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Quantum Entanglements, Part 1 | Lecture 1

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program is brought to you by Stanford University please visit us Ed Stamford died EDU I almost always begin with the same
sermon about especially when teaching about quantum mechanics a relativity sermons always it's the fact that we as animals have inherited rule the process of evolution certain intuitive ways I'm thinking about the physical world and if you don't believe that you think that maybe on ordinary animals are not physicists you watch a lie in chasing an antelope and you'd notice that that wine in the minute the antelope that the relative velocity between the antelope lie in changes sign the lion just stopped dead somehow he did some calculation you choose to Ishida line did some calculations some physics calculations involving some very complicated concepts are velocity direction all kinds of complicated complications like that our 80 a primitive a Romanian man for many Merriman the fall who comes to achieve and sees it became was blocked by a boulder and tries to push the we can push the Boulder decides aim was body that were a flight so gets a bigger component of 4 so in that direction has yet a herd of forces component way get a set of components that not about signs signs yes somehow he did know about signs of course is that things which were inherited our biological origins and they are the basis but Our in the wish about 6 are 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 our from the ordinary world has to do with them with the ranges of parameters which away a outside the range of parameters that humans or animals ever experienced for example it's not surprising that human beings didn't know how to to deal with velocities approaching the speed of light that they got the wrong ideas about how they had velocities when nobody in 1900 had ever probably had never probably never moved faster than 50 or 60 or 100 miles an hour well there that they probably did little falling off cliff but they didn't live to talk about that maybe got 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 intuition that their way of thinking about getting velocities himself for the theory of relativity was and how you synchronize clocks all that stuff like good stuff that Einstein's theory that it was outside the framework of their ability to think about through intuitive pictures so intuitive mathematics column they had to invent new mathematics and mathematics was abstract leading to say you couldn't visualize it four-dimensional space-time You can I can be yours for the region's I've learned trickster visualize this is to some extent rewire themselves for people who learned physics door process of rewiring themselves to some extent to develop intuitions to be able to deal with these new ranges of parameters but still foreign their Italian peculiar even to make quantum mechanics deals with the range of phenomena which is also outside the experience of ordinary humans for which evolution simply didn't provide the means the visualized evolution did not provide the means the visualized an electronic visualize the motion of an electron visualize the uncertainty principle when you think of a particle moving was a part a particle is a thing with the position at every incident kind has a position if it every incident China has a position and has a trajectory has a trajectory you can calculate the velocity along a 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 wears a come from it comes from thinking about rock throwing rocks shooting arrows all kinds of things that human beings normally do so we never developed the need it wouldn't very bizarre if our brains have been wired to understand the uncertainty principle why would the darling have given us a pot of incidentally if you prefer to think of a movie intelligent designer go right ahead I prefer I prefer labor Darwin but why would either of Darwin's ideas or be 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 answers it and so quantum mechanics for that reason appears extremely weird to us physicists Aziz said rewire themselves and developed ways of thinking about it which are intuitive our but still quantum mechanics is much much more on intuitive incidentally the special theory of relativity what went trying to do here is exposed some of the weirdness of quantum mechanics the weirdness Of the logic of quantum mechanics the weirdness all quantum information works this is not a class for conventional class in quantum mechanics at the mental class in quantum mechanics which stressed such things of the Schroedinger equation wave of how particle sometimes behave like waves in store for us us we may may not get to a bit of that but that's
not be important subject that would concentrate on both the concentrate on the basic logic of quantum mechanics the basic logic of quantum information theory visitors information when you say something about a physical system is saying something some 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 where are you you let you usually get it in the form of real numbers the position of Australia set a real numbers and quantum mechanics sometimes you use real numbers but very very often and you give discrete information discrete information such as yes or not or up or down or male or female that's probably not such a good example the differences before he proverb I think I withdraw that had the of tales as a tale according to darken calling for corn head head tail a foot the Corning that's a piece of information that could be tale a tool valued sir EDS and opera down details or sometimes that other logically all the same course that logically all the same whether we're talking about heads of tales opera down or whatever it is a logically the same and simply decisions which have 2 possible questions which have 2 possible answers and a bit of information which has 2 possible answers His cold he bit score bit and it can either be up classical quantum day all real bits in nature are quantum bits obviously since nature is made up of quantum mechanics but some kind of quantum aspects of it don't manifest themselves in ordinary computer be quantum aspects Of they don't really manifest themselves from reasons that will come through our which just called a classical bit classical bit of information as head of the coin flip Berryessa out the quantum it is also is quantum analog of the flip Corning BSO no type question but it is Mike much more subtle and the 1st thing that I want to explore is what it 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 0 hour 1 is we could also use 1 minus 1 or we could use 5 and 15 doesn't matter but 0 1 1 is a convenient location for the tow possible values 0 could step ahead 1 could stand details himself for or sold with thinking on the thinking about this with thinking about some physical system was thinking about information with thinking about a physical systems such as a Corning and this is the information contained in Poynor 0 or 0 1 5 is a notation this seems like a ridiculous and redundant notation importance will only become clear when we start to think about quantum bits but we're going to use the Dirac notation the Dirac notation describes the bones other not whether it's California are gone but the configuration of