Lecture 1 | Quantum Entanglements, Part 1 (Stanford)

This program is brought to you by Stanford University, please visit us at

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.edu. I almost always start with the same sermon, especially when I teach about

quantum

mechanics or relativity, the sermon is always the same, it is the fact that we, as animals, have inherited through the process of evolution certain intuitive ways of thinking about the physical world uh and if you don't believe it you think that maybe common animals are not physical you see a lion chasing an antelope and you realize that that lion at the moment the antelope the relative speed between the antelope and the lion changes sign the lion just stops dead somehow he did some calculation or she usually it's she the lion did some calculation some physics calculation involving some very complicated concepts of velocity direction all kinds of complicated calculations like that uh a uh A primitive chroman man uh, not a chroman man, a nandal who comes to a cave and sees that the cave is blocked by a rock and he tries to push the rock and he can't push the rock, he decides to point his body in that direction H Why to get a larger force component in that direction?

Has he ever heard of strength? Have you ever heard of components? Where did you get this idea for components? Did you know about signs and comfort? Yes, somehow, he knew about signs and comfort. These are things that were inherited from biological origins and are the basis of our intuitions about physics, our intuitive picture of the world, much of physics has to do with those things, in fact all of modern physics, everything in physics modern has to do with those things. Things that are beyond the intuitions we could glean from The Ordinary World have to do with parameter ranges that are far outside the parameter range that humans or animals ever experienced, for example, it's not too surprising that beings humans did it.

They didn't know how to deal with speeds close to the speed of light and had misconceptions about how to add speeds when no one in 1900 had probably ever moved faster than 50, 60, or 100 miles per hour, well there they are. They probably did it when they were falling off cliffs, but they didn't live to talk about it, maybe they reached up to 200 miles per hour, maybe, but no one had ever experienced anything like speeds close to the speed of light, so it's not like that. it was surprising that his intuitions that his way of thinking about adding velocities and so on, the theory of relativity was uh and and how clocks were synchronized, all those things, all those good things that Einstein did, that were outside the framework of their ability to think through intuitive images through intuitive mathematics um they had to invent new mathematics the new mathematics was abstract, that is, it could not be visualized in four dimensions.

Space-Time, you can't. I can't visualize all four dimensions. I have learned tricks to visualize them, as well as physical ones to a certain extent. they rewire themselves or people who learn physics go through a process of rewiring themselves to some extent to develop intuitions that allow them to deal with these new ranges of parameters, but still they are strange, they are strange, they are peculiar even to me ,

quantum

mechanics offerings. with a range of phenomena that are also outside the experience of ordinary humans for which evolution simply did not provide them with the means to visualize.

Evolution did not provide them with the means to visualize an electron to visualize the movement of an electron to visualize the uncertainty principle. when you think about a

part

icle in motion what is a

part

icle a particle is a thing with a position at each instant of time it has a position if at each instant of time it has a position it has a trajectory if it has a trajectory you can calculate the speed along of that trajectory by just knowing the separation between points and what the time interval is, you can calculate the speed and that is the intuitive image of a particle and where it comes from, it comes from thinking about rocks, throwing stones, shooting arrows of all kinds . of things that human beings normally do, so we never developed the need, it would have been very strange if our brains had been wired to understand the uncertainty principle, why would Darwin have given us the uh, by the way, uh, yeah Do you prefer to think about the smart ones? designer, go ahead, I prefer, I prefer to think of Darwin, but why would Darwin's ideas or the intelligent designer have given us the ability to understand the uncertainty principle when it is never something that is part of our ordinary experience?

The answer is no. For that reason, quantum mechanics seems extremely strange to us physicists, as I said, they rewired themselves and developed ways of thinking about it that are intuitive, but still, quantum mechanics is much more unintuitive than the special theory of relativity and what we're going to try to do here is lay out some of the weirdness of quantum mechanics the weirdness of the logic of quantum mechanics the weirdness of how quantum information works this is not a class a conventional class on quantum mechanics In a conventional quantum mechanics class we emphasize things like the Schinger equation and waves and how particles sometimes behave like waves and so on, we may or may not get to a bit of that, but that's not the important topic we focus on.

Let's concentrate. focus on the basic logic of quantum mechanics the basic logic of quantum information Physics theory is information when you say something about a physical system, you are saying something, you are giving some information about it, you give the information in various forms, usually in the form of numbers in classical physics that you usually give. I'll give you some examples, but usually you give them in the form of real numbers. the position. speed. a set of real numbers and quantum mechanics. sometimes you use real numbers but very, very often. you give discrete information discrete information like yes or no or up or down or man or woman well, that's probably not a good example, the difference is between each other, I probably think I'll take that head or that, heads or such, I have a coin no uh gonin thought of coin coin head tail head tail just when I flip the coin heads that's a piece of information it could be tail it's a two value system yes or no up or down heads or tails or sometimes they are logically anyway of course, logically they are all the same, whether we are talking about heads up or heels up or down or whatever, they are logically the same and they are simply decisions that have two possibles or questions that have two possibles. answers and a bit of information that has two possible answers is called a bit and can be a classical bit or a quantum bit B.

