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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
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
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
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
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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