friday. And the first example will be sometime in the week of September 29. In fact, we'll have to find the time. I don't want to have any classes. I'd like to have more time for you to do the exam. I'd like to do it in the evening. But I know that people have other courses. So I'll try to find that. And that will cover material through the next problem set. Okay. And basically through the lectures this week. All right. Very good. So last time, so we were talking about quantum measurement as we had for a while now. And we introduced this one standard paradigm that helps us to understand a particular example of quantum measurement, the Stern-Gerlach apparatus, which is a way of measuring the projection of angular momentum along a certain axis. Okay. And in particular, if the total angular momentum is spin angular momentum, then it's a measure of the spin projection of some axis. So silver atoms are, atoms that are in their electronic ground state have only spin angular momentum. And that's coming solely from the spin of the valence electron. And so when you create a collimated beam and you send it to this device, what you find is that it separates into two beams, corresponding to the two possible eigenvalues of the observable Sz. Okay. And so if you, say, block one of these beams, then you've done a projected measurement with probability a half coming out of the southern. The atoms in this beam are projected into the state spin-up along this axis. Okay. So this is an example. Sort of a rejected measurement. And it's not quite a project of the paradigm of the projected measurement in the sense that in a standard paradigm, it's projected into one of the eigenvectors corresponding to the eigenvectors of the observable being measured with a certain probability here when we block this beam. If we didn't block that beam, did we really do a measurement of spin-up and spin-down if we didn't book that? Well, the answer is no. Not really. It kind of depends. I mean, if this were in the vacuum and there was no other perturbing noise, then in some sense the atom is in a superposition of this beam and that beam. It's like sending it through a beam splitter. And as we said, an atom could be here or here, but we really shouldn't say that it's more of a lieutenant coherence in a superposition of the two. And if we combined those two paths, we would get interference. Okay. And the other thing we talked about briefly in this one is to mention again that it's a concept that is perhaps less familiar, but nonetheless important. And that's the notion of a weak measurement, a measurement that doesn't distinguish necessarily between the eigenvectors of the observables completely. It partially distinguishes between those alternatives. So, for example, if we put, we have the same device, but we put a screen too close in to the point that the beams haven't really separated well enough. And we've collected a histogram of the counts. We would see this kind of double hump structure, but the tails overlap. So there's no way to completely distinguish whether the atom is in this hump or that. If it lands here, well, I can't really say what its state was, right? Whereas if I had these two, and I put the screen over here, then these two alternatives, if I, you know, if I like this kind of lines once in a while, these two alternatives are, for all intents and purposes, other than a tiny, tiny bit in the tail that we can neglect, completely distinguishable. So that lands here, we would say, yeah, if the land was spin up along Z, if it lands here, we would say it's been down along Z. But we can't perfectly correlate where it lands on the screen here with what spin state it was, right? We can kind of do it but only with some confidence. So the measurement outcomes here correspond to, say, taking this data and fitting it with these two curves and saying if it lands in this thing, I'll call it the plus alternative. And if it lands in this guy, I'll call it the minus alternative. So the true alternative is, but they're not orthogonal. They're not completely distinguishable. And so each one of those outcomes is associated with a peel-bian element where, say, for example, this guy is with 95% confidence, I would say it was spin up, but maybe 5% I would say were spin down. And if it lands in this guy and say with 95% confidence, it was spin down, but 5% confidence, it was spin up. So it's a measurement that says maybe. Maybe it's been up or maybe it's been down. I can't tell you for sure because my measurements can't tell the difference. It lands here, what do you say? So in this case, this is a resolution of the edge entity. That's to say if you look at this, the sum of this plus this is the identity. And we would say we can predict the probability the spin is going to land in this yellow pump is equal to the probability that it lands in the minus home is that. And that's a generalization of the corner, right? So last time we were discussing, we were discussing a little bit, at the end of lecture, how do we think about the state that comes out of this, not the old way. Before I get to that, let me raise the following question. We loosely defined a pure state. We said a pure state was a state of the quantum system where we have maximum possible information in system and quantum mechanics. There's no information missing. We have all these possible paths. That's when we have that, then we assign to the system a pure state in such states where it could be mathematically represented as the kets or vectors in Hilbert space and those vectors can be normalized and the overall phase is irrelevant. So question, how do I prepare the system in a pure state? And yes? Sure, when you block the lower part and you only have the pluses, you're certain about that. Exactly. So