All real bits in nature are quantum bits, obviously, since nature is made of quantum mechanics, but sometimes the quantum aspects do not manifest in an ordinary computer the quantum aspects of the bit do not really manifest for reasons that will come to uh and it is called simply a classical bit of classical information this head is the coin toss uh, yes or no, the The quantum bit is the quantum analogue of the coin toss, the yes or no question, but it's much more subtle and the first thing we're going to want to explore is what a quantum bit is now. but before we do that, let's talk classic bits.

Classic bits can be described by writing a zero or a one. These we can also use one and minus one or we can use five and 15, it doesn't matter, but zero and one. is a convenient notation for the two possible values ​​of uh, zero could represent heads, one could represent tails, etc., so when we think about this, we think about some physical system when we think about information. We are thinking of a physical system like a coin and this is the information contained in that coin, whether it is a zero or a one. There is a notation.

This seems like a ridiculous and redundant notation. Its importance will only become clear when we start thinking about quantum bits, but let's use direct notation. Direct notation describes the state of a bit, not whether it is California or Oregon, but the configuration of the bit and is usually labeled with a zero or one notation or some other information. Otherwise you decide to think about the bit, these are the two states in which a bit can have a zero or a one and it is represented, I don't know if I don't know if I drew it, well, let's draw it again, zero or one, these it's the two states of a bit, all this extra junk here is excess, you don't need it, it doesn't tell you anything, it just says you're putting it inside a bracket, by the way, this pointy object in brackets here. the thing that contains the information inside is called Kat, it's called a cat because it's the second half of something we'll later learn is a cat with a bra or a stand, there's another half that we haven't exposed yet, what about multiple? -bits assuming you have more than one bit and we are talking now about classical physics so far we are not talking about anything quantum mechanics assuming we have several coins and I line them up I label them so that we know which is which actually just so as not to confuse coins, let's make sure they are different coins Penny Nickel Tell me quarter half dollar silver dollar okay so we have a bunch of coins we can't confuse them and we can expose some information by saying head tail tail head head tail that would be information about a collection of bits right How would you label that?

You would label it with a string of zeros and ones, so, for example, if we always take zero to represent head, it's easy to remember. zero means heads and one means tail for obvious reasons um right, so my coin string heads head head head tail head tail head I would label zero for heads zero for other heads one 0 1 one for example, that's a configuration of 1 2 3 4 five six coins, okay, let's say it was six coins, right, that is a configuration of a multibit system, in this case six bits again, for reasons that add absolutely nothing to this description, we are going to put it inside a k, we glue it. inside a k, which is just a type of notation, it might be a good idea to put some commas between these, but maybe not, maybe it's better to leave it like that, it's a specification, it could be the specification of the bits of information inside of a computer, it could just be a series of heads or tails or something, but before we do anything else with this, let's ask a very simple question: how many possible configurations, how many possible states are there, well, let's start with one bit if there are only one. so there are only two states, what if there are two bits?

Well, then you can have up up down down up down four 2 * 2, so for two bits we have 2 squared What if we have a What if we have 100 bits? the answer is 2 to the hundredth power 2 * 2 * 2 * 2 100 * so if you have an inbit system, the number of possible configurations of classical figures is just two to the power of N, let's write that, uh, let's put a notation, let's write the number of states the number of states N Subs the number of states of a system of n small Bits is 2 to the power of N suppose that we are going to invert that first of all let's invert small n is what small n is the number of bits N big is the number of states little n is the number of bits, okay, so if the number of bits is four, then the number of states is 2 to the power of 4, which is 16, etc., we can reverse this and we can write If we knew the number. of states of a system, then we can take the logarithm of this equation. log in base 2 is particularly convenient if we take the log in base 2 of the number of states that is equal to the number of bits, you can generalize this, not all systems have the number of states to a power assuming that I have a state a system um a uh a dice you know things that you use in Las Vegas to throw your money with um it has six possibilities from one to six that are not two for to any particular power it's just six, okay, but we can still generalize this definition of the number of information bits, in fact, the number of information bits that a system can contain is, by definition, the logarithm to base two of the number of states. what for D would be logarithm to base 2 of six what is logarithm to base 2 of six is ​​it an integer no it's a stupid irrational number I don't even know what it is how big it is about two to the power of 2 to the power of 2 is 4 2 to the power of 3 is 8 2 to the power of 2.5 3 7 9 8 6 14 or whatever, so the amount of information which is always the logarithm of the number of states does not have to be an integer, but we are Let's consider systems that are composed of a certain number of bits.

