 What did you say to ask her a real quick question about the homework? Would it be a right or no? Let's talk about it. OK. OK. All right. So the fun keeps coming. We do have a work to do today, and there's still more coming. All right. So last time we were laying out, you might say, cautionism, axioms, or just what is the structure of the fun theory itself. It continues to be a challenge to see if the structure can be thought of as arising from some basic principles in the same way that Einstein's relativity sort of just arises from just a few basic principles, right? The speed of light is constant in all reference points, and it's equivalent to very different things. Given that, you can derive the branch transformations and derive it all. Whether the same thing can be said of quantum theory is another story. But with that said, let's just say what that structure is. And the way we're laying out that structure is to think about this from the point of view of what's commonly known as the Copenhagen interpretation, which is to say, we're just a set of rules for doing what physics should do. And what physics should do is predict the outcomes of experiments. And of course, you can only do those predictions with some confidence. We can assign probabilities to the outcomes. So that's what we demand of our theory, that it should be able to allow us to make those predictions, assign probabilities to outcomes or measurements. And so the fundamental mathematical object in that is the state. The state is the mathematical object, which allows us to do this calculation. And I've defined, although I haven't quite made this clear what I'm really doing by this, but it'll become clearer over the course of this week what I mean by a pure state, not to be confused with eigenstate. They're not the same words. They don't mean the same thing. So a pure state is a state that we would assign to the system. If we have maximum possible knowledge, it's consistent with quantum mechanics. And if we have all the possible knowledge we can, we assign a pure state. And that pure state is a vector in the Hilbert space. Really, we can say it's a ray in Hilbert space. The overall length of the vector, it doesn't, is something we can define to be whatever we like. And by convention, we choose it to be normalized in this way. And as we discussed, that still doesn't really pin it down because the overall phase that says we can multiply this by any phase factor, a unit magnitude complex number, that doesn't affect in any way this. So that's one object, the most important object in our structure. We also, although it's traditional to define this in the Copentabian interpretation, really, this is not a necessary partial, if you like, of quantum theory. We'll come back to this over time. But let me just state what it is. We would state what we say is that we talk about observables. These are physical quantities, like energy, momentum, decision, angular momentum. Things that we're familiar with in classical physics, but it also includes other quantities that have no classical description of spin. And what we can say is that our mission operators are observable. Let's just say we can measure them. We can find out the value of these things. But the compass isn't true. There are things that are observable that are not mission operators, contrary to what you might have been told. And we'll come to that. We gave you an example of time. Or for another example is phase of an electromagnetic wave thought from a bottom field theory. There's no permission operator for phase. But we measure the phase. So contrary to what you might have been told, there's not this one-to-one relationship between observables and mission operators. But what we can say is that, and again, this is not really a necessary postulate that we need to understand quantum theory. But it's something that's traditionally discussed. And so I'll put it here, and we'll see why this isn't quite a necessary postulate. But as stated, if I do a measurement of a permission observable, then the results that I will see, the possible results I can see in such a measurement are the eigenvalues of that. What is important is this, which is what we require of our theory to begin with, which is, all right, you have a state. So if the state is a pure state, and that state is then described by this vector in Hilbert's Basin taking, unless we say otherwise, we'll always assume that we've normalized that vector, then if we do such a measurement, and as we said, we can only find one of these eigenvalues, and if that eigenvalue is a, and that eigenvalue is non-degenerate, so there is one eigenvector corresponding to that eigenvalue, then the probability to see that result in the measurement is given by the Born rule. The Born rule is psi star psi. OK? And it's the square of the probability amplitude, right? So in this case, the state generally can be expanded in the basis of eigenvectors with probability amplitudes. This is just the square of the amplitude associated with that eigenvalue. As you said, if things get a little bit more complicated, if we are looking in more general contexts, for example, suppose the eigenvalue was degenerate, well then it depends what the probability of seeing that result really depends on what the device is doing. Because for example, we know if we have a degenerate eigenvalue, then there is an eigenspace associated spanned by all the degenerate eigenvectors, all of which that have that same eigenvalue. There's not one vector. There's an eigenspace, right? Now, whether or not the probability depends upon whether or not the measuring device knows about which of the eigenvectors in that space we've observed. So for example, as we discussed in previous lectures, we can fully resolve which eigenvector or a vector in that space by specifying other