Lecture 1 | Quantum Entanglements, Part 1 (Stanford)

this program is brought to you by Stanford University please visit us at stanford.edu I almost always begin with the same sermon about especially when teaching about quantum mechanics or relativity the sermon is always the same it's the fact that we as animals have inherited through the process of evolution certain intuitive ways of thinking about the physical world and if you don't believe it you think that maybe ordinary animals are not physicists you watch a lion chasing an antelope and you notice that that lion the minute that the antelope that the relative velocity between the Antelope and the lion changes sign the lion just stopped dead somehow he did some calculation or she it's usually as she the lion did some calculation some physics calculation involving some very complicated concepts of velocity Direction all kinds of complicated computations like that a a primitive cro-magnon man I'm not a cro-magnon man and they end the thal who comes to a cave and sees that the cave was blocked by a boulder and tries to push the boulder and can't push the boulder decides to aim his body that way hmm why so that he gets a bigger component of force in that direction has he ever heard of force is you ever heard of components where'd he get this idea of components that he know about sines and cosines yes somehow he did know about sines and cosines these are things which were inherited are biological in origin and they 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 modern physics has to do with those things which are beyond the intuitions that we were able to get from the ordinary world has to do with with ranges of parameters which are way outside the range of parameters that humans or animals ever experienced for example it's not too surprising that human beings didn't know how to deal with velocities approaching the speed of light but they got the wrong ideas about how to add velocities when nobody in 1900 had ever probably had never probably had never moved faster than 50 or 60 or 100 miles an hour well there are they probably did when they were falling off cliffs but they didn't live to talk about it maybe they got up to 200 miles an hour maybe but nobody ever had experienced anything like the velocities approaching the speed of light and so it was not surprising that their intuitions that their way of thinking about adding velocities and so forth a theory of relativity was and how you synchronize clocks all that stuff all that good stuff that Einstein did that it was outside the framework of their ability to think about through intuitive pictures to intuitive mathematics arm they had to invent new mathematics the new mathematics was abstract meaning to say you couldn't visualize it four-dimensional space-time hmm you can I can't visualize four dimensions I've learned tricks to visualize it so physicists to some extent rewire themselves or people who learn physics through a process of rewiring themselves to some extent to develop intuitions to be able to deal with these new ranges of parameters but still they're foreign they're alien the peculiar even to me quantum mechanics deals with a range of phenomena which is also outside the experience of ordinary humans for which evolution simply didn't provide you the means to visualize evolution did not provide you the means to visualize an electron to visualize the motion of an electron to visualize the uncertainty principle when you think of a particle moving what is a particle a particle is a thing with a position at every instant of time it has a position if at every instant of time it has a position it has a trajectory if it has a trajectory you can calculate the velocity along that trajectory just by knowing the separation between points and what the time interval is you can calculate the velocity and that's the intuitive picture of a particle and where does it come from it comes from thinking about rocks throwing rocks shooting arrows all kinds of things that human beings normally do so we never developed the need it would have it very bizzare if our brains have been wired to understand the uncertainty principle why would Darwin have given us the part of the incidentally if you prefer to think of the intelligent design or go right ahead I prefer I prefer to think about Darwin but why would either of Darwin's ideas or the intelligent designer have provided us with the ability to understand the uncertainty principle when it's never anything that's part of our ordinary experience the answer is it didn't and so quantum mechanics for that reason appears extremely weird to us physicists as I said rewire themselves and developed ways of thinking about it which are intuitive but still quantum mechanics is much much more unintuitive incidentally than the special theory of relativity and what we're going to try to do here is expose 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 in quantum mechanics a conventional class in quantum mechanics would stress such things as the Schrodinger equation and waves and how particles sometimes behave like waves and so forth we may or may not get to a bit of that but that's not the important subject that we're going to concentrate on well going to concentrate on is the basic logic of quantum mechanics the basic logic of quantum information theory physics is information when you say something about a physical system you're saying something some you're giving some information about it you give the information in various forms usually in the form of numbers in classical physics you often give I will give you some examples but you you usually give it in the form of real numbers the position the velocity is set of real numbers and quantum mechanics sometimes you use real numbers but very very often you give discrete information discrete information such as yes or no or up or down or male or female well that's probably not such a good example the difference is because yeah that's prevalent up I think I withdraw that heads or tails heads or tails do I have a coin no Gedanken coin thought coin head tail head tail right when I flipped the coin head that's a piece of information it could be tail it's a two valued system either yes or no up or down heads or tails or sometimes they're logically all the same of course they're logically all the same whether we're talking about heads or tails up or down or whatever it is they're logically the same and they're simply decisions which have to possible or questions which have two possible answers and a bit of information which has two possible answers is called a bit it's called a bit and it can either be a classical bit or a quantum bit all real bits in nature are quantum bits obviously since nature is made out of quantum mechanics but sometimes the quantum aspects of it don't manifest themselves in an ordinary computer the quantum aspects of the bit don't really manifest themselves for reasons that we'll come to and it's just called a classical bit a classical bit of information this head the coin flip yes or no the quantum bit is all is the quantum analog of the flip coin the yes or no type question but it is much much more subtle and the first thing we're going to want to explore is what is a quantum bit now but before we do that let's talk about classical bits classical bits can be described either by writing down a 0 or a 1 these are we could also use 1 and minus 1 or we could use 5 and 15 doesn't matter but 0 and 1 is a convenient notation for the two possible our values 0 could stand for heads 1 could stand for tails and so forth so we're thinking over thinking about this we're thinking about some physical system when we're thinking about information we're thinking about a physical system such as a coin and this is the information contained in that coin either a 0 or a 1 there's a notation this seems like a ridiculous and redundant notation