Each of which has two states, so for Simplicity we are going to talk about systems, the number of states is always two raised to a power, that is simply, for Simplicity there is nothing special, but almost all Systems can be represented that way or approximately that way. Let me give you an example. Suppose we have some physics question that has a real number as the answer, but we are only interested in that real number. a certain approximation the temperature the temperature in the room I am interested in the temperature inthe room to a certain number of significant figures I can represent the temperature or any other number by writing it as a number in base Two, right, what is the temperature in this room?

By the way, it's about 300° um from absolute zero, so it's 300. I can write 300, not like 300, which is what 300 means. 300, you know what it means, it means 3 * 10 2 plus nothing. 10 1+ 0 * 10 0 but we write it in things in basedos, right, I don't know what 300 looks like in base two, can anyone figure it out, uh, base two, everyone knows what arithmetic and base two is like, anyone doesn't know arithmetic and base two, okay, so everyone knows arithmetic and base two, we write any number. which we like like a series of zeros and ones 1 0 0 1 0 1 0 0 1 that's a number raised to uh uh some particular integer is an integer so if I'm interested in the temperature and I'm not interested in being uh too much careful to define fractions I want to know if it is 72° or 73° I don't care about 72. 4069 I can write it as an integer and that integer can be represented as a sum of bits, not as a sum of bits, but as a collection of bits, every number, every number, if you're willing to truncate the number of decimal places, approximate that number and say, I'm interested in that number just to 35 decimal places or whatever, or 35 places in base two for that number. simply if both are represented by and represent a collection of bits, so either and, by the way, if you want to have a finer description of temperature than integers in centigrade, just use a more refined notion of degree, go down and ask how many degrees but not in centigrade units but in units of 10 Theus 100 centigrade again you can give it as an integer and integers can always be represented as sequences of zeros and ones so almost any information in physics can be represented in terms of bits. in particular, the measurement of quantities, such as temperature, for example, let me give you another example.

This is a more complicated example of the same thing. Suppose I am interested in a field. A field means something that can vary throughout space. Well, temperature can vary throughout space. Temperature is a field. It varies throughout space. It's not one of the most interesting fields from a fundamental point of view of Al particle physics or anything, but it's certainly a field that varies from place to place. place and how we can represent that, we can represent that in terms of bits, yes, if we are willing to tolerate certain approximations and we are always willing to tolerate some degree of approximation, what we do is divide the room into many tiny little cells I won't try to draw a three-dimensional room in my notes.

I drew a three-dimensional room. It took me about half an hour to put all the lines into just a two dimensional room and this is what we do. first of all we arrange the cells we make the cells small enough so that the temperature does not vary much from one cell to another so we could fill this room with several billion cells we label the cells these are the first cells 2 1 2 3 4 5 6 I don't know up to a third in 1.00 and 2.3 1004 and we can label all the cells and number them once we have numbered them we can write the temperature of the first cell, there is the temperature of the first C I'm putting a small comma just to distinguish between cells and then we can write the temperature in the next cell 0 0 1 1 2 3 4 5 6 7 8 9 I have kept nine decimal places in in in uh in the base in arithmetic and base two uh one uh one Zer one however until I finish then I go to the next cell I do the same temperature there is 1 1 0 1 0 0 and so on eventually all I have is a list of zeros and ones, this long list of zeros and ones, if anyone knows how Using it is equivalent to knowing the temperature at each point in the room, okay, the same goes for the electric field, the magnetic field, anything that varies from one place to another.

To put it, almost everything I can think of in physics can be represented in terms of bits, so if you know everything about how bits work, you basically know everything about how physics works, of course you might not know which ones. They are the rules to manipulate them. things, but this is the basic configuration of physical information in the form of a series of questions. Each of which can be answered yes or no. Now, of course, you may want to refine your description. To refine your description, you may want to add more decimals. to the temperature to the temperature specification and you might want to make your lattice finer, that's just a better approximation, so the correct thing to say is that most physical systems that we know, as far as I know, all physical systems can be represented. at least approximately and perhaps with an ever increasing approximation in a series of bits, that is why we can use computers to do physics, if this were not true we could not use a computer, we could not use a digital computer in any case to do physics, we would have to use analog computers or something, so let me give you another example, another example of how you could use bits to represent another, so far they are all classical systems, like I said, I don't want to redraw the network, but I want get rid of this top row because I have already mutilated it here is a network uh and what I am interested in is the movement of the particles, this network is just an artificial network imposed that I have imposed the room here just to divide the room into mathematical cells and what interests me is the movement of the particles that move in this room at any given instant.