that's it. So that's how you make a pure state. So a preparation procedure. Do a projected measurement, permission observable, keep the state corresponding. So, non-degenerative. Then we are guaranteed, if we've done that, if we could do such a measurement, then we have totally prepared the system that much information we can. This is as fine-grained a measurement as we could possibly do. We can't get any more. This is of course not such a measurement. We did that and it landed somewhere. We blocked, say, we blocked, you know, this guy over here. We wouldn't know for sure. So we wouldn't have prepared. There's missing information still. But if we did this, we prepared the system in a pure state. So, as I said last time, we reduced the simplified with the patient's set of drawing, you know, graded magnetic fields and pointed north and south poles. And we could say there's some kind of drawing on the opposite of the black box. And it measures, for example, SD. And so, whatever comes into there, it has these two output ports. We're going to block this guy, then we have prepared the system in that pure state. Now, the fact that we have prepared the system in that pure state doesn't mean that every measurement we do, we're going to be able to predict its outcome with perfect certainty. Not every measurement we do on a pure state has an outcome that is deterministic with probability one or zero. I mean, some are and some aren't. So, for example, if I measured this again in SD, I mean, I know with unit probability that it's going to come out absolutely here and nothing's going to come out here. So, that's with unit probability. So, with probability one, that happens. If I prepared it in a pure state and I can know the probability of this happening and getting this more. In contrast, of course, if I measure in some non-commuting observable so I prepared the pure state up along z but then I measure along the x-axis then the measurement outcome is random, even though it's a pure state. And that is the bizarreness of quantum mechanics. That randomness, even though I have much information I can, still I can't predict the outcome of every measurement I do. And so I'll get spin up along x or spin down along x, each with probability. So, what about the state of the system coming out of the oven? Well, let's say we did a series of measurements. So, I mean, here I have my oven, Adam's becoming out of it and I put it through an SD, Snervalic Analyzer. And we said that we get suppose that I took this oven and I took the Adam's and I put them through a Snervalic Analyzer along the x-axis. What would I expect? Same thing, right? I mean, with 50% probability of go in the x-up and 50% will go in the arm B. In fact, of course, there's nothing special about x, y, or 0, any direction in space. We expect just intuitively, and we can then think about how to get this more formally, that if I analyze this with a spin analyzer along any direction in space, it doesn't matter x, y, z, or along this. I mean, there's nothing assumed in the oven. The oven is isotropic. It doesn't have, it doesn't know what this is versus that. There's nothing to tell it, assuming that that symmetry is not broken. And so no matter what I did, no matter what I would find for this state, spin up along that axis with 50% probability and spin down along that axis for no matter what that axis was. Now, what is the quantum state that we would assign to the silver atoms coming out of this oven which would have this kind of probability? I mean, if it was spin up along z, that would agree with that, but it wouldn't agree with that. If it was spin up along x, it would agree with that, but it wouldn't agree with that. If it was spin up along some other direction, well, it wouldn't agree with that. So it's spin up along no direction. It's been down along no direction. So what the heck is it? Well, the answer is it's not a pure state. It's not a state. These measurement results are not consistent with any pure state. Any pure state of a spin on that particle will spin up along some axis. So there's got to be something more than pure states in physics. And the answer is yes. Here's the one I'm going to find. It's a somewhat contrived example, but not completely nuts. Suppose that I have a preparer, and the preparer has access to both quantum randomness and classical randomness. What do I mean by that? So inside, now I'm going to draw my little Stern-Gellic, and I'm just going to emphasize this. And I'll just forget about the width of the beam and say that each atom in here goes up, depending on the table to measure them. Sometimes they come out here, sometimes they come out here, sometimes they come out here, sometimes they come out here. Let me just instead draw it that way. I'm going to draw that. So this is my Stern-Gellic apparatus, and it's, say, oriented along the z-axis, for example. Now, suppose the preparer, this is such a dental beam that's coming through that the preparer can measure this one atom at a time. And the preparer then can determine whether the atom in number has been down. She collects all her spin-up atoms in some magnetic bottle, and she collects all her spin-down atoms in another bottle. She has prepared two collections of pure states. She has a bunch of spin-up atoms, and she has a bunch of spin-down atoms in her possession, in her bottles. And now what she does is she flips a coin. So she flips a coin. So she flips a coin, and it could be either heads or tails. And if she gets heads, she spits out of this device, which is now inside a big black box that you can't see inside. She spits out an atom, and she chooses whether to send me one of these or to send me one of these, or whether she gets heads or tails. Does everyone understand the idea? So