the bay and it's usually labeled with the notation 0 1 all or whatever other information with other way you decide to think about it these are the 2 states they can have a 0 0 1 and its representatives the foot of a jury that the well which draw again 0 or more these are 2 states that all of this extra John access you don't made it it doesn't tell you anything just as it to putting it inside the bracket incidentally this Our . be bracketed object of year the thing that contains the information only inside is called it kept its quota kept because it's the 2nd half of something which we later learned is a brawl are kept or bracket is not hair but we haven't exposed here now amount multi bits suppose you have more than 1 billion with talking of classical physics so far away not talking about anything quite mechanical supposedly have several calling around line them up I label them so we know which 1 is which in fact factors that cannot confuse Corning's let's make sure the different points Penny nickel dime quarter half power so Bala appraisal have a bunch of coins we can confuse them and we can lay out some information by saying they had Taylor Taylor head said sales that would be some information about a collection of dates and how would you label that will you label that with a string of zeros and ones so for example a fleet of what's 0 always stand here it's easy to remember 0 spans the hated and 1 spans the pale for obvious reasons of right some restraint of Collings heads head a tale headed they'll head I would label 0 ahead 0 for another ahead 1 0 1 1 for example that's that's a configuration a 1 2 3 4 5 6 Corning let's say were
6 . that's a configuration of a multi bid but in this case 6 again for reasons that an absolutely nothing of this description a stick it inside kept figure inside a cat which is just the kind of notation but might be a good idea put some commas between knees but maybe not maybe just leave it that way that's a specification could be the specification Oliver of the bits of information inside a computer it could be just a series of heads of tales assault but before we do anything else would let s go very simple question how many possible configurations how many possible states are aware of what's with 1 bit it there's only 1 bit then there were only 2 states whatever right to bits well you can have up up up their own barren rock found them for 3 times to show bits we have to squared for a way out or however have a 100 it's the answer is no but a total of 100 power through time and the times 200 times so have a bit system the number of possible classical figured configurations is just totally In let's right down with so much for the notation let's write the number Our state the number of states ends up at the number of states of a system of little in the Bates is to area let's suppose we all invert that 1st of all but ruled the what the the number of bits begin is a number of states Little is the number of bits of the number of bits is for then the number of states too the the 4 which is 16 himself or we can convert this awaken right if we know the number of states ever system then we can take a lot further of this equation what the based who would is particularly convenient that we take a lot to the basic tool of the number of states that's equal to the number of steps you can't generalize there's not every system has as number of states toad poor power supposing they have a state they system on radio a dicey known for use in loss they they used to throw were you money with bomb got 6 possibilities 1 through 6 that is not to do that so any particular power it's just 6 Bolick still generalized this definition of the number of bits of information in fact the number of bits other information but a system can contain is by definition the logarithm deliveries to 0 hour the number of states which for the diary would be are the stew of sex was based tool of 6 is imminent danger now with some still be a rational number at voters have big about the took to the toes bought 2 of 3 is a total 2 . 5 3 7 9 8 6 14 so be amount that information which is all the ways of 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 how our of some number of bits each of which has 2 states sofa simplicity would be talking about systems the Member states is always to lure power that's just the the simplicity is nothing special but almost every system could be represent bid that way your approximately who represented that way but we give you an example of supposing we have some question of physics which has as its answer a real number but were only interested in their real number to a certain approximation the temperature the temperature in the room and I'm interested in the temperature in the room so far after while 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 based to have what's the temperature in this room incidentally calm it's about 300 degrees From absolute 0 3 100 I can write 300 not has 300 which is worse 300 means 300 yards 3 times tent show but nothing constant of a 1 . 0 per cent at 0 but we radical things and they still have North 300 looks like and they
still somebody can figure it out based so you have given no 1 arithmetic and base to be may not arithmetic and still focus of minnows arithmetic and may still where I have any number that will light as a series of zeros and ones 1 0 0 1 0 0 1 0 0 0 1 at some numbers food on our Our some particularly interjected integer so if I'm interested in the temperature and I'm not interested in being a tool she have followed the the fine fractions I wonder whether it's 72 degrees of 73 degrees I don't care but 72 2 . 4 0 6 9 I can write it as an integer and that integer can be represented as a sum a bit last summer but as a collection of bits every number of every single number if you're willing to truncate the number of decimal places approximate that number and I'm interested that number only 35 does more places or whatever what a 35 places the based tool that number simply is it both is represented by and represents a collection of so I think the and incidentally if you want to have it a fine grained description of the temperature and then integers in grade you just you lose we find notion of degree you go down as how many degrees is it but not centigrade units but in units of 10 the minus a 100 agreed again you could give it as an integer and our integers can always be represented as sequences of zeroes in war so almost any information and physics can be represented in terms of beer in particular the measurements of quantities such as temperature for example we give another example is a more complicated example of the same thing supposedly I'm interested in a field of field means a thing which can vary throughout based which can double the temperature could vary throughout space the temperature is a field very store up space it's not 1 of the more interesting fields from the point of view of fundamental fundamental particle physics are anything but that's certainly is a fueled varies from place to place and how could we represent that can we represent back in terms of yes if for willing to tolerate certain approximations always willing to tolerate some degree of approximation of what we do is