I can ask the question, let's take a very simple case. Take the case of a particle where not more than one particle can be introduced into one of the cells. We can imagine that the cells are approximately as large as a particle, in which case no more than one can be introduced and then into each cell. either it has a particle or it doesn't have a particle we can label the cells that have particles with an with a zero in that case this becomes a specification of where the particles are in the lattice it is no longer the temperature but the same long sequence of zeros and ones, now the number of zeros and ones would simply be equal to the number of cells On the network, what would this number mean? it would mean that in the first cell there is a particle in the second cell there is no particle in the third cell there is no particle in the fourth cell there is a particle in the fifth cell no particle in the sixth cell particle and so on and so on and so given such a string of numbers you are given a specification of where the particles are in this room um that way again motion of particles motion of fields temperature almost anything in physics can be represented in terms of bits any question right a little a little is by definition a question about a system that has only two possible answers that you can always take as yes or no it used to be a game 20 questions Don't you uh uh where uh someone would think of a category um uh uh and then you would stand there and say uh yes you don't ask and until you try to figure out what is the category, what is the category, then that was using the bit idea yes ask oh, I just said arbitrarily assuming that we are interested in temperature to a certain degree of precision, like this that I'm interested in temperature precisely, eh, but now I'm not talking about temperature, I'm just giving another example, these are just examples meant to show you something that is more or less clear, otherwise we wouldn't be able to use computers to simulate problems physicists, classic physics problem, yes, sure, but you need for the general real number you need. an infinite number of bits, right, any rational number can be represented by a finite number of bits and the rule, well, that's not entirely true, you have to remember to repeat them, but if it's rational, it will repeat itself later.

Yes, if it's rational, it's I'll repeat after some point, but if it's an irrational number, then you need an infinite string of bits, but in general we'll allow infinite strings of bits, uh, although not on a genuine computer, well, until now remember that we are doing classical physics. Alright, so far there's no quantum mechanics, so I'll go, let's see where, um, yeah, we were going to get to that very, very soon, well, let me tell you how fast, uh, an electron, first of all, we're not talking about movement, still. We are talking about configuration, configuration means the state of a system at a given instant, so the presence of an electron at a given instant, suppose we know that the nucleus is right here and we are not going to ask about the nucleus, the nucleus It just sits there, it's a lump, okay, so we could say that at instant number one, when we start the experiment, the electron is here, in which case we would write a string of zeros with a one somewhere. pure zeros an electron pure zeros except for one place in the uh in the sequence where there is a one.

Well, now if we wanted to describe the motion of the electron, we would say that starting with this configuration we move and let's use this symbol here to indicate that The next thing we could do was divide the space into a bunch of little individual cells. We were also able to split time. I thought I had my watch, but no. We could also divide our clock into a digital clock that uh, which uh digitizes time again as a convenience or an approximation and we could say that if at digital time number one the electron was or the system was described by an electron located in this location, so what happens next?

It moves to some new configuration in this case it could move to a place 1 2 3 4 five six it moves to the sixth place one two and so on, so the movement of a system is described by an information update rule about how update it from one instant to the next, then physics basically consists of two, a physical system consists of two things, it consists of a collection of possible states that can be labeled by a collection of bits and it consists of a temporal evolution which is an update that you tells how to take a collection of bits and replace it with another collection of bits at a slightly later time.

I don't know if that really works. An orbital motion orbiting around here gets confusing because when you jump from one layer to the next, uh, if this is one and this is 100, then 101 is here, so don't jump from 100 to 101, you could jump from 100 to here, that would be 200, so the update procedure can be complicated. It may seem complicated but, nevertheless, it is an updating procedure that simply updates your state of knowledge, at each instant of time, that is classical physics. Now there are some rules and we're going to get to them, but before we do, let's define the space. of State, this is and I want to emphasize that we are still doing classical physics, there is nothing quantum mechanics, although we are talking about discretizing systems and converting them into systems of individual bits, so far we are dealing with what should be called classical bits.

C bits, I think they are called unlike qits qbit is a quantum bit, these are classical bits so far, okay, so let's take all the configurations and in an abstract way, in a purely abstract way, we take all the configurations, for right, what are they? This is approximately 10 by 10. This is approximately a 10x10 grid. The 10x1 network has 100 sites. How many states does it have? If I'm not talking about one particle, I'm now talking about any number of particles that can be in this network. many different configurations there are two two to the power of 100 a very very very large number 2 to the power of 100 that is the number of different ways we can arrange zeros and ones in this network or specify whether there are particles in various positions a large number of possible states , but let's think abstractly about all these states and draw them as points.