she can send me one atom at a time, either a spin-up along the z, or a spin-down along the z. So she decides, with probability, so she has a probability for spinning me a spin-up, if hence, and she sends me with some probability which she'll send me spin-down, if tails. So these atoms are coming at me. And now I can analyze them. Suppose, for example, I do a spin-down analysis along the z-axis. I'll get one of two possible outcomes, assuming I separate my means sufficiently to resolve them. What probability will I see spin-up along the z? So if it's a fair point or not, suppose that that point has this probability to show heads, and this probability to show tails. So with what probability will I see spin-up, this probability, because if she sent me spin-up, I'm going to get spin-up. If she sends me spin-down, I'm going to find spin-down. I don't know which one she sent me. So with probability, this is the probability of heads. I'll see this. And with the probability of tails, I'll see this. Is that clear? There's nothing to do with the boring rule. It's just that there were these two alternatives. She might have sent me a spin-up at him. Now, of course, this is subjective. I don't know what she sent me, but she knows. So she, with probability one, can predict which way it's going to go. But I don't know. So from my point of view, I see a bunch of random up and downs. From her point of view, she goes, oh, that one's going to go down. She knows whether she got heads or tails. But suppose now, instead of analyzing analyzing the state of the spins that come out of this contraption with a Stern-Griller-Copper lattice along x. Thank you. And I suppose they do along x. So there's these two possibilities. What is the probability that I'm going to see x and what is the probability I'm going to see spin up along x and what is the probability I'm going to see spin down along x? Let me say this slightly differently just so I don't confuse notation. Let me call this the probability of heads and the probability of tails. So what is the probability in this kind of analyzer to find, say, spin up along x given that the pair is spinning? Here's how we calculate it. There is a probability to find up along x given she, the preparer sent spin up along z times the probability she sent me back or she might have sent me spin down, right? And if she sends me spin down then I have to say, well, there's a probability of finding spin up along x given that she actually prepared spin down along z I have to wait that by the probability she actually sent me. Is that clear? This is logic, right? This is just classical conditional logic. She might send me this she might send me that I don't know, but she's going to send me one of those that's the promise. And I would say, well, I know she also showed me her coin whether it was a fair coin or how many times it's more likely to come up heads versus tails, right? So, Howard, what is this? The probability to find this given that she says, what is that? How do you calculate that? It's a half, and how do you get that? The absolute square of the... Right, you have the square of the use the bornable times the probability she actually sent me that that was the probability of heads, right? Plus, if she sent me down along z and that happens with the probability that she got heads, okay? Or, if I wanted to say the probability she sent me this either way. So this is the probability to get plus x given that she sent this times the probability she sent this times the probability to get up x given that she sent down along z times the probability she actually sent. That's logic. And in this particular case what is the answer? As was said, this is a half and this is a half in this particular instance. And this thing is equal to a half then times the sum of the probabilities of heads or tails together is one and a half. So in this case I'll see random 50-50 chance, okay? So this state the state of the can I assign to the spin of the atom here is not a pure state. It's not a pure state because I'm missing information. That missing information is whether she got heads or tails. The information's there I just don't know it. If I knew it then I would say this guy's up along z this guy's down along z this guy's down along z this guy's down along z depending on the sequence of heads and tails. But she knows it but she's keeping that information from me and I'm missing that information, okay? So this kind of state the state of this spin is what we would call a statistical mixture. She has these bottles of spins and she can make something out of it any way she likes. She can make stuff. This is what we would call this is an example we call a mixed state. Okay? So this state is a statistical mixture weighted by different probabilities. So I have some probabilities of this guy in the mixture and some probability of this guy in the mixture. Okay? This should be counter-attractive with a what we would call coherent superposition. What is a coherent superposition? A coherent superposition for example is this. This state is not a mixed state. It is a pure state. Does it mean that every measurement I do will have a definite outcome but it's a pure state? And that state can have the same probabilities for measurement outcomes along say the z-axis. The probability of z instead of along z could be this and the probability of z instead of along z is this and that could be the same as the statistical mixture I have of these guys. So if I measured this guy along the z-axis I couldn't tell the difference from this. Right? It could have been that we're going to flip. So what's different? The difference is interference because in the statistical mixture case here we saw in this case the outcome of seeing up and down along x is a half, right? Suppose the probability in the head equal the probability in the tails is a half. We have an equal statistical mixture of spin-up and spin-down along the z-axis. Okay? So the probability to find up along x or up along z and down along z equals a half. And the probability to find up along x and down along x is also a half. As we just saw if I have a pure state then suppose I choose this guy. This state the probability to find up along z and probability to find down along z is a half. Just like this. But the probability to find up along x is what? 1 or 0? 