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Quantum Entanglements, Part 1  Lecture 5
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different assigned to make her particular the sign that occurs we studied and tangled state of collector arms we call up down and then at possibilities Miles Morimoto possibilities but they would lose interesting situations with a plus or minus purity take a choice we study Blossomwood study miners plusminus bound up so this is still electrons In the state of arriving there was a linear superposition states 1 electron 1 is up for sworn to electron what is up for Toronto was down that like like best and the other component of the state just flip from all over and we can superpowers them with a plus sign or a minus sign sense of different and either case there's a good deal of correlation between 2 Spain's if 1 of them is up and the other 1 is surely down 4 members down the hill is surely up in both cases the probability incidentally nothing right now is that depend on whether we we should normalize it correctly that but normalization squares too tired to make be probabilities and the 1 In both cases these sums wealth more lawyers in both cases In both cases the bends Our opposite to each other or the 3 the 3rd component of spinners opposite to rich ever so what is the big difference between these states in a quite different quite different and Aruba of time explaining that before I do let me write a list begins a list actions of the various Sigma operators to send this thereof same table a road less time 1st of all signal 1 of our up equals down will make reference to this regularly 1 on down equals up with will honor was similar to warm up February of apartheid times down stigma tool on their own as minors I went up and stigma 3 armed bobbed up and with 3 on their own that that's the actions of the matrices born to submit on the Eigen states of Sigma 3 itself so will will make reference to that list as we now in both of these states the expectation value of In component of the Spain let's say sexual remember now that Segner refers to the 1st electron power which is an identical family of operators identical except acts it acts on 2nd spearmint the 1st stigma and Powell on to spin operators once in the referee men either of these states the expectation value did component of stigma or cow 0 so there is no direction In which Our either the 1st or the 2nd spin is definitely polarized it was not their actions our if you compare it you will know that the Spain has to become out in that direction that is of course different than any state that you would make for a single electronic and so this is some sort tangled state that is not a simple are juxtaposition spin in the state plans another spend own state mixed together in some some entangled away OK so 1st of all every component in his state of the 0 and how y 0 in this particular state check battle epic which I last time we think we that my guest also true but plus 91 of Dewitt there we can do it let's just as it's it's quite obvious forcing the signatory is 1 up electron 1 down electronic notes but but reverts to fivestory have to take a brought down let's just right up down plus through which when the user miners were sort of course of plus down for brighter around that squiggle around it and now let's take them woods 1st takes Sigma 3 and then OATS pop down plus down we got a right Brackettville awake that is Querida not here is 0 hour will be 0 4 multiplied by square who are whatever OK so on what happens if we multiplied the Sigma 3 saying 3 hits up and leaves it up saying the 3 kids down and changes the sign right so this is just gonna be um up down my newest down up actually episode 3 now because of his back it would certainly 3 the times of downed plus down out OK so the inner product of up their own there was 1 of the product a downward up 0 different opposite the store right here we get all 1 from India and an a minus 1 from he bound up with them again 1 but as a might assign so Saugus 0 now this does not mean that these states are eigenvectors of Sigma 3 with eigenvalue 0 means is the expectation value for certain 3 0 hour was no eigenvectors of Sigma 3 what I value 0 you Connecticut 0 form the spear you could only a plus or minus 1 year so it doesn't mean that when you measure you get 0 what it means is better too measuring spinners affair . co corn toss where you get the Alps as much as you get down in equal amount equal probability OK now I work on the different and the difference you concede most easily by considering an operator which is the sum of both lowered the Spain's represent vectors will look pointers excuse me signal Mike that direction power might point that direction they represent little magnetic moments we can't think about the magnetic moment of the combined system which is simply the vector some of the tool to magnetic bomb Our and so it's interesting and tools will look at the operator Sigma warning signal How to amend the 3 components of it all 3 components of it I let's see what the be let's see what signal plus due to these 2 states 1st of all what the action of Sigma 3 plus how 3 arm down plus founder of them sorry plusminus down but the answer is 0 why is that of personal 1 reason is obvious I I end up this spin the 3rd component of 3 years up in down these component of tells freeze is down so it up 0 but let's just jacket our signature acting on our dues plus pop Tower 3 acting on down his mind so 3 must have really acting on their own will give 0 signatory acting on up plus out 3 acting on their own use mind this endeavor sourcing with recourse 3 definitely equals 0 the left state and also for the