its importance will only become clearer when we start to think about quantum bits but we're going to use the Dirac notation the Dirac notation describes the state of a bit not whether it's California or Aragon but the configuration of the bit and it's usually labeled with the notation 0 or 1 or whatever other information whatever other way you decide to think about the bit these are the two states that a bit can have either a 0 or 1 and it's represented I don't know that I don't know if I drew that well let's draw it again 0 or 1 these are the two states of a bit all of this extra junk here is excess you don't need it it doesn't tell you anything it just says that you're putting it inside the bracket incidentally this point the bracketed object over here the thing that contains the information on the inside is called a ket it's called a ket because it's the second half of something which we will later learn is a bra ket or a bracket there's another half that we haven't exposed yet now what about multi bits supposing you have more than one bit and we're talking now classical physics so far we're not talking about anything quantum mechanical supposing we have several coins and I line them up I label them so we know which one is which in fact just in order to not confuse coins let's make sure the different coins penny nickel dime quarter half dollar silver dollar okay so we have a bunch of coins we can't confuse them and we can lay out some information by saying head tail tail head head tail that would be some information about a collection of bits all right how would you label that well you would label it with a string of zeros and ones so for example if we let's take 0 always to stand for head it's easy to remember 0 stands for head and one stands for tail for obvious reasons umm right so my string of coins heads head head head tail head tail head I would label 0 for head 0 for another head 1 0 1 1 for example that's that's a configuration of 1 2 3 4 5 6 coins right let's say it was 6 coins right that's a configuration of a multi-bit system in this case 6 bits again for reasons that and absolutely nothing to this description we're going to stick it inside our cat and stick it inside a cat which is just a kind of notation it's might be a good idea to put some commas between these but maybe not maybe it's just best to leave it that way that's a specification or it could be the specification of the bits of information inside a computer it could be just the series of heads or tails or so forth 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 of well let's start with one bit if there's only one bit then there are only two states what if there are two bits well then you can have up up up down down up down down for two times two so for two bits we have two squared what if we have what if we have a hundred bits the answer is 2 to the 100th power 2 times 2 times 2 times 2 100 times so if you have an in bit system the number of possible classical figure configurations is just 2 to the N let's write that down let's so let's put a notation let's write the number of states the number of states n sub s the number of states of a system of little n bits is 2 to the N let's suppose we are let's invert that first of all let's invert that little N is what little n is the number of bits big in is the number of states little n is the number of bits ok so if the number of bits is 4 then the number of states is 2 to the 4 which is 16 and so forth we can invert this and we can write if we knew the number of states in a system then we can take the logarithm of this equation log to the base 2 is particularly convenient if we take the log to the base 2 of the number of states that's equal to the number of bits you can generalize this not every system has as its number of states to to a power supposing I have a state a system um a die you know the things you use in Las Vegas to throw away your money with it's got six possibilities one through six that is not to to any particular power it's just six but we can still generalize this definition of the number of bits of information in fact the number of bits of information that a system can contain is by definition the logarithm to the base 2 of the number of states which for the die would be log to the base 2 of 6 what is log to the base 2 of 6 is it an integer no it's some stupid irrational number I don't even what is it how big is it about the 2 to 2 to the 2 to the 2 is 4 2 to 3 is 8 2 to the 2 point 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're going to be considering systems which are made up out of some number of bits each of which has two states so for the simplicity we're going to be talking about systems the number of states is always 2 to a power that's just the simplicity there's nothing special about but almost every system can be represented that way or approximately represent it that way let me give you an example supposing we have some question of physics which has as its answer a real number but we're only interested in that real number to a certain approximation the temperature the temperature in the room I'm interested in the temperature in the room Co after to a certain number of significant figures I can represent the temperature or any other number for that matter by writing it as a number in base two right if what's the temperature in this room incidentally it's about three hundred degrees from absolute zero so it's three hundred I can write three hundred not as three hundred which is what is 300 min 300 you know it means it means three times ten to the two buts nothing times 10 to the 1 plus zero times 10 to the zero but we write it two things in base two all right I don't know what 300 looks like in base two somebody can figure it out base to you everybody know how to rhythmic taken base to anybody not know arithmetic in base 2 okay so everybody knows arithmetic in base two we write out any number that we like as a series of zeros and ones one zero zero one zero one zero zero one that's some number to the some particular integer it's an integer so if I'm interested in the temperature and I'm not interested in being too careful to define fractions I want to know whether it's 72 degrees or 73 degrees I don't care about seventy two point four oh six nine I can write it as an integer and that integer can be represented as a sum of bits not a sum of bits but as a collection of bits every number every single number if you're willing to truncate the number of decimal places approximate that number and say I'm interested in that number only 235 decimal place or whatever or 2:35 places to base to that that number simply is both is represented by and represents a collection of bits so anytime and incidentally if you want to have a finer grained description of the temperature than then integers in centigrade you just use a more refined notion of degree you go down to ask how many degrees is it but not in centigrade units but in units of ten to the minus 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 supposing I'm interested in a field a field means a thing which can vary throughout space all right a thing which can well the temperature can vary throughout space the temperature is a field it varies throughout space it's not one of the more interesting fields from the point of view of fundamental of fundamental particle physics or anything but that certainly is a field it varies from place to place and how can we represent that can we represent that in terms of bits yes if we're willing to tolerate certain approximations and we're always willing to tolerate some degree of approximation what we do is we break up the room into a lot of little tiny cells I won't try to draw a three-dimensional room in my notes I drew a three-dimensional room it took me about a half an hour to put in all the lines just a two dimensional room and here's what we do we first of all