1. This guy is 1. This guy is up along x. And the probability to find up along x is 0. Now how the heck does that happen? It happens because of interference. Even though this is a 50-50 chance of up along x it's not a statistical mixture of these two. So the pure state would be the person sending the particles out. So this is an important point. States are subjective. From the point of view of a preparer the state is pure. Because she has complete information. From the point of view of the observer the state is mixed. Because he's missing the information. It's like the chalk in my hand. I know where it's going to be so it's a deterministic outcome. But for you it's a random outcome. But that's about that what's weird what happens even when you have complete information you still can have randomness. So let's talk about this interference stuff once again. So let's say we have a pure state sample. Which we said is a coherent to a position of spin up along z and spin down along z. But that particular set of probability amplitudes the state's been up along x. Okay? The probability to see according to the Bourne rule that and that is equal to just writing it out in this superposition just to emphasize a point which is equal to contentic and slow here. And now let's square out the term I have. The square of this term plus the cross terms. So that's equal to the square of the first term and then the cross terms. So the cross term I have this times the conjugate of that. Right? Let's take a close look at this expression. These first two terms are exactly what we had for the statistical mixture. This is the probability of seeing spin down along x given that the state was actually spin up along z times the probability that it was spin down along z. And this is the probability of finding spin down along x given that the state was actually spin up along z. Times the probability that the state really was spin down along z. And that's what we would have if we had a 50-50 statistical mixture of these two terms. But that's not the whole story. I have these terms which are the cross terms which are the interference between these two alternatives. So we don't know it's in a coherence of a position and those two terms interfere. Okay? So these are the classical alternatives. So this is classical statistical and this is quantum interference illogical. And what is it in this case? Well here we have to go through and do all that. This guy is equal to 1 over root 2 this guy is actually minus 1 over root and so this is the conjugate minus 1 over root 2 and plus 1 over root 2 you can do it out. And so this is equal to this is a half so I have a quarter plus a quarter is a half minus a quarter minus a quarter which is minus a half. Which is zero. Complete destructive interference and that's what we expect. If the same spirit's been up along x the probability of being spin down along x is zero. And the reason is there's this destructive interference which is one way to interpret it the reason is but a way of thinking about it is that whereas classically you would have these two alternatives quantum mechanically those two alternatives destructively interfere. So the distinguishing feature of the pure state from the mixed state is the ability to see interference between those alternatives. Questions so far? So now if we want to remember what we said was the job of our quantum mechanical formalism is to allow us to calculate the probabilities of outcomes of measurements to predict the outcomes of measurements to the best of our ability. And we developed this formalism so far to do that. We had the state vector of quantum school. When we have statistical mixtures life has become more complicated after they say well if she got heads then she sent me this and this is the probability and I have to go through all these extra steps. And there's it seems like I should be able to encapsulate this all. There should be one thing I call the state of the system that allows me to calculate the probability of outcomes whether or not the state is pure or mixed. So there should be some generalization of the state from this the ket. There should be some more general state of the system and that more general state of the system is what we call the density operator. How many people have seen the density operator before or in their previous studies of quantum mechanics. Not a common deal. But you know the state that comes out of the other is a density operator. That's to say that's our state of the system. We can't just sign up your state because we're missing information. We just don't have it. It's like a coin flip. So what do I mean by that? So let's look at what we did in this case with the heads and tails. So what we said was so suppose we have a statistical mixture of spin up a long arc or z that doesn't matter and spin down along z. Prepare the probabilities e plus and minus. Within those coins. That's the coin has that. And I want to find what is the probability to find say spin up along an arbitrary direction. I'm going to have a Stern-Gerlach analyzer. It's going to analyze it along some random care. X, Y, Z 45 degrees between the X and Y axis. Some axis in space. How do I calculate that? Well