we break up the the room and little tiny cells but I won't try three-dimensional room and my notes that drew a three-dimensional room about a half an hour just a two-dimensional room and he is a window we 1st of all order the cells make sells small enough so that the temperature doesn't vary much because from so we might fill this room with several billions woes label ourselves as the 1st to want to 3 4 or 5 6 dropped a thousand thousand 1 thousand 2 thousand 3 thousand 4 allocable label all of the cells and list at once with listed them we can write the temperature on the 1st there's the temperature some caboodle comic in justice distinguish between cells than we can write the temperature in the next cell 0 0 1 1 2 3 4 5 6 7 8 9 kept my best places in a new in the bases in arithmetic in base 2 1 1 0 0 1 however from that I got the next step for the same thing temperature there was 1 1 0 1 0 0 so what eventually all I have is a list of zeros in 1 this law on list of zeros and ones if somebody not how you lose it is equal and the knowing the temperature at every point in the role of the same is true of the electric field the magnetic field anything which varies from place to place sold almost everything that I can think of in physics can be represented in terms of picks so if you know everything about how bits work you basically know everything about how physics works of course you may not know what the rules are from the from manipulating these things but this is the basic set up of physics information in the form of a series of questions each of which can be answered yes or no now you call you may wanna refined your description refined description you may not want to add more decibel places the temperature of the specification of temperature gave 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 toe always increasing approximation by a series of accept only get to use computers to the physics if his want true we could use a computer we could use a digital computer in any case to the physics refuse analog computers or something OK so that's a really give you another example another example of how you might you lose bits to represent another these also for classical systems are I don't redraw the lattice but I do want to get rid of role here since sincerity mutilated that is allowed us Our and what I'm interested in is the motion of particles this lattices is just an artificial imposed lattice imposed on the room he just but divided the room into the mathematical sells what I'm interested in is the motion of particles moving around in his room at any given a instead I can ask the question let's take a very simple case let's take a case where a particles where you can squeeze more than 1 particle into 1 of these cells we can match and that the cells are about figures particles in which case you can squeeze in more than 1 in every cell it has a particle or it doesn't have a particle little label the cells that have articles were X weakened label cells don't have a particle nothing or better yet Wickham label the cells that have a particle with a 1 and the ones that no particle with Zero Hour in that case this becomes a specifications we're the particles or are in the lattice along with temperature but the Wong sequence of zeros and ones now the number of zeros and ones would just be equal to the number of a number of cells in a lattice what does not mean it would mean that the 1st sale was a apart in the 2nd not particle and their itself was not part what sells particles in the 5th small-particle 6 private and so forth and so on and so given such a string of numbers you are given a year specification of where the particles are this room bombed in that way again in motion of particles fields temperature just about anything in physics can be represented in terms of request right a bit it a bit is
by definition a question about a a system which has only 2 possible answers which you could always take Tobias not Mr. big gained 20 questions to the wearer of somebody would think a category over and then you would stand there and Astra yes no questions and to you try to figure out what to category repair or to category so that was using the idea that yes questions or I just I just arbitrarily said supposing we're interested in the temperature was certain degree of accuracy right so are registered the temperature factors two-year her from now on begin at temperatures and is doing another example is just examples intended to show you something which is really little more less clear otherwise we could not use computers to were to simulate physical problems classical Ford was it yes wracked by duty of
a bird through the general real number you need in infinite number of dates any rational number can be represented by a finite number of bits and the role well that's not quite true you have that you have to remember to repeat them but rational that Yao rational and repeat at some point so but the city national number than you need an infinite string of its but in general we will allow infinite strings of our although not in a genuine computer but a cell so far
remember we doing classical physics foreign no quantum mechanics I will call research see where yes come to that there very shortly he let me tell you how very quickly the electron 1st 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 so by the presence of an electron at a given instant of time let's suppose the nucleus is not going to be right over here when I can ask about the nucleus-nucleus just sits there so long on her I'm so we did say at least No. 1 when we begin the experiment electrons over here In that case write down a string of zeros with 1 someplace pure 0 1 electron pure zeros except for 1 placed in a sequence where now wanted to describe the motion of electrons we would say Starting this configuration when he moved use this symbol he indicate that at the next thing we we could would we be broken up space and a lot of little individual cells because a break-up time watch we could also break up our watch into with digital watch which showed the richer Her digitizers time just began as either a convenience store approximation and we could say if at Digital current No. 1 in the electron war policy or system forced described by 1 electron located at this location then what happens next next most of them no configuration In this case it might move 1 place 1 reform 5 of 6 Somoza over 6th place want to think and so forth so the motion of a system is described by our rule of updating of updating information how you updated from 1 instance of the next metaphysics basically consists of 2 0 physical system consists of 2 things it consists of a collection of possible states which can be labeled by a collection of debts end it consists arm and kind evolution which isn't updating which tells you how to take 1 collection of bits and replace it by another collection of bits and slightly later instant of Prime some of that and still actually worked out an orbital motion orbiting around he gets confusing because when you jump from 1 lay at the next if this is 1 now and this is a hundred the 101 is over here so you don't jump from a hundred or 100 1 you might jump from a hundred over here which would be 200 sell it can be complicated the updating procedure to look complicated but nevertheless it's an updating procedure that just updates you're your state of knowledge our at each instant time at classical physics there are some roles however come to them but before we go let's define