order the cells we make the cells small enough so that the temperature doesn't vary very much because from cell to cell so we might fill this room with several billion cells label the cells this is the first cells - one two three four five six seven or up to a thousand thousand one thousand two thousand and three thousand and four and we can label all of the cells and list them once we've listed them we can write the temperature of the first cell there's the temperature of the first I'm putting a little comma in just as distinguish between cells then we can write the temperature in the next cell 0 0 1 and 1 2 3 4 5 6 7 8 9 I've kept 9 decimal places in you in in the basis in arithmetic and base 2 1 1 0 1 however till I'm come on finished then I go to the next cell do the same thing temperature there is 1 1 0 1 0 0 and so forth eventually all I have is a list of zeros and ones this long list of zeros and ones if somebody knows how to use it is equivalent to knowing the temperature at every point in the room the same is true of the electric field the magnetic field anything which varies from place to place so almost everything that I can think of in physics can be represented in terms of bits so if you know everything about how bits work you'll basically know everything about how physics works of course you may not know what the rules are the myth I'm manipulating these things but this is the basic setup of physics information in the form of a series of questions each of which can be yes or no now of course you may want to refine your description to refine your description you may might want to add more decimal places to the temperature to the specification of temperature and you might want to make your lattice finer that's just making a better approximation so the right thing to say is that most physical systems that we know about as far as I know all physical systems can be represented at least approximately and perhaps two always increasing approximation by a series of bits that's why we get to use computers to do physics if this weren't true we couldn't use a computer we couldn't use a digital computer in any case to do physics we have to use analog computers or something okay so that's a let me give you another example another example of how you might use bits to represent another these are all so far classical systems as I said I don't want to redraw the lattice but I do want to get rid of its top row here since I've already mutilated it here's a lattice and what I'm interested in is the motion of particles this lattice is just an artificial imposed lattice that I've imposed on the room here just so that I've divided the room into mathematical cells and what I'm interested in is the motion of particles moving around in this room at any given instant I can ask the question let's take a very simple case let's take the case where a particle where you can't squeeze more than one particle into one of these cells we can imagine that the cells are about as big as a particle in which case you can't squeeze in more than one then every cell either has a particle or it doesn't have a particle we can label the cells that have particles with an X we can label the cells that don't have a particle with nothing or better yet we can label the cells that have a particle with a1 and the ones that have no particle with a zero in that case this becomes a specification of where the particles are in the lattice so the longer the temperature but the same long sequence of zeros and ones now the number of zeros and ones would just be equal to the number of number of cells in the lattice what would this number mean it would mean that in the first cell there's a particle and the second cell is no particle and the third cell is no particle in the fourth cell is a particle and the fifth cell no particle in the sixth cell particle and so forth and so on and so given such a string of numbers you are given a specification of where the particles are in this room in that way again motion of particles most of the fields temperature just about anything in physics can be represented in terms of bits any questions right a bit it a bit is by definition a question about a system which has only two possible answers which you can always take to be yes or no used to be a game twenty questions didn't where somebody would think of a category and then you would stand there and ask yes/no questions and until you try to figure out what the category or what the what the category was so that was using the idea of death yes question oh I just I just arbitrarily said supposing we're interested in the temperature to a certain degree of accuracy right so I'm interested in the temperature to accuracy but now I'm not speaking about temperature I'm just giving another example these are just examples intended to show you something which is which is more or less clear otherwise we could not use computers to to simulate physical problems classical figure physic pop yes right but you need to provide for the general real number you need an infinite number bits right any rational number can be represented by a finite number of bits and the rule well that's not quite true you have to you have to remember to repeat them but yeah if it's rational it's going to repeat after some point right so but if it's an irrational number then you need an infinite string of bits but in general we will allow infinite strings of bits although not in a genuine computer well so so far remember we're doing classical physics all right so far no quantum mechanics so I will come on let's see where yes we were going to come to that very very shortly let me tell you how very quickly an electron first of all we're not talking about motion yet we're talking about configuration configuration means the state of the system at a given instant of time okay so the presence of an electron at a given instant of time let's suppose the nucleus is known to be right over here and we're not going to ask about the nucleus the nucleus just sits there it's a lump on all right so um we could say at instant number one when we begin the experiment the electron is over here in that case we would write down a string of zeros with a 1 someplace pure zeroes one electron pure zeroes except for one splice in the in the sequence where there's a 1 now if we wanted to describe the motion of the electron we would say starting with this configuration we move and let's use this symbol here to indicate that at the next end we we could we've broken up space into a lot of little individual cells we could also break uptime I thought I had my watch but I don't we could also break up our watch into a digital watch which the witch digitizes time just again as either a convenience or an approximation and we could say if at digital time number one the electron was or the system was described by one electron located at this location then what happens next next it moves did some no configuration in this case it might move over one place 1 2 3 4 5 6 it moves over to the sixth place one two and so forth so the motion of a system is described by a rule of updating of updating information how you update it from one instant to the next all right so physics basically consists of two a physical system consists of two things it consists of a collection of possible States which can be labeled by a collection of bits and it consists of an time evolution which is an updating which tells you how to take one collection of bits and replace it by another collection of bits at a slightly later instant of time I don't know if that answer to actually work out an orbital motion orbiting around here gets confusing because when you jump from one layer to the next if this is one and this is a hundred then 101 is over here so you don't jump from a hundred to 101 you might jump from a hundred to over here which would be 200 so it can be complicated the updating