we just said we have to argue there's the probability I'll see spin up along Z where the error sent me spin up along Z. There's some probability that she did it. And if she sent me spin down along Z then this is the probability of mine being spin up along that direction and I have to weight that by the probability she actually sent that to me. This is a statistical mixture. There's no interference between these two alternatives. So let me write this. This is equal to this times its complex conjugate. And the complex conjugate is reversing the bras and the caps. And so I'll write this. And because the inner product is linear I can rewrite that probability in the following way. This is equal to, this probability is equal to that product. And then I'm going to write this as the probability she sent me up times that projector. Plus the probability she sent me down times that projector. You have to read that those two things are equal. Right? So look what I've done. Everything about the preparation procedure is in the brackets. Is in this thing. Everything about how I'm analyzing the system is outside. So this is about the preparation procedure. This is about the state of the system that I know. So this operator is an operator we call rho. And this operator is the density operator. The term density is such a arcane and it's a stupid name at this stage. We really should call this the state operator because this is the state of the system. Now let me emphasize a fact. This is an operator but it's the state. Okay? You have learned in the past that states are vectors and observables are operators. But that's just the way you learn it. We can define the state as an operator. And we do and we should. And it's what do we want from the state? What do you want from these? What you want is you want to be able to calculate the probability of outcomes of measurements given the state. And you can do it. If I want to find the probability of an outcome I just take the expectation value with respect to the outcome. And that is the probability of seeing that. Because this we just calculated. It's just a new piece of mathematical tools and formulas you need to learn. Alright? So the density operator is the most general state of the system that you could possibly have in quantum mechanics. It includes pure states and a row. Mix states. If it's a pure state it would be if one of these probabilities were one and the other was zero. Then it would be a definite state we know. But if these probabilities are not then it's that. So let me look at my cheat sheet here and remind myself of the order I might have thought about doing these things. So this is my I'll continue here. This is my state. Let's say we have the following things. Say things a little bit more generally now than just a thing one half or even a two size coin. I could say a general statistical mixture can be thought of as the following. The preparer sends one of an ensemble pure states and she does so with probability. So now she doesn't have just two bottles she has as a bottle of as many different states as she likes. They didn't even have to be orthogonal states. She has a bottle spin up along x, a bottle spin down along y, a bottle spin up along this direction as well. Whatever she likes. And she sprinkles from it. She rolls the dice. She has a 12 sided dice. She has 12 different pure states she can send and she sends them off. She mixes with that probability. What is the state of the system? Can you guess this? So here she had two choices. She had a coin with two and she had a bottle of spin up along z and a bottle of spin down along z. Now she has all kinds of bottles of all kinds of spins along all kinds of axes. Would be a 12 by 12 matrix pure is it a two level states? Look into the matrix business in a bit. Or a sum of both. Exactly. It's a little salty there. I'll come to the question of matrices in a moment. It's a sum over all the members of the ensemble weighted by the probability times these projectors. That is what we would call a statistical mixture of these size. If this is a spin one half particle even if I have 12 different members of the ensemble this is still a two by two matrix because it's still a two dimensional pure space. Okay. There's a 7. Okay. So what is a pure state? A pure state what is the density matrix for a pure state? This is one. It's just this. There's some sign. Looks like that. That's a pure state. A pure state is a projector along that side. Okay. So if it's a pure state there exists a side for which this is true. Notice one of the nice things about the density operator formalism is that it gets rid of the phase ambiguity because the overall phase of side cancels out. Alright. What about a thermal state at temperature T? What you've learned in your statistical physics course that if you're at thermal equilibrium at temperature T we don't know for sure what specific state in the system is we can only say with a certain probability we have a Boltzmann distribution of possible energies the system can be in. We know that probability the probability of having a certain energy is the Boltzmann distribution over the partition function of temperature T. So what is the quantum state of the system associated with this? Well statistical mixture of different energies it's not a coherent superposition of these different energies it's a statistical mixture and so what we would say is a sum over all energies let's say there's the nth energy V. And that's what's coming out of the other so so now what we want to do is delve into this a little bit more deeply to understand the nature of the dancing opera trying to develop a little bit more intuition about it and we use it to do calculations maybe I should say the following let's just say let's