eigenvalues of mutually commuting observables. But if the measuring device in some sense doesn't measure what the other values are, then the probability I see is just the sum of the probabilities of all of the possible eigenvectors overlap all of the probabilities for all the different degenerate eigenvectors with that degeneracy. Summon, because I don't know which one happened. If somehow the device resolved which of these eigenvectors it is by specifying other quantum numbers, then it would just be one term in the sum. So it depends what's going on. What we decided, though, is we could write this in sort of a more general way by saying the probability of measuring A is this overlap, the overlap between the projection on the state and the state. Where the projection, so if it's not degenerate, then there's just one term, and then I get this. But if it's degenerate, then the measuring device doesn't resolve which one happened, then I have to add them all up. And of course, all the probabilities sum to 1, the total probability of having something is 1, because we've normalized that. That's equivalent to the fact that this sum of all the projectors is a resolution of the eigenvector. Because then if I take the expectation value of this with respect to psi, I get the sum of the probabilities of this 1. So another point we made last time was the fact that really what is necessary in order to define probabilities to measurement outcomes is not that I do what's called a protective measurement. And protective measurements are specified by measuring apparatuses that distinguish yes or no questions. Are you in this eigenspace yes or no? And then you assign some probability to being in that space. And these different eigenspaces, the spaces associated with the different eigenvalues, are orthogonal. Because we said that the eigenvectors or eigenspaces of permission observables or permission operators or different eigenvalues are orthogonal. But more generally, as we will see later in this lecture, any quantum measurement is defined not by a set of projectors, but a set of operators that I call E, where E are positive operators. You've seen positive operators now in your homework. Positive operators are permission operators whose eigenvalues are positive numbers, but actually none they are. What we call them. And if I have such a set of operators, such that they form a resolution of the identity, then I can assign probabilities to outcomes. This resolution of the identity, this is what has this horribly heavy-handed name that still sticks up P O V M. And what it says is that the probability of finding outcome A is this. This is a number. And because of this and because of these two facts, these things are positive numbers that lie between 0 and 1, and that's our probabilities. And they sound to one. So if I have such a mathematical structure, I can assign probabilities, and that's what I need. And every measurement that we could possibly do is defined by this. This is a special example. These are our projection operators, positive operators. What are the eigenvalues of projection operators? So if I have this, and it's some lambda, it's 1 or zaneral. If psi is in the eigenspace, when you're projected, you get back. The state doesn't change it. If it's not in it, and I come back to itself, then 0. Is that clear? Of course, if that, these are the eigenvectors. So projection operators are like yes or no answers. Are you in that space? Yes or no? They have fewer faults. So they're very sharp measurements. General measurements aren't that sharp. They can't be, because most devices can't resolve so perfectly, argue this or do that. They have finite resolution. We all know that. Telescopes, what happened? They always have finite resolution. And so whatever we're doing can't always be a projected measurement. We'll see an example of that later in the lecture. But maybe you never heard about do-vampi-should. And this is the more important rule. This is the generalization of psi star psi. So a few of the things we discussed last time, I just wanted to reinforce these things, because they're very important concepts. We talked about the expected value, and we said that if I had a state, the expected value, the mean value of an observable, I've given by statistics, we proved that this was equal to. And this is equal to, for a pure state, the sum over a, little b, fine, OK. The other thing we talked about was the uncertainty. Uncertainty is to say the following thing. I want to know how certain am I that when I do a measurement, I'm going to get the expected value, OK. So the uncertainty, which depends on the state, is given by the square root of the variance, which is the expected value of the difference between the man and his expected value. And that's the same thing as the average of the square minus the square of the average. And what is true is that what we have is the uncertainty principle, which is easy to prove. It's proved in the notes, which says the product of the uncertainties associated with two observables. So this is really equal to the average of a squared, times the average of b squared, the average of b squared. And when we say that, also, the square root, that those, that quantity must be bigger than or equal to one half expected value of the commutator absolute value, which says that if the two operators don't commute, if the two permissions are both, don't commute, then in general, I can't simultaneously be certain about a measurement outcome I will find if I measure a or I measure b. So the uncertainty that in my measurement outcome that I'll find in a as measured by the RMS is given here and similarly for b. If, for example, the state is an eigenstate of a, what is the uncertainty in the measurement outcome? If the state is an eigenstate of the observable a, what is my