the space of states visitors and I want to emphasize we are still doing classical physics there is nothing quantum mechanical even tho we're talking about disparate pricing system and making out of their systems of individual bits so far we are dealing with what should be called classical CBX I think they're called as opposed to Cuba Cuba does a quantum bid Miss these classical bit so far OK so let's take all the configurations and just abstractly him a purely abstract way we take all of the configurations incidentally what about can by 10 is roughly 10 by 10 ladders you can buy can lattice as a hundred sites finally states doesn't have a three-year I'm not talking about 1 particle book about any number of particles can be on the slightest under a different configurations off their total delivery of very very very big numbers Due to the 100 hour that's how many different ways we can arrange zeros and ones on the slightest or specify whether the particles In the end various position during a large number of possible states but let's just abstractly think about all these states and just throw those points if there are tender hundreds perhaps joint can't 100 points which are not about to do these other various state is not the lattice points these arm the various states for example for 1 bit if I had only 1 bit In the space of states would consist of only took point us up and down and I would destroy 2 points this would be the a space estates of a simple 1 big sister now wants S R 1 other possible laws of updating words 1 the laws of motion laws of motion of the laws for updating configuration 1 of the possible loss of updating well here's 1 possible loss of updating this could the heads this could stand tales slabs Oates think about it from recordings a moment as Costanza heads this could stand Pateros if we sell heads Fatah corn we would do it 2 tales kids sales I want possibility is very simple if start with heads of states nothing happens if you start with tales they nothing happened that's a lot of updating its not a very interesting lover How would you draw that well here I will draw our heads goes ahead McCann arrows which start with heads of state and we start with tales it stays tales so we draw an arrow from what you start with to work you and with what's another possible loss of update and there's a lot of trading scales context becomes stale become can becomes sales of the little would not really much of a little more arresting slightly more interesting system to flip back how would we draw that who drove out again heads tails and pales if you start with head You Got Talent and historic tells you go ahead so the law of updating in this case is just described by such a diagram basically a diagram which tells you if you start a given state what will be in the next instant Pires a clear this is 1 way of describing the laws of physics write down all the states keep in mind what they stand for of course remember that this case once since heads once their tails or whatever happens to stand for the male-female this could be an interesting case of that would be an interesting is this would be a very interesting their law motion In that case the guy was don't forget 1 exploded if this for my under graduate class I would never brought that up this is more likely if this is the more likely loss of updating Ferrer sexuality female male so you see simple lawyers sometimes applied sometimes a more complicated for you think of an interesting system that flip flops like this offhand I can't think of everything it's obvious that applies to a lot of things that never
ever right right right right because of the yes but if you just that you're not intervening but would you think that this is the system by itself if he'll have some peculiar lights which by itself or for backup quarterback for the back and forth in a regular way back if that's slowed down with what they have none of this is that it's With where a lot of states that pendulum and between yes you got the idea it's as hard to think of a simple example about that will go home we came back six-week every 1 of us would have an example of up we should call was the flip flop this is a flip-flop motion this is beyond on the on emotion well we can extend this if we know what the space of configurations it is and we lay them all out either abstractly reminder actually just write them on the blackboard Then the motion of the system could be represented by a series of arrows where I and getting tired but there and so forth so Warren E. yes it Armed wet still Let's go the possible let's think of some possible emotions a 2 bit system but do bid system simply has 4 states that's been our has 4 states while 1 possible motion if we start with this configuration removed that configuration for star with this configuration removed that configuration himself for us if we watch actually happened with time the system would move from 1 configuration the next around the closed but a close look is not necessarily a close look in space it's a closed loop in these logical space of possibilities geological space over of configuration that's 1 possible thing he is another wanted perfume regret what this is is is a pair of course it's a pair of such which are separately undergoing flip-flopped beach or undergoing flip-flops this 1 was flipping and flopping this was simultaneously flipping of flopping if we start here but see what that stands for that stands for example for both head could both heads then we go kept were both voted for 4 wickets followed 1 had tale of it that's what this this is a pair of systems flipping in flopping there are other possibilities so that different laws emotion but the system whatever happens to be could have so you specify system you not only have to to specify what the states of the system are but you'll have to specify how it moves and how it move is a rule for jumping From 1 configuration of the next nominee give you an example of a logically perfectly sensible ruled by which is defective from the physics port of your never happens in physics E begin I think I would go with 4 with would 4 states with 4 states but she her yes theories of you start yet he go here if you go here start here you are here such too stewed your moment for some odd reason and my notes are drawn considerable dying in shape by drama square that may go back as my 4 states start idea here here you start here New Grolier juror you already have a majority here you over here Tyson I can't say what happens wherever you start you start over here and you don't hear job he a joke hear a job that he had a job here you go here you go here you will hear Soviet loci never come back here With this particular law is something different about this lot than there is about the other examples and all the other examples but but can anybody spot what's wrong with this will not what's wrong with it but with different about it but doesn't consist of this is true 1 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 came from here or the you came from here if you're your lost a piece of information this is a motion which loses information they loses information in the sense that you can't tell where you came from is no way to reconstruct the past but you can reconstruct the future while constructive future where do you want your told they go next but wherever you are you don't know how get back a over here Well you know you came from here but then you don't know whether to go back year of Slovakia so at this it is what is called in a reversible history it's a history or a year-old law which loses information and at a fundamental level of physics of fundamental level where you're really keeping track of everything not really euro not where you're coarse grain up looking carefully but we have carefully looking at every degree of freedom of system classical physics and never allows the loss of information right there is a unique future point wherever you want and there's is a unique past point wherever you are that is 1 of the laws is not necessarily a law logic it is something which is true of all physical that they are reversible and that's young but said change they