If there are 10 out of 100, I have to draw 10 out of 100 points, which I'm not going to do, these are the different states, these are not the points of the network, these are the various states, for example, for a bit, yes If you had only one bit, then the state space would consist of just two points up and down and you would only draw two points, this would be the state space of a simple one. bit system now let's ask what are the possible laws of updating, in other words, what are the laws of motion, the laws of motion, are the laws for updating settings, what are the possible laws of updating, well, here's a possible update law that could support. for heads this could mean tails let's think of this in terms of coins for the moment this could mean heads this could mean tails true if we start with heads if I had a coin we would do heads and tails heads fails very well one possibility is very simple if you start with heads it stays heads nothing happens if you start with such it stays such nothing happens that is a law of updating it is not a very law of updatingmolecule, which means you will have to give a set of real numbers. involves, as you say, an infinite number of bits, so the answer is for a collection of real moving particles that you actually try to follow classically, depending on how long you wanted to follow it, you would need more and more bits to describe . that oh oh oh yes yes yes that is correct, if the room was really sealed, let's idealize this room so that nothing can enter or leave the room, all the particles bounce and reflect off the walls of the room so that it is completely sealed . above the room then the room can be described discretely course of quantum mechanics at least up to some energy if we know that the energy is not arbitrarily high then we can describe it by a discrete collection of variables that has no output so let's say um, yeah, then we could do such a thing.

We could just reverse all the arrows here, an example, back to itself. No, this one has no way out, not only does it have a way out for itself. Well then, if you would like to come out yourself. we have to do this we could do that but as I drew it there was no way out but let's think about what it means, I'm not really interested in what the question is here. What happens when you are here? Well, when you are here you have two paths you can go and you don't know which path to take, so it's not deterministic, you don't know whether to go this way or this way. way it might be half the time this way half the time this way you might need some statistical rule 50% of the time or 30% of the time it goes this way 30% sorry 70 % of the time with random statistics that would be non-deterministic, okay? so it seems that the real laws of nature are deterministic forward and backward in time, that is the implication of not having loose ends floating around like this, that they are deterministic either way, so that wherever you are you can trace forward uniquely or backwards uniquely and that's all classical physics in a nutshell, now you have taken a complete course in classical physics.

There's nothing that doesn't fit that pattern, at least to an arbitrarily high degree of approximation, let's take a uh, let's take a 7 Minute Break, well, I was going to move on to Quantum Mechanics, but before I do that, I want to do a little math. Elementary math, most of you know this, but anyway, let's lay out matrices and vectors, um, I'm not right now. I'm not going to mathematically define a vector in any kind of sensible method that you know of, not even in a roughly rigorous or abstract way. I'll just tell you that a vector is a sequence of numbers, a finite sequence of numbers, and you can represent it. in a variety of ways, but I'll give you two ways to represent a sequence of numbers.

The first way is to write them one after another, let's just give them names. I don't want to, I don't want to call and my numbers now, right now I'm talking about real numbers as opposed to complex numbers. I don't mean zeros and ones, I mean arbitrary sets of real numbers, they could be zeros and ones. Zeros and ones are fine, but they are just general numbers. so I just told them what should we call them um uh components H components yes no yes they are called components but I want I want a letter for them a a a is good so a A1 A2 A3 A4 and just put something around them to surround them so we know that this would be a four-dimensional vector, right, why four dimensions, because it has four components?

Forget it, don't try to visualize the vectors now, it has no value for our current purposes and try to visualize them. like point in space or something, they're just a list of numbers, okay, lists of uh, that's one way, okay, there's another way we can list the same set of numbers, put them in a column A1 A2 A3 A4, same information on them, I mean. and I'm not talking about information in the abstract sense that I wrote about the same thing before. Sometimes it is useful to write it this way. Sometimes it's helpful to write it that way.

You'll find out as we go when it's written this way. It's called a row vector when it's written in this form it's called a column vector what we're actually talking about now is notations, ordered notations for uh, to do certain arithmetic operations that involve collections of numbers, it's not one of those daggers. when we get the complex. numbers, then we will use complex conjugate notation, yes, but for the moment, let them be just real numbers, okay, now there is another concept called a matrix and you think of a matrix in the following way: a matrix is ​​something that acts on a vector to give another Vector, okay, so it's some kind of machine.

You put the vector into the machine and another Vector comes out according to a particular rule. Oh no, sorry, before we do that, let's imagine a particular Column Vector and a different Row Vector. The row vector has different entries, not the same set of numeric entries, but a different set of numeric entries ENT, so let's call them B B1 B2 B3 B4, these could be 6.01 5.97 3.04 and A1 could be 7.8 A2, none of them could be the same. or the A and B may not be the same this is a particular row vector and a particular column vector there is the notion of multiplying a row vector by a column vector and the notion of multiplying a row vector by a column vector is a simple is the following simple operation: take the first input, oh, by the way, the dimensionality of the row vector and the dimensionality of the column vector must be the same, which means they must have the same number of entries, not necessarily four could be five, six, seven, uh in which case they would be five dimensional vector spaces.