uncertainty in what I will find in the measurement of a? Zero. Zero. Exactly. It's an eigenstate of a. It's an eigenstate of a, then the uncertainty is zero. Clear? That's obviously true because it has a sharp value and everything, man, this is a lot. This is a definite value. There is no uncertainty. Now, suppose I measure b. Do I know what outcome I will find if I measure b? Well, if it's eigenstate a, then it won't be eigenstate b. It depends. If it's an eigenstate of b as well, then you're absolutely correct. Then we know for sure what we'll find. But it's not true. The sign is not necessarily an eigenstate of b. If a and b commute, then we can be guaranteed. But if it doesn't commute, we don't know. But what we do know is that this uncertainty product must be bounded by this. But if a and b commute, then it's possible that my uncertainty will be zero. No. This doesn't say... So what it says is that it's supposed... So here's an example. Suppose a and b commute. It's greater than or equal to... So this is true in this case. This is greater than or equal to zero. Now, we said that this is an eigenstate of a. That means delta a to zero. So this holds. But it doesn't tell me what delta b is. It's possible that delta b can be zero as well. So for example, if we had a non-degenerate eigenvalue an a and b commute, I'll get to your moments. An a and b commute. Then we know that psi is certainly also an eigenstate of d. But if we have a degenerate eigenvalue, then as we discussed a couple of lectures ago, that state need not be an eigenvector of d. We can find there exists one, but not every one is. So just be careful. Just because the commute doesn't mean that every possible measurement outcome is perfectly predicted with zero uncertainty for all measurements I do of those community observables. Yes? Looking at the understanding principle, the bottom one. It seems that if psi is an eigenvector of a or b, then a and b either commute with each other or they expect the value of the commutator is zero. Automatically. Yes, that's correct. So if a and b commute, then this right hand side is zero. If psi is an eigenvector of either a or b, then this left hand side is zero because one of these uncertainties is zero. And then for the right hand side, the one is zero. Very good. So, so far we've talked about states, we've talked about observable, we've talked about how we calculate probabilities and outcomes and predictions of measurements. But what we haven't talked about yet is dynamics. The state. Sort of two very different dynamics that we have in quantum theory. Okay? Here's what I'll call closed system dynamics. So what do I mean by that? Well, in this case what I mean is the following. Suppose whatever the system is, it's completely closed off to all, to everything else. Okay? So, system is completely isolated. It doesn't mean that it doesn't have lots of stuff in it. It could be, you know, a solid state material with Avogadro's number of particles in it. But it's completely isolated from everything else. Now, if it's completely isolated, we neither gain nor lose. That's true. If we started in a pure state, meaning we had maximum possible information about the system. So let's say at time t0 we assign this state. At a later time, the state is also pure because we neither gain information or lose, really, in a certain sense, nor lost information. So if we start in a pure state, we stay in a pure state. That means at a time t later, we should assign this state. So that means that what quantum mechanics says that the state at a later time is obtained from the state at an earlier time by some kind of map. That takes me from time t0 to t0 plus t. This is my time evolution map. And we can ask of it some nice physical properties like it's continuous in time. Take this to be a differential. So if that's the case, and if this is pure, and we haven't either gained nor lost information, then we didn't lose any probability of anything happening. Total probability of some, that is to say, the total probability, which is the norm of the vector, the sum of all the squares of any probability amplitude, has got to be one. And given this, that's the dagger of that, is this. That's this vector. And this is its equals one, which is the same thing as what it was initially. So given that fact that the system is isolated and we're neither gaining nor losing information, what it says is that the time evolution map must be a unitary map. So what it says is that if we have an isolated system the time evolution of the system must be a unitary map because the property of unitary properties is that they preserve the inner product. And preserving the inner product in this case means preserving the total probability. So that's one thing we can say. If the system is isolated, its evolution must be unitary. And this is a deterministic evolution. That's to say, given this, I know what is going to be in a later time. I know what this guy is. And physics will dictate. But there's something else we know. There's another kind of way in which the state changes. There's a state. And that's where all the attention is. So, let me get my hands here because I want to thank you, sir. So now we're going to come to the measurement possible. A gentleman said this is some detail for the remainder of the lecture. So where does the measurement possible if it tells us? The measurement possible, it says, the typical way in state is as follows. If we measure permission, we find the non-degenerate item value, little a, then following the state such that if I had a pure state, which was a superposition of these possible item vectors with these probability amplitudes, then I find one of them with probability given by this, then immediately after, this is the new state. So what can we say? How does this kind of evolution distinguish