twice Umuahia 1 1 to offer came from your right we could get this property and mean hers we give it the uniqueness of the future point the uniqueness of the past 4 Inc could invent name for we could call you'll need parity do you know what the quantum version of unique parity is has remained it's called unit parity unique carries a name I just made up you how is quantum equivalence which tells you that you can always reconstruct the past from the future are state of a quantum system you can eat run forward uniquely all run backward uniquely and you come to some unique previous state for or future state and that's calls you parity inquiry can actually have quite text nothing quite mechanical yet but it's a prime time at the current time reversal symmetry your as it's actually Miller time reversal OK so it's not time reversal symmetry exactly Our it's a tiny reversibility I would say this diagram has a sense of orientation are if I start something over here it goes around this way it definitely does not go around this way on lesser reversed I look at it backward in time so it's it's not precisely what you would call kind reversal symmetry time reversal symmetry means that you could either go in going into the future you could go either way but this case you wanna go 1 way but said it it's the reversibility of the laws that that you can find the reversal walk given a lawyer you confined the reverse law which will pick you In the backward direction and I think that's the right way to
last coming away from points is I don't know which way to go out of my way the out of the rat so it's a it's it's clear that that's not all law motion your branching ratios well what's it would dollars the model don't mechanics or were at a classical mechanics really gets more complicated with a branching his ship was a classical
mechanics is less fundamental on old we'll systems a quantum mechanical the question why some of them to press the quantum mechanics and you don't see it is a question which would try to answer as we go along but we might ask it at the quantum level 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 loss that that basically says forward time and backward from all over the world Our Ross equivalent to each other but that is no preferred really preferred sense in which forward in time is different and backward and you know it feels like periods with this
this is this is the question that took 20 years in an hour of the answer is in my opinion no it is not possible but it was 1 of the great questions of physics for a long time to to answer and I think it would now but it might be interesting thing for us to explore for the in after we've talked about Kwan mechanics 3 we have to acquire mechanics before that makes sense that question I think Dole
stands in the sense the Tunney and to me he simply means that there's a definite state for a system that we've all the time according to a definite loss Our up determine stick determine a stick that's that that's the right word but it really think more complicated situations you just could start drawing some diagrams itself and see what makes sense what what's reversible what's not reversible error but yes you're right stroke play loses the information as to where you came from persistently back 1st that yet we're not part of the system a system started here like a hero and they'll they'll in I your find that point ever again right right for the main point is you've lost a distinction between the 2 possible starting point wasn't all the other situations if you know where you are and you know how many steps you made you could say where you work well I think I think for a long long time Mr. Stephen Hawking but the the way black gold were not so clear so clear OK the yes but good so it's lots of that harmed if I take are a bunch of molecules in bath around work work good examples of the molecules in this room and I myself stop an all-out them a certain configuration very definite configuration for them all up and left hand corner of room there an 11 girls after a while the room will be full of air circuit but wound up in that corner of room over there and go after a while saying they Primack corner after warm and things or looks like we lost information but in fact that's not true we followed every single molecule followed it In a detailed with infinite precision which we don't do of course cause then then we could reconstruct a running everything back would weaken reconstruct the fact that the molecule may have come from the corner of the it April Hibbard of lead impossible to do practice but principle following every single detail remark that what really happens in the real world is we lose information because we lose the ability followed the details not be costly information gets lost but because we lose these because we lose the ability of fire information that's where the 2nd war can't when you start losing the ability to distinguish different states so we don't distinguish whether they're not coarse-grained picture we don't distinguish the different detail of the level of a molecular detail so it looks like different configurations become the same configuration but that's only because we simply don't look carefully enough because what lazy union infinite number of bits I mean in these in the real world where there's no because of quantum mechanics because of quantum mechanics now but if it were not for quantum mechanics yes you would need an infinite number of bits that wasn't me that means that you have to specify a bunch of real numbers precisely with infinite with with tremendous precision after precisely prescribed the locations and also the velocities but particularly the locations of every single molecule with a tremendous amount the precision and the longer that you want to track the system the more precision that you need so ultimately to track system for a long time you need to specify with infinite precision the exact positions of every point a molecule that means you have to give a set of real numbers a set of real numbers involves as you say an infinite number of it so the answer is forever a collection of real particles moving around that you really try the Our classically depending on how long you want a follower you would need more and more bits describe all yes yes yes that's right if the room really sealed let's let's that idealized room so that nothing can get into or out of the room all particles bounced off reflect off the walls of the rule so it's entirely sealed up then the moon can be described discreetly because of what can at least up to