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right 1 over here so the conclusion is that either of these states factors told separate components here are both eigenvectors of stigma 3 3 Eigen values 0 that means measure signatory passed out 3 will always get 0 either of these states that's not surprising since war was after more was that pay now or about signal 1 was how 1 works so that the exercise signal 1 plus how 1 this is some of these 1st component Herbie X component of spend but see if we can work out what it does to us down plusminus down OK let's start with City 1 what the stigma more to top it turns it down right similar wonder what's what's right out everything that come here signal 1 on up down gives us the film value and signal 1 and how 1 enough down gives my not sorry Taiwan on damages gives up stars tell warned up up up so we get by applying similar warned plus Dow warned about down singing 1 flips up that makes them Taiwan's flips down and makes up about with plus sign because with plus not fast plus sizes assigned here then we have plus or minus the earlier plus or minus let's see what 1 does Soon water tanks on down to give up up and then plus Martinez yeah metal tell what flips else and use you down plus minus the same plusminus that's here French so quite clear that if we take a B plus sign we don't give 0 Whitacre plus I we don't get 0 we just get twice down round twice prop up implication is if you would measure well the implication is that this is not and eigenvectors of the sum of signatory tells you will not get 0 every time you measure it even know its expectation value happens to be 0 God if you measure it sometimes you get toward the possible values you could get Cigna 3 Canoe plus or minus 1 cow 3 commute plus a minus 1 sort quickly answered 2 0 my tool 0 0 minus so that's a but there would be no reason and in fact it's quite clear that you can't have it you don't get 0 every single time because this is not In eigenvectors I loved Tower 1 plus signal 1 with Eigen value 0 so we could be quite sure then this operated here had not give 0 every time you measure just because it is not a bargain in this state just because unlike the 0 what about for the minus sign the minus a minor Saarinen here elusive but might have to lose and yes it does down down my used them above murders but equal 0 so the state of the minus sign here concentrate on that 1 narrow state with a minus 1 here every single time you measure signal 1 plus tell 1 year 0 every single time you measure signatory 3 plus how 3 months to separate electrons 1 of them you measure signal the other 1 you measure towel U.S. had the answers no 1 plus Taiwan or submit to post out toward you measure you'll always gives 0 for some of the 3rd component and for the sums of the 1st components were about the 2nd component let's check that won an award of plus or minus now with already checked that signal warrantless how 1 does not kill down plus downdraft let's see what happens if we put a signal to but on the same on the same of their own miners down I 1st stigma DeRosa similar to multiplies be up here and mix of all icons down so the 1st entry will be i.e. kind down down that's from Sigma 2 times it's now about how to plan this how to give the pigs down input up with my aside minus I up now about how to count to takes down and puts it up with the side as minors found down and then with assigned murmuring assigned here and out to takes up down Matsumoto where I wanted to harm where my theory have to down bounced have 0 which armor trying to do a day earlier pop up that's this 1 Yap and that should be by 8 times up up I turned jargon 2 0 all components of do 0 furthermore if I would take but scored signal that town Little vectors remember city Terrel Nov . and got into some unit vector to take the component in indeed directions while those things are built up out of signal 1 asked how 1 sigma to post out to answered with 3 Forestal 3 so if we basically with show that all 3 components of Sigma Pi power 0 and therefore less stigma plus pound indeed directions whether direction measured and you will get 0 . 4 the negative sign the negative sign is called the singlet state why it's called a singlet state maybe will get too but for the moment it's called single state and it has the property that the son of these little magnetic moments the sum of spins of the 2 particles are exactly equal 0 every time measured in any direction it happens that the property rotationally invariant no matter what direction new measured in it always looks the same but we see that particular specifically from from looking at various components of signal past how they're all 0 12 had to be book a show about courses that if you measure a component of Sigma you'll instantly know what the scene component of how is the rear electoral Rivera Alice's electron over their barbs electron over here and Bob measures component of the space instantly knows what Alice will get if she measures the same component of spin 2 or more necessarily know what the what's gotten by merging some other component of but what he does know is if he measures any component and gets an answer he instantly knows what the there Alice will get when she measures be others White use some of the spends in any direction are equal to 0 so that's a good that in 1 month for porcelain for death a routine but every time we visitors because suspected I did sector of sea plus towel I guess that for that tells you that every time you it In that state you get are you get to say mentioned equal to the ideal value whatever Hussein which they got but not for the younger ones called triplet state why another commercial component of the day as assailants say has a special property that you measure once