procedure it can look complicated but nevertheless it's an updating procedure that that just updates your your state of knowledge at each instant of time that's classical physics now there are some rules and we're going to come to them but before we do let's define the space of states this is and I want to emphasize we are still doing classical physics there is nothing quantum mechanical even though we're talking about discretizing systems and making out of them systems of individual bits so far we are dealing with what should be called classical bits see bits I think they're called as opposed to qubit skew bit as a quantum bit this these are classical bits so far okay so let's take all of the configurations and just abstractly in a purely abstract way we take all of the configurations incidentally what it is about ten by ten is roughly a ten by ten lattice ten by ten lattice has a hundred sites how many states does it have if we I'm not talking about one particle now I'm talking about any number of particles can be on this lattice how many different configurations are there two to the hundred a very very very big number two to the hundredth power that's how many different ways we can arrange zeros and ones on this lattice or specify whether there's particles in various positions a very large number of possible states but let's just abstractly think about all these states and just draw them as points if there are a 10 to 100 I have to draw a 10 to the hundred points which I'm not about to do these are the various states these are not the lattice points these are the various states for example for one bit if I had only one bit then the space of states would consist of only two points up and down and I would just draw two points this would be the space of states of a simple one bit system now let's ask our what are the possible laws of updating in other words what are the laws of motion laws of motion are the laws for updating configurations what are the possible laws of updating well here's one possible law of updating this could stand for heads this could stand for tails let's let's think about it in terms of coins for the moment this could stand for heads this could stand for tails if we start with heads if I had a coin we would do it as it tails heads tails all right one possibility is very simple if you start with heads it stays heads nothing happens if you start with tails it stays tails nothing happens that's a law of updating it's not a very interesting law of updating how would you draw that well here's how we'll draw it heads goes to heads will make an arrow if we start with heads it stays heads and we start with tails it stays tails so we draw an arrow from what you start with to what you end with what's another possible law of updating here's the law of updating if its tail's it becomes heads that it becomes tails then it becomes heads then it becomes tails that's a that's a little more not very much but a little more interesting a slightly more interesting system it just flip-flops back and forth how would we draw that we would draw that again heads tails heads tails if you start with heads you go to tails and sour tails you go to heads so the law of updating in this case is just described by such a diagram basically a diagram which tells you if you started a given state what it will be in the next instant of time is that clear alright so this is one way of describing the laws of physics write down all the states keep in mind what they stand for of course remember that in this case one stands for heads one stands for tails or whatever it happens to stand for if it's that male female this could be an interesting case of this wouldn't that would be an interesting this this would be a very interesting the law of motion in that case I don't think I want to I don't think I want to explore that any further if this was my undergraduate class I would never have brought that up this is more likely this is the more likely law of updating for sexuality female male so you see simple laws sometimes apply sometimes they're a little more complicated can you think of an interesting system that flip-flops like this Oh hand I can't think of anything I mean it's obvious that it applies to a lot of things but I offhand I can no no no right right right right but this is the thing yes but if you just tell you're not intervening I don't want you to intervene this is the system by itself if you had some peculiar lights which by itself we're back and forth and back and forth and back and forth in a regular way but that's stay and you know this is the Excel which well a lot of states to the pendulum in between but yes you've got the idea it's hard to think of a simple example but I bet by that if we'll go home and we came back next week every one of us would have an example of a we could call this the flip-flop this is a flip-flop motion this is the the the uh the uh notion well we can extend this if we know what the space of configurations is and we lay them all out either abstractly in our mind or actually just write them on the blackboard then the motion of the system can be represented by a series of arrows where I'm getting tired but and so forth and so on yeah um let's do let's do the possible let's think of some possible motions of a two-bit system a two-bit system simply has four states that's all we have to know it has four states well here's one possible motion if we start with this configuration we move to that configuration if we start with this configuration we move to that configuration and so forth if we watch the what actually happened with time the system would move from one configuration to the next around the closed loop now the closed loop is not necessarily a closed loop in space it's a closed loop in the logical space of possibilities here logical space of configurations that's one possible thing that K here's another one perfectly good what this is is it's a pair of systems a pair of it's a pair of systems which are separately undergoing flip-flops each one undergoing flip-flops this one is flipping and flopping this one is simultaneously flipping and flopping if we start over here let's see what that stands for that stands for example for both heads could same for both heads then we go to both tails then we go to both heads then we go to both tails or we could start with one head one tail and do this that's what this is this is a pair of systems flipping and flopping there are other possibilities so there are different laws of motion that the system whatever it happens to be could have so when you specify a system you not only have to specify with the states of the system R but you'll have to specify how it moves and how it moves is a rule for jumping from one configuration to the next now let me give you an example of a logically perfectly sensible rule but which is defective from the physics point of view never happens in physics we can do it I think well let's do it with four with with four states with four states let's see how this went yeah here it is if you start here you go here if you go here start here you go here excuse me one moment for some odd reason in my notes I've drawn instead of a diamond shape I've drawn a square let me go back there's my four states okay if you start here you go here if you start here you go here if you're over here you go over here and if you're over here you go over here all right so now we can say what happens wherever you start if you start over here you jump to here you jump to here you jump to here you jump to here you jump to here you go here you go here you go here you go here you go here notice you never come back to here okay with this particular law there's something different about this law then there is about the other examples and all the other examples well can anybody spot what's wrong with this well not what's wrong with it but what's different about it well