some energy if we know that the energy is arbitrarily high then we can describe it by trade a discreet collection of variables that has no exit so let's say yup so we could serve to make such a thing we did just reversal the Arabs here is an example does not as high as No I what if I want they to itself I have to do this we could do that as age-related had no exist but but Quellette similar machines that but let me a metallurgist interested when he the question what happens when you're here when you're over here you have 2 ways it you could go you don't know which way to go so it's not determinist it doesn't know whether to go this way for this way and Michael half the the time this way have the Times is what you might need some statistical rule 50 % the time of 30 % time goes this way 30 % The Arts Arts and because of the time with random statistics that would be None chairman Mr. Pink so it seems that the real real laws of nature are both determine stick forward in time and backward in time that's the implication of not having ends floating around like that determine a stick either way so that wherever you are you commuter trains forward uniquely of backward you and that is although love classical physics in a nutshell you know I think in a complete costs and classical physics there's nothing that does not fit that pattern at least 2 were arbitrarily high degree of approximation but state Dalia said the both go don't acquire mechanics but before I do harm where do a little bit of mathematics elementary mathematics most of you know it's Romero for way about matrices vectors for armed on that at the moment I'm not going to mathematically defined a vector in sensible math even approximately rigorous way or abstract way is that that there is a sequence of numbers of finite a sequence of
numbers and few could represent a variety of ways but do you 2 ways to represent a sequence of numbers the 1st players to write them 1 after another but his just given name but I want I want quality and my numbers now 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 zeroes in 1 0 someone defined but they're just general numbers so it is planned I watch or a quorum now yeah that core compounds but I want to remove letter for them on it is good for a while 8 2 0 0 8 3 8 4 and they're just put something around them there were to surround them so that we know this would be a four-dimensional vector why four-dimensional because has 4 components forget don't try the visualized vectors now is no value at all president purposes and trying to visualize these as pointing in space or anything like that just lists of numbers less that's 1 way of McCain there's another way that we can list the same set of numbers but the a column May 1 2 0 33 a for same information and I'm not talking about information and the abstract and before saying things sometimes it useful riotous way sometimes it useful to write it that way you'll find out as move along when written in this form the core of a role that there were written in this form core column vector were actually talking about now is not patient meat notations doing certain arithmetical operations involving collections of numbers after when we get the complex numbers to 0 then use complex conjugate notation yet but for the moment let this be real numbers until now is another concept now call the Matrix and Finkel matrix is a following a matrix of things which acts on a vector to give another vector so it's a kind Xinhua put the vector in the machine and up pops a vector according to a particular rule role not sorry before we do that we do that let's imagine a particularly column rector and another a different role vector given role vector has different entries are not the same set of numerical entries but of different performed so what column B 8 b wonder beat 2 0 P 3 before 4 Kirby 6 . 0 1 5 . 9 7 3 . 0 4 and 81 could be 7 . 8 8 2 9 of them could be the same whether might not be the same reason the bees this is some particular rolled back there and some particular column effect the notion of multiplying a row vector by column the notion of multiplying a role that the buyer column but there is a simple the following simple operation you take the 1st entry always the dimensionality of the role actors and the dimensionality column vectors should be saying that means that they should have the same number of entries not necessarily could 5 6 7 Our in which case they would be 5 dimensional vectors spaces 6 dimensional vector spaces Ms. extends to any number of entries in columns and rows of the Rose in the column should have the the same number of entries is the notion of the product of a row vector and column that it's Colby eater product and it's very simply constructed you take the 1st entry of the role and multiplied by the 1st entry of a column you would enter the 2nd entry times a 2nd entry plus a 3rd entry times a 3rd entry plus the 4th entry times before so the product of these through which it just right B they say that truck in a prior years be won 81 grossbeak beat tool in to be 3 3 Cosby for 84 it's a number it's not itself the product of these 2 sectors be a product is none other than it's not a matrix it is just a number of numerical value is just gotten by up Collins of the saga road plans a column injustice for be 181 for speed 2 8 2 Cosby 383 Busby 484 they're clear but as bets that condition yet young if we were talking about ordinary vectors space who would be the but from young yet more abstractly for abstract vector spaces call the inner proper our but yes a news about product three-dimensional ordinary vectors space with these would be the components of the vector OK now has the concept of a matrix matrix as I said its operation but you couldn't do on a vector to give it's not any old operations their particular family of operations characterized matrices matrix is represented by a school where ready of numbers let's call the entries and I saw the 1st place we put and no 1 wants to indicate that it's in the 1st 4 0 in the 1st column then huh and want to then M 1 3 1 4 area for truck always work so what it's in the 2nd row of the 1st in the 2nd row 2nd column 2nd row 3rd column X 1 3 1 M 3 2 M 3 3 and the reform earned him for 1 and 4th 2 and 4 3 and 4 for now the set Our I've chosen for their mentions just arbitrarily 4 is about as many years as bigger doesn't want a handle on the blackboard and it's big enough to be a little
abstract us over the