from this kind of evolution? A few obvious important ways. One thing we can say, first of all, is that this is, this kind of evolution is random because which state this goes to is not predetermined. It goes to that state with probability piece of it, depending on the warning. Moreover, it's kind of discontinuous. Now, what can we say about this? Firstly, what about the case that we described earlier in the class where we said that the eigenvalue was degenerate? What happens then? The probability. If you're not sensitive to the different values, what does this say in the superposition? Right. That's a big one. So let's say, let's think about this. What about degenerate? What about the degenerate case? So let's say that psi is a sum. It's generally a superposition. Well, let's do a little bit of algebra for fun. We know, we want to represent this in the basis of eigenvectors of the observable A. The way we do that is we insert a resolution of the identity. Right? One way to do it. And a resolution of that identity is a sum over all the projectors. But each projector itself is a sum over all the degenerate eigenvectors all of which have that same eigenvalue. Right? This is a sum over A. Sum over I equals 1 to the degeneracy of that. U, A, I, U, A, I, psi. So this is equal to a sum over A. Sum over all vectors within that. I'll call it C, A, I, U, A, I. Where C, A, I is the probability amplitude to be in that vector. Is that clear? So now I measure, I do a measurement. I'm going to talk about that. And that measuring device measures the observable A and finds a particular outcome A. But it doesn't resolve. It can't tell the difference. There's nothing in the device that distinguishes which one of these use A. So what is the state afterwards, guys, in the gender neutral sense? Yeah. Is it just the superposition of all the degenerate states? Exactly. So afterwards, we have this. Well, sort of. This state is not normalized. Right? Why isn't this state normalized? Yeah, what we had was that normalization before was that the sum over A and I of C, A, I squared 1. Right? But now I'm just summing over all the I for fixed A. So I have to divide this after we normalize it. Now it's normalized. So that's what happens. This kind of, we can unify this altogether to say the following thing. A projective measurement, under a projective measurement, let's say psi less than is the state just before the measurement then the state just after the measurement, I'll call it T greater than, psi greater than, is what I get by projecting the state onto the eigenspace associated with that and the normalizing. This is the projection possible that it's sometimes called. Does this, the same thing as what we wrote down in the case of the non-degenerate situation? Let's take a look. Pose the state just before I measure it, as opposed to that where it is. Okay? As opposed I find A. Now, because I'm going to take this to be a dummy index and call it A prime. This is a sum. I don't want, I'm finding a particular value. This is a sum over all possible values. So you shouldn't use the same symbol. Okay? All right. And so there is a projector onto that which is a one-dimensional projector. So the state just after the measurement according to this projection postulate is I find this guy to this and then be normalized. Well, according to this, this is sum over A prime C A prime. And what's the top? Well, this guy over here is what? It's a chronicle delta, right? So it picks off the value where A prime equals A. So this is equal to C A A divided by the norm of that. And what's the norm of this vector? Well, I can write it many ways. You just have to go through this formally. You can just look at it. This is a unit vector. This is a candidate, right? So the norm is the spread of root of that, right? And what's the denominator then? Yep. So this is equal to C A divided by the absolute value of C A. So it's some phase times that which is irrelevant. So this works for the case of a non-degenerate eigenvector. It projects onto that eigenvector period. More generally, it projects onto the eigenspace and then we have to re-normalize. Now, obviously this is kind of weird, right? I mean, this thing is an idea that the state collapses. And this collapse continues to be a source of confusion and some mystery. Now, why is this, in some sense, mysterious and strange and confusing? Well, one reason is when I first... I'll tell you one of the reasons. I'll tell you why I was confused when I first learned this. There's an undergraduate student. So quantum mechanics is supposed to be the fundamental theory of everything, of the universe. And everything, in some sense, emerges after me, right? So the measurement process itself should be described by quantum mechanics. But yet the measurement process is doing something different than that general kind of evolution that we have. And this is in which physical situations correspond to measurement and which kinds of physical systems are not. How do I tell the difference? And if I can't tell the difference, how do I know when to apply this rule and when I apply that rule? Let me try to be a little bit more precise about that. Suppose I have a spit, and I want to measure it. We'll talk about that in just a moment to get a little more in there. So I was about to share a paragraph. And suppose I have some kind of meter. Now, why isn't it true that this together is an isolated system? In which case the whole thing together should evolve according to a unitary evolution. After all, the meter is made up of atoms and molecules and solids and should be described. Why isn't there a joint quantum state, pure state of the whole thing, and it evolves according to unitary evolution versus the situation where I say, okay, I have my spin. My spin is isolated, maybe. And then I peek into it, and