it doesn't consist of loops this is true you can't figure out necessarily where you came from you may be able to tell where you go next but you can't always tell where you came from for example if you find yourself over here you don't know if you came from here or whether you came from here if you're over here you have lost a piece of information this is a motion which loses information it loses information in the sense that you can't tell where you came from there's no way to reconstruct the past but you can reconstruct the future or construct the future wherever you are you're told where to go next but wherever you are you don't know how to get back if you're over here well you know you came from here but then you don't know whether to go back here or to go back here so this is what is called an irreversible history it's a history or a law which loses information and at the fundamental level of physics fundamental level where you're really keeping track of everything not where you're not where your coarse graining or not looking carefully but we are carefully looking at every degree of freedom of a system classical physics never allows the loss of information like this there is a unique future point wherever you are and there is a unique past point wherever you are okay that is one of the laws it's not necessarily a law of logic it is something which is true of all physical systems that they are reversible in that sense yeah let's say I change state twice all right I'm over here one two or one two I don't know if I came from here or here right we could give this property a name we give it the uniqueness of the future point and the uniqueness of the past point we could invent a name for we could call it uni parity do you know what the quantum version of unique parity is it has a name it's called unitarity unique parity is a name I just made up unitarity is the quantum equivalent which tells you that you can always reconstruct the past from the future of the state of a quantum system you can either run forward uniquely or run backward uniquely and you'll come to some unique previous state or future state and that's called unitarity in quantum mechanics we haven't done quantum mechanics nothing's quantum mechanical yet it's a kind of time it's a kind of time reversal symmetry or it's actually not a time reversal okay so it's not a time reversal symmetry exactly it's a time reversibility I would say this diagram has a sense of orientation to it if I start something over here it goes around this way it definitely does not go around this way unless I reverse it unless I look at it backward in time so it's it's not precisely what you would call time reversal symmetry time reversal symmetry means that you could either go in going into the future you could go either way but in this case you only go one way but it's it's it's the reversibility of the laws that that you can find the reverse law give it a law you can find a reverse law which will take you in the backward direction I think that's I think that's right yes yes well two arrows coming away from a point since I don't know which way to go oh boy I go that way or do I go that way so it's it's it's clear that that's not a law of motion okay no branching ratios oh well let's we do at the moment we're doing quantum mechanics so I mean classical mechanics so it gets more complicated with the branching ratios in the classical mechanics is less fundamental in quantum mechanics all real systems are quantum mechanical the question why some of them suppress the quantum mechanics and you don't see it is a question which we'll try to answer as we go along but we might ask it'll at the quantum level and at the quantum level I think we can give a better answer but ultimately at the end of the day it seems to be a law that that basically says that forward in time and backward in time are our I won't say equivalent to each other but that there's no preferred really preferred sense in which forward in time is different than backward in time even though it feels like there is well this this this is this is the question that took 20 years to answer and the answer is in my opinion no it is not possible but it was one of the great questions of physics that took a long time to answer and I'm not going to get into it now but it might be an interesting thing for us to explore toward the end after we've talked about quantum mechanics we we have to talk about quantum mechanics before that makes sense that question yeah I think they're all Newtonian in a sense in the sense Newtonian to me simply means that there's a definite state for a system that it evolves with time according to a definite law of a deterministic deterministic that's that that's the right word yeah but yeah I think you can think of more complicated situations you just can start drawing some diagrams yourself and and see what makes sense what what's reversible what's not reversible and but yes you're right that is true it loses the information as the way you came from the system yep we mean it's not part of the system the system started here at went to here I went to here aren't they yeah whatever hmm you won't find that point ever again right right but the main point is you've lost the distinction between two possible starting points whereas in all the other situations if you know where you are and you know how many steps you made you can say where you were well I think I think for a long long time mr. Stephen Hawking thought that this is the way black holes work so not so clear not so clear okay yeah yeah good good so let's let's talk about that um if I take a bunch of molecules in a bathtub I don't what's a good example well let's take the molecules in this room and I sell to start them all out in a certain configuration very definite configuration I put them all up in the left-hand corner of the room there and I let them go after a while the room will be full of air just like it is if I put them up in that corner of the room over there and let them go after a while same thing put it up in that corner after a while same thing so it looks like we've lost information but in fact that's not true if we followed every single molecule and we followed it in infinite detail with infinite precision which we don't do of course ah then then we could reconstruct by running everything backwards we can reconstruct the fact that the molecules may have come from that corner of the room it's prohibitively impossible to do in practice but in principle following every single detail of every molecule know what really happens in the real world is we lose information because we lose the ability to follow the details not because the information gets lost but because we lose the because we lose the ability to follow the information that's where the second law comes when you start losing the ability to distinguish different states so we don't distinguish whether in our coarse-grained picture we don't distinguish the different detail at the level of a molecular detail and so it looks like different configurations become the same configuration but that's only because we simply don't look carefully enough it's because we're lazy do you need an infinite number of bits - ah ah ah you mean in the you know in a real room like this no because of quantum mechanics because of quantum mechanics no but if it were not for quantum mechanics yes you would need an infinite number of bits now what does that mean that means that you have to specify a bunch of real numbers precisely with infinitely with tremendous precision you have to precisely prescribe the locations and also the velocities but in particular the locations of every single molecule with a tremendous amount of precision and the longer that you want to track the system the more precision that you need so ultimately to track a system for a long time you need to specify with infinite precision the