understand your dance suppose let's just talk about things a little bit more so what we just said is that given the state row say for spin one half the probability to find say spin up along some arbitrary direction was equal to this that's what we just showed you can think about this as a diagonal matrix element what do they tell us what do they tell us well let's take a look at a pure state consider the state which let's say we have a state which I'll write as a tan which is some superposition of up and down an arbitrary pure state is of that sort and let's write it as a dancing operator so now we just got plug in so let's multiply that first two terms look like the statistical mixture but these terms the off diagonal term of the matrix elements don't look anything like statistical mixtures because these are the terms that are related to what we call the coherence so written as in a matrix representation in basis of say spin up and spin down along c let's look at two different cases suppose I have in one case a statistical mixture these two guys with mixing probabilities both is equal to this what is the matrix as a 2 by 2 matrix where this in this order P plus P minus 100 diagonals and what about the off diagonals 0 this guy has an off diagonal matrix element better 0 contrast pure state the one we wrote down over here which is some arbitrary superposition the matrix representation of this at c plus where the c minus on the diagonal c plus on the diagonal c minus c plus c star minus on the top right so we can have two states which have the same diagonal matrix elements but very different diagonal matrix elements where they would have look the same as if we measure them x and z but if you measure them in x they would show very different and these off diagonal matrix elements are what we call the coherences as opposed to the populations so we could say are you down and they can up and down to the fear with one another the off diagonal elements of the density matrix are things that are telling me whether or not those two alternatives can interfere of course that's a basis dependent question it's interfere in that respect to that so that's an important point so let me this part of the discussion we do something a little bit more general yes please can we diagonalize the matrix yes it's not a done question and then like perform I guess the classical probability absolutely so yes it's a very good intuition something we'll talk about probably on Thursday of course this is a permissioned matrix and so we can diagonalize it and its eigenvalues and eigenvectors are important ways of characterizing the state of the system you can always think about the density matrix and the statistical mixture of its eigenvectors weighted by its eigenvalues if it's a pure state it has one eigenvalue one and the rest is pure if it's a mixed state then it has more than one non-zero eigenvalue which represents statistical mixtures of more than one pure state cool alright but before we get there I just want to say just to emphasize this idea of the coherences versus the populations so let's talk about things just a little bit more generally not just about how it's been when it happened let's just say let's consider consider a general state in a d-dimensional over a space some it's got d dimensions it could be a spin j with 2j plus 1 so let's say if I have a pure state I can write it as a k I don't have to write it as a density matrix and that is a sum over all the elements of some basis written as a density operator this pure state is these two indices so the elements of the density matrix which are called the density operator which is called the density matrix people use that term interchangeably people talk about the density matrix even when they're talking about the operator so the matrix elements so again the diagonal elements to this which is equal to the probability of elements let me write this in terms of a magnitude of each complex number and a phase so I bring each complex number c in polar it has a magnitude and a phase is that clear so notice these octagonal elements have something to do with the phase relationship the octagonal elements capture the interference because they have something to do with the relative phases so they are the coherences what we call them these are sometimes called the populations so what about when I have a mixed state then I have a statistical mixture of different pure states so if I have a statistical mixture of different psi i's each of which has some potentially different superposition so I'll say psi i is some superposition for the i-member of these then my state here is a sum over i p i mixing together these pure states and that is equal to sum over alpha and theta sum over i c i alpha c theta weighted by the probability so in this case what we have is the elements of the density matrix have to be averaged they have to be averaged over the mixing probability this is confusing we have two different kinds of probabilities we have the normal here we have the mixing probabilities and then we have the probability amplitudes they're not actually as separable from one another as I've made them appear nonetheless we can go forward this way and what we see here is that in general this matrix element I can think about as an average over the ensemble of all the different elements this bar is this average so what you can see here is that if I have a mixing of different states all of which that have random phases relative phases relative to one another that when I mix it this is going to generally average out to zero which is why statistical mixtures wash out the phase differences and wash out interference the loss of coherence is called decoherence and that's the subject we'll talk about in the next lecture