general enough for us to what's going on 5 that's automaker exists at has made him think of it you could think of it depending on each column as a combat was components are labeled by the 1st entry here within each 1 of these committee thought of as a contract there were the 1st entry labels the column entry will you could think of that as a collection of Rome sectors where it's the 2nd entry which labels component it way they have it both ways the same kind of a collection of column vectors or a collection of raw vectors but altogether if forms and matrix now matrices can multiply vectors the sort of vector over here at 8 1 8 2 0 should lined them up more carefully 1 8 2 8 3 8 4 and when you I don't know what I have done a reasonable job of keeping rows and columns underneath the next whichever but if you like throw some imaginary lines to separate them into rows and columns this matrix acts on this vector to give move vector what is the knows that he is the rule I May the vector Hawaii because each entry is gonna be a fairly complicated expression but it is just another vector of other similar to my column column which I've had to draw wide in order to be able to fit in everything right is a leader with you wanna find the 1st entry Inter this column study into the brought into this world you take the 1st rolled and you multiplied by the column In a product of the 1st row with a column he is that that an 1 1 times anyone plus M 1 2 times in 2 hours and won 3 times and 83 and 1 for former ability to call this a new multiplied by according to you product rule and that gives you the 1st role and 1 1 8 1 pass and 1 2 0 8 2 0 plus M 1 3 8 3 M 1 1 8 4 now you want the 2nd entry into this new vector already done exactly the same way except you go to the 2nd World War and you take the 2nd World and multiply by column that's going to give you an not only do toward arrest you can do yourself and to want and so once again plans 81 plus em 2 times 8 plus and 3 83 forced them to 4 times before and of the other 2 entries you could figure out you get them by multiplying the next world by the columns and finally the 3rd row column that gives U.N New Vector jet it's a way of processing a vector to produce new vector I will give you some examples are as we go along and it's a rule multiplication of which is very useful the reason it's defined is because useful and see how it useful by using it are we do example How will make tricks how we idea of a matrix can represent tying the evolution of the configuration of the system supposing again we have made our arms are configurations let's label what's label limits figures label the 1st configurations 2nd configuration their configuration configuration is not points of spaces bees are configurations system which has 5 distinct state and let's take a very very simple of evolution the first one to start here you go to hear that he had you go to hear start here Rhodia India in European nor go from here go back now plays down want that no good that's that's disallowed I think that disallowed I think that's the sort of the that's the solar our have that this allow would be chorus of you find this year doesn't that's not reversible that's not reversible that's not what I want to hear arriving here is due back here Dick so this is just a year 1 goes 2 0 2 0 2 3 3 4 although 5 5 was back 1 to cycle another way to represent the same thing we can represent the state of the system by a column and the contractor we simply insert 81 someplace if I wanna represent the 1st state over here I put on 1 and then a bunch of zeroes 1 2 3 4 5 1 2 3 4 part of this simply represents the 1st state what about the 2nd state the 2nd states are represented by 0 1 0 0 0 the 3rd state by 0 1 and so forth so the states of the system can be represented by column but a particular kind a column Ocalan with all zeros and ones whereas the 1 namely which ever state Europe focusing on few focusing on the left but the 1 in in the 5th century here now what is this will of evolution the role of evolution says that if you have all 1 someplace than in annexed this and the time the 1 moves down so you start via the 1 here in the next instant in was here next incident was down years next incident was happy hear them wears a Co after top right so there is a procedure that you'd do work on this
calm to tell you where the system goes in the next instant From 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 of which you could think of in this case as up to baby operation the operation which updates the vector so here it is a student of 0 1 0 0 5 dimensional Sony fired 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 I'm so that the Lakers for the rich getting sick of my life plans to many for me right 1 0 0 0 let's try it out let's try out on this vector right over here this represents the 3rd state what happens if we packed with this matrix on the 3rd statements just try it out so we get well the 1st entry up on the top is guarded by taking the pound vector and multiplying column 0 10 0 plus 1 kind 0 0 times 1 0 times 0 was Xerox next placed 0 times 0 0 0 0 1 time White will work swept Scudder give the architect says that at that instead of going down go work but that is OK up now which is journals over How would you prefer but that's just the yeah right but he did 0 0 0 0 with heavily for over a year warned 1 Warren warned them of Kim OK so let's start oriented What's up on the top 0 . 0 0 times 0 0 0 times 1 1 0 we still 0 x 1 1 tie 0 0 10 0 0 0 2 1 0 times Iraq still 0 1 the third-place please please please dropped 0 . 0 0 1 5 0 0 0 0 times 1 0 0 0 0 but nonetheless place I have 0 times 0 0 times 0 1 times 1 0 times 0 also warned columns moved down 1 step you could check for yourself is your homework jet that's a place that you put this 1 it will move down by 1 step gets the bottom and recycling go up the top OK so vessels the check and that you could put in any numbers Our zeros and ones they make sense but we could put in any number I ADB BUC D come out here is everybody will move down staff a B C D but that the EU will move so if you put 1 in any 1 of the places it will slide down 1 unit and your but the point is that the evolution Our systems can be represented by matrices matrices are particular kind made in classical physics in this kind of classical physics there always is when 1 lives here quantum mechanics of more complicated more difficult but the classical physics but sprinkling of zeros and ones so as to who make these each state shifting to the next 1 that's a that's an example of the use of matrices in classical physics so far no quantum mechanics just pure classical physics abound there is there is an interesting well I guess that this is what before a planned being come from back it so that for example of matrix algebra matrices multiplying vectors not matrices multiplying matrices hailed wine might do we want to multiply matrices by matrices well and he has the idea supposedly wanted upgrade or update a 2nd time to take a 2nd time we would do away with Beetle apply the same matrix of the resultant that we got another words what's right is right for Friday abstractly whenever a matrix M which we multiplied by a veteran V you forget a newly elected prime as of abstract notation for writing a matrix of that shirt and getting the new vectors that updating the Victor Vito a move that lets up did again let's go 1 more interval of time we do that well what we do is we write and times prime equals be double prime we will do the the same updating trick except now updates prime instead of the EU we would get a double prime double prime being the state of the system after 2 units of time Bullock could also write that realizing that a prime time XVIII with the right this as end times am times V Z would to be double prime this just means we apply the Matrix quite so we can also think of it as square in nature him and then multiplying it how do you square matrix or had multiplying 1 matrix by another 6 facility what you would do if you would want update twice once with 1 matrix in woods with another make direct or the same matrix have you multiply matrices the answer is basically the same kind of rule I will do it now for to buy 2 matrices