exact positions of every point wherever every molecule that means you'll have to give a set of real numbers a set of real numbers involves as you say an infinite number of bits so the answer is for for a collection of real particles moving around that you really try to follow classically depending on how long you want it to follow it you would need more and more bits to describe it oh oh yes yes yes that's right if room were really sealed let's let's let's idealize this room so that nothing can get into or out of the room all particles bounce off reflect off the walls of the room so that it's an entirely sealed up room then the room can be described discreetly because of quantum mechanics at least up to some energy if we know that the energy isn't arbitrarily high then we can describe it by a discrete collection of variables that has no exit so let's say yeah so we could so to make such a thing we could just reverse all the arrows here is an example no this one has no exit well then if I wanted to exit to itself I have to do this we could do that but as I drew it it has no exit but but let's let's think about what it means I'm not too interested in we're here the question is what happens when you're over here all right when you're over here you have two ways that you could go and you don't know which way to go so it's not deterministic it doesn't know whether to go this way or this way it might go half the times this way half the times this way you might need some statistical rule 50% of the time or 30% of the time it goes this way 30% so it's our 70% of the time with random statistics that would be non-deterministic okay so it seems that the real real laws of nature are both deterministic forward in time and backward in time that's the implication of not having loose ends floating around like this that they're deterministic either way so that wherever you are you can either trace forward uniquely or backward uniquely and that is all of classical physics in a nutshell you now think in a complete course in classical physics or there's nothing that does not fit that pattern at least two arbitrarily high degree of approximation let's take a let's take a seven-minute break well I was going to jump to quantum mechanics but before I do I want to do a little bit of mathematics elementary mathematics most of you know it but nevertheless let's lay it out matrices and vectors I'm not at the moment I'm not going to mathematically define a vector in any sort of sensible method you know even approximately rigorous way or abstract way I'm just going to tell you 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 I don't want to koala and my numbers now I at the moment I mean real numbers as opposed to a complex numbers I don't mean zeros and ones I mean arbitrary sets of real numbers they could be zeros and ones but zeros and ones are fine but just general numbers so I just lay my what should we call them hmm yeah yeah the called components but I want to I want to letter for them a a a is good so a a 1 a 2 a 3 a 4 and just put something around them to surround them so that we know this would be a four dimensional vector all right why four dimensional because it has four components we get don't try to visualize vectors now there's no value at all for our present purposes and trying to visualize these as pointing in space or anything like that they're just lists of numbers okay lists of that's one way okay there's another way that we can list the same set of numbers put them in a column a 1 a 2 a 3 a 4 same information in him I mean then I'm not talking about information in the abstract sense that I used before same thing sometimes it's useful to write it this way sometimes it's useful to write it that way you'll find out as we move along when it's written in this form it's called a row vector when it's written in this form it's called a column vector well we're actually talking about now is notations neat notations for for doing certain arithmetic 'el operations involving collections of numbers when we get the complex numbers we will then use complex conjugate notation yes but for the moment let them just be real numbers ok now there's another concept now called a matrix and think of a matrix as the following way a matrix is a thing which acts on a vector to give another vector all right so it's a kind of machine you put the vector into the machine and out pops another vector according to a particular rule oh no sorry before we do that before we do that let's imagine a particular column vector and another different row vector different row vector has different entries not the same set of numerical entries but a different set of the miracle enter entry so let's call them B B 1 B 2 B 3 B 4 these could be six point oh one five point nine seven three point oh four and A one could be seven point eight a two they none of them could be the same or they might not be the same the A's and the B's this is some particular row vector and some particular column vector there's a notion of multiplying a row vector by a column vector the notion of multiplying a row vector by a column vector is as simple as a following simple operation you take the first entry Oh incidentally the dimensionality of the row vector and the dimensionality of a column vector should be the same that means that they should have the same number of entries not necessarily four could be five six seven in which case they would be five dimensional vector spaces six dimensional vector spaces this extends the any number of entries into the columns and rows but the rows and the columns should have the same number of entries all right there's the notion of the product of a row vector and a column vector it's called the inner product and it's very simply constructed you take the first entry of the row and multiply it by the first entry of a column you add to that the second entry times the second entry plus the third entry times the third entry plus the fourth and three times the fourth entry so the product of these two which you could just write as B next to AE that product the inner product is B 1 a 1 plus B 2 a 2 plus B 3 a 3 plus B 4 a 4 it's a number it's not itself the product of these two vectors the inner product is not another vector it's not a matrix it is just a number the numerical value has just gotten by adding up the column though sorry the row times the column in just this form P 1 a 1 plus P 2 a 2 plus B 3 a 3 plus B 4 a 4 is that clear don't ask me why that's definition yeah yeah if we were talking about ordinary vectors in space it would be the dot product yeah yeah more abstractly for abstract vector spaces it's called the inner product but yes it is the same as the dot product for our three dimensional ordinary vectors in space where these would be the components of the vector yeah ok now there's the concept of a matrix and a matrix as I said is it it's an operation that you can do on a vector to give a new vector all right but it's not any old operation there 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 all right so in the first place we put m 1 1 to indicate that it's in the first row in the first column then m 1 2 then M 1 3 then M 1 for M what should I call this 1 2 1 it's in the second row for the first column this is in the second row second column second row third column next one m3 1 m3 2 M 3 3 m3 4 and m41 m4 to m4 3 & 4 for now as I said I've chosen four dimensions just arbitrarily four is about as many as bigger as I want to handle on the backboard and it's big enough to be a little abstract so that it's general enough to see what's going on alright that's what a matrix is that's all it is now you can think of it you can think of it you can think of each column as a column vector whose components are labeled by the first entry here okay each one of these can be thought of as a column vector where the first entry labels the column entry or