I measure it. And now it's open to this meter. Stephen, what do you think? Well, in the first case, the isolated case, the meters inside the isolated system, and you can't observe it since it's isolated. True. But so that's an interesting point to use. So that puts you at the center. Well, if you're in the system as well, so you can observe it as well, the meter just starts looking at the spin, chains the state itself. Say that last question. If you assume that you're in the system as well, so you can observe it without disturbing it, what did the meter look at, observing what state of spin it is in, chains the state itself? So what we're trying to get at here is how we describe, under what circumstances do we describe it? We initially had, there's some state of the spin, and we wanted to measure it. If the rule is when you bring a meter, whatever a meter is, in contact with the spin, then the meter does some particular kind of physical thing that is no longer described by unitary evolution on the whole thing, but instead described by this. I mean, to have the spin somehow affect the meter without touching. But the question is when, how does that work? I mean, under what circumstances, what does the meter have to be? A spin out of her. But what is that? You know, in terms of being a particular amount of stuff. So, I mean, when you were saying I was wondering too, like, so if the meter measures the spin, it should store in the meter when you look at later. So, I mean, how complicated a device does the meter have to be, is it a simple quantum mechanical system, or is it this, you know, Amogadro's number of... So, all right, so now we come to some of the interpretations of quantum mechanics. So, how does the Copenhagen interpretation deal with this problem? And this really comes to your point. What the Copenhagen interpretation would say is that what distinguishes these two kinds of evolution is that the meter, and really this is more, the meter itself is governed by different laws of quantum mechanics. The meter itself operates, the meter operates according to classical mechanics. So, in the Copenhagen view, in order to extract information about a quantum system, and therefore us to learn about it, we need to bring it into the macroscopic realm. That's Zeke's point about the meter being sufficiently complicated in some way, having Amogadro's number of particles. I don't know, Amogadro minus 10, Amogadro divided by the 300. I don't know, but somewhere there's some, at some point, we would say that the meter is a classical object. And what it's pointing on is not a quantum on a certain state, but it's actually realistic in some sense. Now, maybe there's something we could say, so this question that the meter itself lives in the classical world sets up a new conundrum for us. Yeah, before a picture. What if the meter was like a bunch of photons? Exactly. I mean, sometimes this quantum mechanical. Sure. So that's exactly the point. Where is the border between the quantum world and the classical world? At what point, if I start building up my meter with, you know, I have buckyballs, I have clusters of buckyballs, I have droplets of buckyballs, at what point does it become macroscopic enough for us to say that it's a classical meter? We can't observe the states anymore. What is it about it? What do we quantify about it that makes it say what aspect, what's going on physically? So this question of where the quantum classical boundary lies is something, the emergence of our classical world out of the fundamental quantum world has been one of the great pursuits, I'd say, of the last, you know, 70 years, trying to understand is there a quantum classical boundary? Is this a sharp transfer? What does this really mean? There is certainly true, and we will see in this course, I mean, we don't see interference between all terms in a macroscopic world. So there is something to be said for the fact that classical experience is different from what's predicted quantumly and somehow that should come out of quantum mechanics in some way, if quantum mechanics, and there is, so there's, I've set up a complete dichotomy here between what I call closed system dynamics and measurements. There's an intermediate case which might be called open quantum system dynamics. Maybe there's part of what's going on in the meter, this thing is interacting, but some of the information is going someplace else that I don't have asked. Quantum mechanics should be able to describe that, open quantum system dynamics. And what we will see is when we have an open quantum system, interference phenomena can be washed out and that's called decoherence. So decoherence is something important that's going on when the measurement is happening. But it still doesn't, decoherence as we say will not give us this. There is no way to go from a deterministic evolution to a random evolution. We have to put it in by hand. We do that, then we can ask the following question. What is collapsing? I mean the nav, not the verb. What stuff is collapsing? It comes to a point that I've kind of avoided, but now I want to address because it's an important point in what we debate right now. Which is, is the state a physical thing? Meaning the following. Is the state like a wave on a string, a wave on waters? Or is it even something as abstract as a wave of the, of a field? I mean electromagnetic waves are already pretty abstract things. But we believe, or when we think about it classically, we believe that electromagnetic waves of stuff is fields are physical things that exist. And if the state is a physical thing and if the state collapses then there must be a physical process which causes it to collapse. The wave becomes a double function somehow. So if that seems inconsistent with this kind of deterministic evolution what else could the state be? Another operative, if yes, not if the state, is the state. If yes, which implies collapse is a physical process. I'll give you a different example. What is the probability that the coin is heads up? What kind of coin is it? It's a quarter. Well you don't know. What probability would you assign to it? 50%. It's a 50-50 chance. Alright Steve, come over here. What is the probability that it's heads up? Zero. No, look at it. Oh that's not tables. That's true of Washington. Look everybody come on. Maybe watch this later. What is the probability that it's heads up? 100%. 