Six-dimensional vector spaces. This extends to any number of entries in the columns and rows, but the rows and columns must have the same number of entries. Okay, there is the notion of the product of a row vector and a column vector is called the inner product and is constructed very simply: you take the first entry in the row and multiply it by the first entry in the column, add to it the second entry multiplied by the second entry plus the third entry. multiplied by the third input plus the fourth input multiplied by the fourth input, then the product of these two, which you could simply write as B next to that product, the inner product is B1 A1 plus B2 A2 plus B3 A3 plus B4 A4 is a number is not itself the product of these two vectors the inner product is not another vector it is not a matrix it is just a number the numerical value is simply obtained by adding the column sorry, the row multiplied by the column only in this form B1 A1 Plus B2 A2 plus B3 A3 plus B4 A4 is that clear, don't ask me why that definition is what you call yes, yes, if we were talking about ordinary vectors in space, it would be the dot product, yes, yes, more abstractly for abstract vector spaces.

It's called the inner product, uh, but yeah, it's the same as the dot product of ordinary three-dimensional vectors in space, where these would be the components of the vector. Yes, okay, now there is the concept of a matrix and a matrix, as I said, is an operation that you can do on a vector to give a new correct vector, but it is not just any operation. It is a particular family of operations that are characterized by matrices. A matrix is ​​represented by a square array of numbers. Let's call the entries M. Okay, in the First we put M11 to indicate that it is in the first row and the first column, then M12, then M13, and then M14.

Ok, M, what should I call it one on one? It's in the second row, but the first column is in the second row, the second column. second row third column next M31 M32 m33 m34 and m41 m42 m43 m44 now as I said I have chosen four dimensions arbitrarily, four is about the size I want to handle on the back board and it is big enough to be a bit abstract to make it general enough to see what's going on, that's what a matrix is, that's it, now you can think about it, you can think about it, you can think of each column as a column vector whose components are labeled by the first entry here, well, each of these can be thought of as a column vector where the first entry labels the column entry uh uh or you can think of it as a collection of row vectors where the second entry is the that label component, either way, can think of both ways at the same time, like a collection of column vectors or a collection of row vectors, but all together they form a matrix.

Now matrices can multiply vectors, so let's put a vector here. A1 A2 I should line them up more carefully A1 A2 A3 A4 and when I don't know, I've done a reasonable job of keeping rows and columns below and next to each other, but if you want draw some imaginary lines to separate them. in rows and columns, this Matrix acts on this Vector to give it a new Vector, what is the new vector and here is the rule. I've expanded the vector because each entry will be a fairly complicated expression, but it's just another Vector. it's another single it's another column it's a column that I had to draw wide to be able to include everything I want to write here this is what you have to do if you want to find the first entry in this column I'm sorry in this row in this row, you take the first row and you multiply it by the column, the inner product of the first row with a column here, so what is that?

M11 * A1 + M12 * A2 + M13 * a 3 + M14 * A4 in other words you take all this and multiply by this according to the inner product rule and that gives you the first row M11 A1 plus M12 a2+ M13 A3 more M14 A4 now you want the second entry in this new vector here to be done in exactly the same way except you go to the second row and take the second row and multiply it by the column which will give you m. I'm only going to make two of these, the rest you can make yourself M21 M21 again multiplied by A1 plus M22. * A2 plus M2 3 A3 + M24 * A4 and the other two entries that you can calculate are obtained by multiplying the next row by the column and finally the third row by the column which gives you a new Vector, it is a way of processing a vector to produce a new vector.

I'll give you some examples. As we move forward, it is a multiplication rule that is very useful. The reason it is defined is because it is useful and we are going to see how it is useful using it. Let me. give you an example of how a matrix how the idea of ​​a matrix can represent the temporal evolution of the configuration of a system assuming again that we have our um our configuration space let's label them let's label them take a let's label them the first configuration the second configuration the third configuration the fourth and fifth configuration these are not points in space these are configurations of a system that has five different states and let's take a very very simple law of evolution the first if you start here you go to here if you arrived here you go to here if you start here you go to here if you come here you go here and what do I do if from here I go back again no good no no that that's no good that's not allowed I think that's that's not allowed uh I think that's not allowed uh I think that's not allowed yeah that's not allowed because if you find your yes that no that's not reversible that's not reversible that's not what I wanted to hear what I wanted to hear is that you come back here Okay, so this is just a um one goes to two two goes to three three goes to four four goes to five five goes back to one is a cycle ok here is another way to represent the same thing, we can represent the state of the system by a column vector and the column vector we just insert a one somewhere if I want to represent the first state here I put a one and then a bunch of zeros one two 3 4 five 1 two 3 4 5 this simply represents the first state what happens? the second state the second state I will represent by 0 1 0 0 0 the third state by 0 one and so on so that the states of a system can be represented by a column, but a particular type of column, a column with all zeros and a one somewhere where that one is, meaning whatever state you're focusing on, if you're focusing on the fifth state, put the one in the fifth entry here, well, what is this rule of evolution? ?