you can think of it as a collection of row vectors where it's the second entry which labels the component either way you can think of it both ways at the same time a collection of column vectors or a collection of row vectors but altogether it forms a matrix now matrices can multiply vectors so let's put a vector over here a1 a2 I should line them up more carefully a 1 a 2 a 3 a 4 and when you I don't know I've done a reasonable job of keeping rows and columns underneath and next to each other but if you like draw some imaginary lines to separate them into rows and columns all right this matrix acts on this vector to give a new vector what is the new vector and he is the rule I've made the vector wide because each entry is going to be a fairly complicated expression but it is just another vector it's another single it's another column it's a column which I've had to draw wide in order to be able to fit in everything I want to write down here's what you do if you want to find the first entry into this column I'm sorry into this row into 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 that's M 1 1 times a 1 plus M 1 2 times a 2 plus M 1 3 times a 3 plus M 1 4 times a 4 in other words you take all of this and you multiply it by this according to the inner product rule and that gives you the first row M 1 1 a 1 plus M 1 2 a 2 plus M 1 3 a 3 plus M 1 4 a 4 now you want the second entry into this new vector over here done exactly the same way except you go to the second row and you take the second row and multiply it by the column that's going to give you and I'm only going to do two of these the rest you can do yourself n - 1 M 2 1 again times a 1 plus M 2 2 times a 2 plus M 2 3 a 3 plus M 2 4 times a 4 and the other two entries you can figure out you get them by multiplying the next row by the column and finally the third row by the column that gives you a new vector it's a way of processing a vector to produce a new vector I will give you some examples as we go along it's a rule of multiplication which is very useful the reason it's defined is because it's useful and we're going to see how it's useful by using it let me give you an example of how a matrix how the idea of a matrix can represent the time evolution of the configuration of a system supposing again we have a our our configuration space let's label let's label them let's take a let's label them the first configuration the second configuration the third configuration the fourth and the fifth configuration these are not points of space these are configurations of a system which has five distinct States and let's take a very very simple law of evolution the first one if you start here you go to here and you've got 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 it from here go back now well you know that one that's no good that's that's disallowed I think is that this allowed I think that's this allowed I think that's this allowed yeah that's this allowed because if you find your yeah that doesn't that's not reversible that's not reversible that's not what I wanted to hear what I wanted to hear is that you go back to here so this is just a 1 goes to 2 2 goes to 3 3 goes to 4 4 goes to 5 5 goes back to 1 it's a cycle here's another way to represent the same thing we can represent the state of the system by a column vector and the column vector we simply insert a 1 someplace if I want to represent the first state over here I put a 1 and then a bunch of zeros 1 2 3 4 5 1 2 3 4 5 this simply represents the first state what about the second state the second state are represent by 0 1 0 0 0 the third state by 0 1 and so forth so the states of a system can be represented by a column but a particular kind of column a column with all zeros and a one someplace where is the one namely whichever state you're focusing on if you're focusing on the 5th state put the 1 in the 5th entry here ok now what is this rule of evolution the rule of evolution says that if you have a 1 someplace then in the next instant of time the one moves down so if you start here the 1 moves down to here and the next instinct it moves down to here next instant it moves down to here next instant moves down to here then where does it go up to the top right so there's a procedure that you do on this column to tell you where the system goes in the next instant of time that process can be represented by a matrix so let me show you the matrix that represents that the matrix is an operation on a vector which you can think of in this case as the updating operation the operation which updates the vector so here it is let's see we put 0 1 0-0 this is 5 dimensional so I need 5 0 0 1 0 0 0 0 0 1 0 0 0 I'm sorry I'm gonna make this for dimension I'm getting sick on I don't like 5 dimensions 5 too many for me right 1 0 0 0 let's try it out let's try it out on this vector right over here this represents the third state what happens if we act with this matrix on the 3rd state let's just try it out let's see what we get well the first entry up on the top is gotten by taking the top vector and multiplying by the column 0 times 0 plus 1 times 0 plus 0 times 1 plus 0 times 0 what's the answer 0 next place 0 times 0 0 times 0 1 time 1 whoops whoops whoops again ok yeah instead of going down it's going to go up by it's okay up down we just turn the whole thing over would you prefer let's let's let's just let's let's get it right let's get it right 0 0 0 0 with a habit be flat over here 1 1 one one and then up here one okay so let's start over again what's up on the top 0 times 0 0 times 0 0 times 1 1 times 0 we still okay 0 next 1 1 times 0 0 times 0 0 times 1 0 times 0 still 0 what about the third place oppa please please please God 0 times 0 1 times 0 0 times 1 0 times 0 it's still 0 but now in the last place I have 0 times 0 0 times 0 1 times 1 0 times 0 so 1 column has moved down one step now you can check for yourself here's your homework check that any place that you put this one it will move down by one step till it gets to the bottom and then it will recycle and go up to the top ok so that's a little thing to check in fact you can put in any numbers here only zeros and ones way make sense but we could put in any number a b c d and what will come out over here is everybody will move down the step a b c but then d will move up to the top so if you put a 1 in any one of the places it will slide down 1 unit and then we appear at the top the point is that the evolution of systems can be represented by matrices matrices of a particular kind bunch of in classical physics in this kind of classical physics there always just one here quantum mechanics is more complicated and more difficult but in classical physics but sprinklings of zeros and ones so as to make the each state shift into the next one that's that's an example of the use of matrices in classical physics so far no quantum mechanics just pure classical physics there isn't there is an interesting well all right this this this will do for the time being will come we'll come back to it so that's an example of matrix algebra matrices multiplying vectors what about matrices multiplying matrices here why might we want to multiply matrices by matrices well here's the idea supposing we wanted to upgrade or update a second time to update a second time what we would do would be to apply the same matrix to the resultant that we got in other words let's write it this way let's write it abstractly we have a matrix M which we multiply by a vector V to get a new vector V Prime ok that's just abstract notation for writing a matrix and a vector and getting a new vector that's updating the vector V to a new vector let's update it again let's go one more interval of time how do we do that well what we do is we write M times V prime equals V double prime we would do the same updating trick except now update the prime instead of V and we would get V double prime V double