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Spain you automatically know the other ones in the opposite direction I mean to say you know you never measure the direction of a spare you measure component of respect any component of the spending measure if you measure the same component of the other spin you'll get out of hand so that's a form of correlation kind of similar to the penny in the dying if verified forgive house 1 Corning Bobby other point now looks Corning change feeling a little Bob but it somehow deeper than that it's not just the 3rd component be up that this but every single component our whichever 1 you have where you happen to pick in any direction Weber to spins will come out the opposite that senior that's a profound kind of correlation that Dutch called Einstein adult Rosen correlation where Einstein the Boesky Rosen the first one discover entanglement not role Her believe shrubbery 1st the 1st discussed it I think I think he called entanglement think that was before but many case let's let's have go wash restoring next want to come the Bells famous here I could tell you the theory famous what I can tell you is whether or not profound but trivial and that those theorem but then Bell's inequality in the factor quantum mechanics violates I cannot tell you have the days of the week I think this is the most profound thing in the world the other half days just purchased another consequence of mechanics but I can tell you is a John Bell had exactly the same ambivalence to it at least 1 final him our anger I can tell also literature finally had the same some of the times it just seems still prefer the so deep enough times it just seems like what trivial consequence of quantum mechanics of course draw you'll have to make up your own mind whether you think it's very profound marked by gets a lot press the most profound things Swiss cheese and sometimes they know they will be In the end it's gonna depend on what the consequences are future physics at the moment the consequences are just corner works its so without further ado let's let's discuss other of itself was quite trio will probably got the excludes Kwan mechanic for exclude pornographic now the piano exclude quantum mechanics and quantum mechanics and confined violations of Bell's inequality Bell's inequality is an absolutely classical theory it's a theory about classical system is completely classical systems and when I do right now I don't want you to remind me anything about corner mechanics it doesn't have to do with choir mechanics derive a theorem about perfectly good classical logic was classical logic classical logic is basically set theory are moved out of what we is it would violate causality if you could use it to send the message if by Bob looking at the state Alice with the instantly know something which he did no before that would violate but she doesn't she doesn't know anything different over Our she has no more information and she had before for example in this state are spend half the time will be up and half the time will be now that'll be Bob May measures they I know anything she'll look ever spend half the Times will be up times will rebound if she gets on message from Bob and message says my up then she knows that her down which is gotta get that message that message can't can't 3rd big got past the speed of light we need made dead ball I love no no whether but whatever Bob does well above the others if Alice doesn't get a message from Bob she doesn't get a message from Bob Dylan no implications for anything for Harp Show spilled she will have learned nothing from the fact that Bobby the measure our to know nothing she has to get a message from him to use the information that he had if he sends a message I got up then she knows it's down about but without that message being sent jitters know anything so this is in no way a violation of causality locality or anything like that that's 1 of the reasons that have the days of the week I think this is so profound that doesn't violate NTV deeper important in the dog in every Tuesday status would you have to be a good idea not plus led to a 0 I have a lot of measures in the last half of any direction any component it's in the triplet state which is 1 with the class where if you measure it you could get a 2 0 0 or 4 minus 1 my excuse OK so that they would go series very simple fare supposedly have some objects all collection of an hour a set of things and we look at the class of the object that have certain properties consider 3 properties were caught a B and C whether they happen to be a because Santa anything any property maybe right down inequality 1st In words each 1 of these little things were systems that and each member of the set has any of the city and has it has the properties right now we can I'm interested in the number of entities which have property AT and not be you might be is whether the office of beers or whether it would have the opposite of the number of things which have property AT and not be plus the number of things yet however property B and Nazi and the classical theorem which is so easy that again I never understood why there long longwinded proves that the basic is greater than or equal to the number of things with property AD and not see a number of things that they not not be plus number things of being in Nazi is greater than equal to the number of things with AT & at April there's a problem by drawing pictures for 1 reason or another in all the literature that I find never draw a picture but let's draw a piece that the leader the on the set let's delineate the set which has property subset which has property is a subset which has property Bermuda with aware however notes or on the girls grow up for years a over here for now wears a subsidy for the subset being may or may not intersect intersect say it means elements in common that doesn't affect don't have elements in common but he is a subset B & B has no elements in common with a it just means you should know it needed just means that this regional here 0 that not that but there's nothing in it here but in general Libby something in here which will be those things which have both a and B finally see against seen may or may not