The rule of evolution says that if you have one somewhere, then in the next instant the one moves down, so if you start here the one moves down to here and the next instant it moves down to here, the next instant moves down to here, the next instant moves down to here, so where does it go? all the way up, okay, so there's a procedure that's done in this column to tell you where the system is going to go next. That process can be represented by a matrix, so let me show you the matrix that represents that.

Matrix is ​​an operation on a vector which you can think of in this case as the update operation, the operation that updates the vector, so here it is, let's see, we put zero 1 0 0, this is five dimensional, so I need five 0 0 10. 0 0 0 0 1 0 0 0 Sorry, I'm going to do this in four dimensions. I am getting tired. I don't like five dimensions. Five is too many for me, right. Z z0, let's try it, let's try it. take it out in this Vector here this represents the third state. What happens if we act with this Matrix in the third state? Let's try it, see what we get right.

The first entry at the top is obtained by taking the top. vector and multiply by the column 0 * 0 + 1 * 0 + 0 * 1 + 0 * 0 what is the answer 0 next place 0 * 0 0 * 0 1 time one whoa, whoa, whoa! I did it right, instead of going down, it's going to go up but it's fine up and down we just turn it all over uh would you rather let's just uh let's do it right, let's do it right zero z z zero where did you have it before? I'm here yes one one one one is so up here one okay so let's start again what's up 0 * 0 0 * 0 0 * 1 1 * 0 we're still okay zero next 1 * 0 0 * 0 0 * 1 0 * 0 still zero what about the third place oh please please God 0 * 0 1 * 0 0 * 1 0 * 0 is still 0 but now in the last place I have 0 * 0 0 * 0 1 * 1 and 0 * 0 so one the column has moved down one step now you can check it for yourself here is your task check that anywhere you place this it will move down one step until it reaches the end and then it will recycle and go up to the top, okay? so that's a little thing to check, in fact, you can put any number here, only zeros and ones can make sense, but we can put any number, A B C D, and what will come out here is that they will all go down one step to b c, but then D will move up, so if you place a one in either spot, it will slide down one unit and then reappear at the top.

The point is that the evolution of systems can be represented by matrices of a particular type. A lot of In classical physics, in this kind of classical physics, there are always a few, here quantum mechanics is more complicated, more difficult, but in classical physics, a little bit of zeros and ones to make each state change to the next, okay, that's an example of the use of matrices in classical physics, so far there is no quantum mechanics, just pure classical physics, um, there's an interesting one well, this will do for the moment, we'll come back, we'll come back to that, so that one is an example of matrix algebra, matrices multiplying vectors, what about matrices multiplying matrices here?

Why would we want to multiply? matrices by matrices well here is the idea assuming that we wanted to update or update a second time to update a second time what we would do is apply the same Matrix to the result that we obtained in other words let's write it this way, let's write it abstractly we have a matrix M that we multiply by a vector v to get a new vector v Prime, okay, that's just abstract notation to write a matrix and a vector and get a new vector that updates the vector v to a new vector, let's update it. again, let's go to one more time interval, how do we do it?

Well, what we do is we write M * V Prime equals vble Prime. We would do the same with the update trick, except now we update V Prime instead of v and we would get V Prime V. Prime is the state of the system after two time units, but we could also write that upon realizing that V Prime is M*V, we could write this as M*m*V is equal to VP Prime, this just means we apply the Matrix twice. We can also think of it as squaring Matrix M and then multiplying it by V. So how do you square a matrix or how do you multiply one matrix by another?

Array, this is what I would do if I wanted to update twice once with one. Matrix and then once with another Matrix or the same Matrix, how do you multiply matrices? and the answer is basically the same type of rule. I'll do it now for 2x two matrices because it's getting too complicated even for 4x4 matrices for a 2x2 matrix that we have. M11 M12 M21 M22 let's call it some other Matrix n n11 n12 n21 n22 the result of multiplying a matrix by a matrix is ​​another matrix, it is another Matrix and we do it in a very similar way assuming we want an entry here we get the one entry by taking the first row and multiplying it by the first column M11*n11+M12*n21 of the same type in a product and we put it here now assuming we want the next input for the next input we take the first row because after all we are interested in the first row here we take the first row but we multiply it by the second column here so what would be here would be M11 * n12 plus M12 * n22.