100% one. So your state of knowledge collapsed. You previously assigned a state to the system. You assigned, because you didn't know, means collapse is subjective. It's about the state is about what you know. It is a state of knowledge. And collapse means that you update your state of knowledge. Condition on what you know. And this is the Bayesian perspective on probability. This is probability assignment. So according to Bayes' rule, suppose I have some prior knowledge. Maybe it's lack of knowledge. I don't have complete knowledge. And I assign a certain probability to a certain random variable being true. So this is my prior probability that x is true. I assign that based on what I know. So for example when I covered this up and even if you didn't look at it you still couldn't tell that was George Washington. But you assigned 50-50 for example. Now you do a measurement and you find something. You find the value y. So now I want to know what's my posterior. What is my probability of x given that I now found y? So this is equal to the probability that I would find y given that x is true times the probability that my prior probability and then re-normalize. That's Bayes' rule. So this is my prior that x was true. And this is what is called the posterior. Now my posterior. That's to say the posterior probability that x is true given that I found y. And what Bayes' rule tells me is that there's a, this is what's called Bayesian updating. I should update my probability assignment conditioned on a measurement result I find. Are those two different types of p's? One is they're all probabilities but this is a conditioned probability, right? This is the probability that y is true given that x is a conditioned one. So that's y depend on x being true? Yeah. Exactly. The probability that y is true given that x is true. Yeah. So if the rate function is that the state is not a physical thing and the class is objective, then an example in the beginning of the semester with the inforometer where we, you know, have 100% going to 1, 0%, but just by, you know, distinguishing the paths, the changes. Sure. But what it says, as we will see and we'll continue this discussion, all measurement outcomes that happen with 100% probability, all observers must agree. Just because it's subjective doesn't mean that there aren't situations where they all have to come to the same conclusion. We'll see that. So what I want to say, and I guess, I won't, we won't go much further than this, unfortunately, but it's okay, there's some, this is really important foundational stuff, so the rest of this too much, is that in some way the projection posture looks like being easy and updating. Right? So let's say, so let's do this with the quantum state. Okay? In quantum mechanics, we have some, my prior, my state, prior state, that's the state that I'm going to start with, sign less than, and the probability that X is true is given by the vulnerable. So now, I'm not, I mean, what X is, I've made some observable that has values X. Okay? Now, I want to know, so this is my prior, I do another, I do a measurement and I find Y. What is the probability that X is true given that I found Y? Well, according to quantum mechanics, I calculate that. This is equal to the posterior state given that I found Y, the probability that X is true. And the posterior state, given that I found Y, is given by the projection pilot, that is a project on Y, prior state, and I renormalize. According to that, this is equal to prior state, PY, PX, divided by the prior probability. Now, that looks kind of like phase rule, but not exactly. In fact, if X and Y are different eigenvalues corresponding to a Hermitian observable, well, this is either zero or one, right? Because if X equals Y, well, this is one, and if X doesn't equal Y, it's zero. But as we'll see when we talk about POVMs, where these are no longer projectors, but measurements would define that resolution, this can look exactly like phase rule. And this is where I want to conclude for the next three minutes. Is quantum mechanics just Bayesian updating? No, there's things in quantum mechanics that have no classical analog whatsoever. And one of them is the question of quantum disturbance, or back-action, in compatible operators. Suppose I measure an observable A. I become more certain about the value of A. If I find an eigenvalue that is non-linear, I become perfectly certain of it. So I gain information about A. A and B don't commute, then I lose information. So for example, suppose at my prior state, was an eigenstate of B, with eigenvalue of little b. My uncertainty in B is zero. My major A, my uncertainty has been increased, and now if I measure B, I'm not guaranteed. That can't happen classically, in general. So this is where the collapse is weird. The collapse isn't weird that I had an uncertainty and by measuring I reduced my uncertainty. That happens classically all the time. We do a more and more resolved measurement, more and more and more where it is, and we update what we know about that. But that shouldn't mean that I am forced to make incompatible non-commuting observables less and less certain. That's weird. That's the part of the collapse that's weird. Okay, I guess we will continue this discussion.