prime being the state of the system after two units of time but we could also write that by realizing that V prime is M times V we could write this as M times M times V is equal to V double Prime this just means we apply the matrix twice we can also think of it as squaring the matrix M and then multiplying it by V so how do you square a matrix or how do you multiply one matrix by another matrix this is what you would do if you would want 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 kind of rule I will do it now for two by two matrices because it's getting too complicated even for 4x4 matrices for a two by two matrix we have M 1 1 M 1 to M 2 1 M 2 2 let's call it some other matrix n n 1 1 n 1 2 & 2 1 & 2 2 the result of multiplying a matrix by a matrix is another matrix it's another matrix and we do it in a very similar manner supposing we want 1 1 entry here we get the 1 1 entry by taking the first row and multiplying it by the first column M 1 1 times N 1 1 plus M 1 2 times n 2 1 same kind of inner product and we put it over here now supposing we want the next entry for the next entry we take the first row because after all we're interested in the first grow up here we take the first row but multiply it by the second column over here so what would be over here would be M 1 1 times N 1 2 plus M 1 2 times n 2 2 I'm not going to write it all out now we will move down to the bottom the bottom if we wanted this entry we would take the bottom row and multiply it by the first column if we want the last entry over here we would take the bottom row and multiply it by the last column so we multiply matrices by the same kind of pattern that we multiplied matrices times vectors we can simply think of it as multiplying this matrix by this vector putting it over here multiply this matrix by this vector put it over here okay so there's a notion of multiplying matrices and what multiplying matrices does is it gives you a new matrix which updates you not by one interval of time but updates you by two intervals of time if you wanted to short-circuit are the problem of updating and you want it to update the state of a system five units of time what you would do is multiply the matrix together five times that's you do it in sequence first the first time the next result times the next one times the result times the next one and you can you can work out what the matrix is which would take you from the state of the system and an instant of time to a state of the system five instants later so matrix multiplication multiplying matrices by matrices is also an important concept one last example of matrix algebra involves row vectors supposing you have a row vector and you want to multiply it by a matrix the rule is you write the row matrix first B 1 B 2 B 3 B 4 and then you write the matrix M 1 1 M 1 2 m 1 3 and so forth M 1 4 dot dot dot dot dot dot dot dot dot dot tired of writing M's well what's the result going to be the result is going to be a row vector and here's the way you get the entries of the row vector the first entry of the row vector you get by taking the original row vector and multiplying it by the first column vector over here that product is the first entry then you take the original row and multiply it by the second column that gives you the second entry then you take the original row and you multiply it by the third column that gives you the third entry over here and so forth you see the pattern it's always multiplying rows by columns and putting them putting the result in the right place in the right row and column in this case a row vector times a matrix is another row vector a row a matrix times a column vector it is another here it is a matrix times a column vector is another column vector and a matrix times a matrix is another matrix um get familiar with that work out some examples work out some examples of your own devising just put some numbers in multiply row vectors times matrices matrices times column vectors and matrices times matrices and get the experience of working out how these things work do it for 2 by 2 for 3 by 3 matrices and you'll get familiar with it because we will use it over and over and over again in fact that's the primary mathematical operation of quantum mechanics is multiplying rows and columns times matrices if you know how to do that and you're familiar with it and you can read off the answers easily you've got all of the basic mathematics of quantum mechanics it would help to have a little bit of calculus to go with it but the basic new thing is matrix multiplication and column vectors and row vectors so please practice with it a little bit that should have made up some examples for you to do but you can make up your own they're very straightforward ok ah we're getting close to 9 o'clock are there any questions next time we're going to start talking about qubits quantum bits and how quantum bits are very different than classical bits question yes my name Leonard Susskind or Susskind you if you like you know polishing the Apple for the professor you can call me Leonardo I like that very much well you mean let's say the other way what restriction does reversibility place on em yeah yeah that it have an inverse right there yeah yeah but I mean in the more if I were earth in a more abstract sense the answer is that it should have an inverse that the matrix should have an inverse the inverse of course is the thing that takes you back not all matrices have inverses so and you want you known inverses yeah okay then we'll look if you don't we'll come to it any other question yes that is a good question yeah the final exam is buying me lunch right that's a lot of lunches out there boy every look nobody asked me about grading the class no but I mean a lot of you have been here before so you know my policies my policies are you here to learn physics there is nobody here who is here for a degree or if you are then I'll be glad to give you a numerical grade if you need one in fact if everybody needs a numerical grade I know that there's an enormous difference in the level of preparation of different people here and to compare to compare you in an A and an exam setting wouldn't make sense because I do know that there's an enormous difference I know that everybody here is here because you want to be here and you want to learn physics and not because you have to be here so my policy is to either not grade the course at all or if somebody needs a grade in order for some particular purpose to give it d-minus low lowest possible grade lowest possible passing behavior all right so it is it's I didn't tell you what it's for yet I gave you that's right I gave you an example of a ha way you can use it to implement the idea of updating a vector from one from one instant tone to another it's one example and but I haven't told you yet why we're doing this I often spend I often spend an hour talking about qualitative aspects of physics in this case it was how do you abstractly think about deterministic physics abstractly in terms of bits and so forth and then spend some time doing some mathematics which really I want to tell you what it's for until the next time but I want to make sure since I'm going to start doing some quantum mechanics the next time I want to make sure that everybody will recognize the little algebraic little bit of manipulations that we'll do and have have the the mathematics for the next time so it's really for the next time that that I set this up yeah I think I think you will see I think it will be clear I think it will be clear uh right I think it will be clear yes I do promise to tell you why ya know

Channel: Stanford

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Keywords: Science, physics, math, theory, relativity, equation, formula

Id: 0Eeuqh9QfNI

Length: 95min 35sec (5735 seconds)

Published: Wed Apr 23 2008

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