I'm not going to write it. everything now we can move down down if we wanted this entry we would take the bottom row and multiply it by the first column if we want the last entry here we would take the bottom row and multiply it by the last column then we multiply matrices by the same type of pattern that we multiply matrices by vectors, we can just think of it as multiply this matrix by this vector, put it here, multiply this matrix by this vector, put it here, okay, so there's a notion. of multiplying matrices and what matrix multiplication does is that it gives you a new Matrix that updates you not for one time interval but it updates you for two time intervals if you wanted to Short circuit uh uh the update problem and you wanted to update the state of a system five units of time what you would do is multiply The Matrix five times, that is, you do it in sequence first the first time the next and the result multiplied by the next multiplied by the result multiplied by the next and you can solve what is the Matrix that would take you from a state of a system and one instant of time to a state of the system five instants later, so matrix multiplication multiplying matrices by matrices is also an important concept okay one last example of matrix algebra. involves row vectors, assuming you have a row vector and you want to multiply it by a matrix, the rule is that you write the Row Matrix first B1 B2 B3 B4 and then you write the Matrix M11 M12 M13 and so on uh M14 dot dot dot dot dot dot dot dot dot dot dot tired of writing M, well, what will be the result? the result will be a row vector and this is the way to get the row vector entries, the first row vector entry you get by taking the original row vector and multiplying it by the first column vector here that product is the first input, then you take the original row and multiply it by the second column which gives you the second input, then you take the original row and multiply it by the third column which gives you the third input here and so on, you see the pattern: it always They multiply rows by columns and put the result in the correct place, in the correct row and column, in this case, a row.

Vector multiplied by a matrix is ​​another row Vector a row a matrix multiplied by a column Vector is another here is a matrix multiplied by a column Vector is another column vector and a matrix multiplied by a matrix is ​​another Matrix, get familiar with it, solve some examples, solve some examples of your own design just put some numbers in multiply row vectors by matrices matrices by column vectors and matrices by matrices and get the experience of finding out how these things work, do it for 2x matrices two for 3x3 matrices and you will get familiar with it because we will use it again and again, in fact that is the main mathematical operation of quantum mechanics: multiply rows and columns by matrices, if you know how to do it, are familiar with it and can read the answers easily .

I have all the basic mathematics of quantum mechanics. It would be helpful to have a little calculus to go with it, but the new basics are matrix multiplication and column and row vectors, so please practice a little. I should have made up some examples for you to do, but you can make up your own, they are very simple, okay, we are approaching 9:00. Are there any questions next time we start talking about Q? bits Quantum bits and how quantum bits are very different from classical bits ask yes what is your name my name Leonard susin or susin you uh if you want uh you know how to polish the apple for the professor you can call me Leonardo that I like it a lot well you I want That is, put it another way, what restriction does reversibility impose on it?

Oh yes, yes, yes, it has a converse. Okay, yes, yes, yes, yes, but I mean, in a more abstract sense, the answer is that it should. have an inverse that Matrix should have an inverse, the inverse of course is what sets you back, not all matrices have inverses, so, and you want to know what an inverse is, yeah, okay, well, and it we will, if you want. Won't we get to that? Any other question. Yes, that's a good question. Yes. The final exam is buying me lunch. many of you have been here before so you know my policies, my policies are: you are here to learn physics, there is no one here who is here to get a degree or if you are then I will be happy to give you a numerical grade Yes I need one, in fact, if everyone needs a numerical score, I know there's a huge difference in the preparation level of different people here, uh, and comparing it in an exam environment wouldn't make sense because I do know. there is a huge difference.

I know that everyone here is here because they want to be here and they want to learn physics and not because they have to be here. So my policy is not to grade the course at all or if someone needs a grade for some particular purpose uh to uh to give a D minus the lowest possible grade the lowest possible passing grade, so that's it, I didn't tell you. What is it still for, I gave you, it's true, I gave you an example. How it can be used to implement the idea of ​​updating a vector from one instrument to another is an example, but I haven't told you yet why we are doing this.

I often spend an hour talking. about the qualitative aspects of physics, in this case it was how to think abstractly about deterministic physics, abstractly in terms of bits, etc., and then spend some time doing some math, which I won't really tell you to What's the point until next time? time, but I want to make sure, since I'm going to start doing some quantum mechanics next time, I want to make sure that everyone recognizes the little algebraic manipulations that we're going to do and we'll have the math for next time, so it's really for next time set this up, yeah, promise to tell us next time what I think, I think you'll see, I think it'll be clear, I think it'll be clear, uh, clear.

I think it will become clear, yeah, I promise to tell you why not, it's Quantum addicts, uh, that's what it's called.