because getting too complicated for but 4 matrices till by 2 matrix we have M 1 1 am 1 2 M 2 1 and 2 2 but for some other make Texas and 1 1 and 1 2 0 0 0 8 2 1 and 2 to the result of multiply in the matrix but the matrix of mother of matrix make bricks I would do it and very similar manner supposedly 1 1 1 entry here we get the 1 1 entry by taking the 1st row and multiplying it by the 1st column M 1 1 times in 1 1 plus M 1 2 times into 1 saying kind of in a proper way put over here now surprisingly want the next entry for the next century we take the 1st world because after all were interest in the 1st row or we take the 1st Rose Bowl multiplied by the 2nd column over here well overseer would be and 1 1 times in 1 tool us and 1 times in 2 to authorize Valley move down to the bottom the bottle if we wanted this century we would take the bottom and multiply it by refers column 1 for less 3 over here we would take a bottom role and multiplied 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 up with you multiplied as matrix by this vector put in over a year I think so the notion of multiplying matrices and multiplying matrices does is it gives you and matrix which updates you will not buy 1 interval of time but updates you buy 2 intervals of time if you wanted to short circuit part of the problem of updating new wiped out of date the state of the system find units of time what you would do is multiplied the Matrix together 5 times that sees you do it and in sequence 1st the 1st time the next result and the next 1 comes a result of the next 1 man Hugh Kenner you can work out what the Matrix which would take you from the state of the system and this the time 1st system find instance later so matrix multiplication multiplying matrices by matrices it is also an important concept 1 last example of matrix algebra involves role vectors supposing you have a role vector and you want to multiplied by matrix the rule is your right from wrong matrix 1st B 1 B to be 3 before and then you write the matrix M 1 1 M or 2 M 1 3 and so full of sorrow him more 4 not that started the back of tired of writing in well what's the result drawing debate resort was gonna be they wrote right and here's a way you get the entries of the role vector the 1st entry of the euro back there you get by taking the original role and multiplying it Private 1st column vector over here that product the 1st entry they did take the original role and multiplied by the 2nd column that gives you a 2nd entry and you take the original Rome and you multiplied by the 3rd column accuse you the 3rd 3 over here and so will you see the pattern it's always multiplying rose by columns and putting them bring a result in the right place at the right wrong column in this case a row vector plans of matrix is another role there there a role on matrix times a column that vector there is another here it is a matrix times a column vector it is another problem matrix Hunter matrix is a modem makers books on get familiar with that work out some examples work out some examples of your own devising just but some numbers and multiply a role vectors and matrices matrices Times column vectors and matrices times makers to use air and get the experience of working out how these things work but by 2 3 by 3 matrices and you get familiar with it because we will use it over and over and over again in fact that's the primary mathematical operations quantum mechanics is multiplying rose and columns times matrices if you know I'll do that and you're familiar with it new career off theaters easily you've got all of the basic mathematics of Kwan can fix it would help the have a little bit of a calculus to go with it but the basic new things matrix multiplication column vectors of rogue vectors so please practice with a little bit that should made up some examples for you do the you commit your very straightforward OK I'll be getting closer 9 o'clock under any questions next time tried to stop talking about cube its quantum bits and quantum bits of very different in classical Barrett's question yes the my name renters Suskind the four-star skinned you a few like polishing Apple for the professors Conley Leonardo I like that much of a new set of what restrictions does reversibility price gap that have in inverse Brent yeah
a button in the month of November In a more abstract sends the answer is that should have been worse of the Matrix should have remembers the inverse of course is finger picture back not all matrices Evan verses Charles Robb annually Bursar's national there are removed will review if you don't will come to it any other questions that that's a good question now final exam is
buying me lunch but it that's a lot of lunches at applicable a every Mark nobody has me about the of a brought view him before you know my policies the policies idea he learned physics there is nobody here who the a year for a dip greedy if you want that I but be glad to give you a commerical grade if you need 1 that if everybody needs in America Grade I know that there's an enormous difference in the level of preparation of different people odd and too compare compare you in area and exam said would make sense because they do know that was the difference I know that everybody here is here because you want a beer you wanna learn physics and not because they have to be soul or my policy but even Greek course of or from early needs agreed in order for some particular purpose to give a D minus but half the most possible I'm so it is it's I didn't tell you what it's for I gave that I gave you an example of how you can use it to implement the idea up dating a vector from 1 from 1 still to another it's for example and but I haven't told you yet why we're doing this are often spend I often spend and now we're talking about a qualitative aspects of physics in this case it was how deal abstractly think about determined stick physics abstractly in terms of bits and so forth and then spend some time doing some money mathematics which really I won't tell you what's for until the next time but I want to make sure it 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 will build with and have view of the mathematics for the next time so freely for the next time they said the shop now I think I think you will see I think it will be clear and it will be clear our I think it will be clear yes I don't from are you why not Ad hoc
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Metadaten

Formale Metadaten

Titel Quantum Entanglements, Part 1 | Lecture 1
Serientitel Course | Quantum Entanglements: Part 1
Teil 1
Anzahl der Teile 9
Autor Susskind, Leonard
Lizenz CC-Namensnennung 3.0 Deutschland:
Sie dürfen das Werk bzw. den Inhalt zu jedem legalen Zweck nutzen, verändern und in unveränderter oder veränderter Form vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen.
DOI 10.5446/15092
Herausgeber Stanford University
Erscheinungsjahr 2008
Sprache Englisch
Produktionsjahr 2006

Inhaltliche Metadaten

Fachgebiet Physik
Abstract Lecture 1 of Leonard Susskind's course concentrating on Quantum Entanglements (Part 1, Fall 2006). Recorded September 25, 2006 at Stanford University. This Stanford Continuing Studies course is the first of a three-quarter sequence of classes exploring the "quantum entanglements" in modern theoretical physics. Leonard Susskind is the Felix Bloch Professor of Physics at Stanford University.

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