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have elements in common with a it doesn't it is more than ever the move over a little bits of the present government come the general situation is where it does have elements in common but you can always you lose nothing by pretending it has elements in common cause you could always be in say that the elements in common adjusted that there is not OK now let's let's and that's also label but of course but also labeled all these various individual Little sub regions here Stewart call OK this trees and a half years as 1 was that that's all out a which is neither being North Sea UNESCO label all the various sub areas in here 3 4 5 6 and 7 and of course there may be some elements which have neither a B or C and out here about need replica of the Wright as boundaries on my interest focus let's take the number of elements that have a and not be there has been a key but does not have B or is that that's region 1 and 2 others a but doesn't have baby that means it's aII but exclude anything that has B and that means that the number that's in 1 plus a number that's until February see that OK so the 1st thing he said it In 1 plus that's an 80 not be mellow that the number of gay Iowa the number of systems which has b but not say so it will be an ASME exclude those perhaps C 5 and 6 have C O in looking at the things which have B but not were vast whichever but not state for every right plus and for Parsons 7 now where about it knots about 80 and Nazi AP and not is 1 of former Nazi is 1 of 4 but but but right so it right here is greater than equal now that get a N Nazi that His work and which 1 which 1 and that and 7 Frank forever right I think so yes well obviously this is a N 1 and obviously and 1 and the Anza a positive means no such thing as a negative number of elements and 1 thousand troopers and what wasn't said certainly bigger and 1 0 0 recall to it at worst it's equal it as Bell's theorem that's all thing that's it was amazing is that violated a could be by a rate will differ from of the violation of salt former character by classical yes I know a set of always been demand that really get on the ballot my diving ability they did what they said and that's what they that's what our stopping it from here OK let's now take our system be to spare us assistant despair and then will define the various properties AT & R B which will think of it's another was still going with going where imagining that still wealth gap will 1st retained that spans are classical and that you could measure them as if they were classical objects will write down what Bell's inequalities of the system of toast the only the Nolan to keep track of the talk about the war photograph 1 but what analysts say is estate things like BOA you might get proposition a proposition might be that spin 1 ports see if I give serve revolvers Spain 1 it is up along zaxis along the 3rd axis but could be some proposition well I can we may give is 31 is not of along the 3rd axis but of course that the same thing is now along with their voters spin 1 is measured along the 3rd axis and you can be sure of the tool is up along with their backs so the negative in the singlet state and the which they were putting electrons into the single state and classical logic of ordinary logic made in he said Italy would do we would measure we will all be of these electron pairs zillions of them millions of Paris which are each correlated in fashion separate from each other any measure the 1st 100 thousand of them in every possible direction you measure signal 1 how 1 new resisting 3 in how 3 the from many many of them and you convince yourself after a long tedious bunch of experiments that every time you measure a component this spin and you measure component of the sayings of Compton component of the others that they always opposite to each other in that case the grew all agree that the negative up saying the opposite of saying is up saying the Beers up because of a is down and use up so the negative of a proposition about 1 spared becomes the positive statement about the others the negative our spin 1 B and was caught a wonderful half of the negative of calling this negative of the spin being up suspend Ireland but then is the same as saying was so it is that to keep track of what the author of the relationship between these various propositions OK we're right now what a B and C R 8 nor I don't talk about in 1 at 1st but then will use Pinto the tell us something about so aII is up for longer free access at 80 B is equal sign you just means is the statement is a statement that's been warned his pop along 3rd axis b Spain 1 is Our along a 45 degree angle the 45 degree angle could be in the in Lindsey explained playing for example are expected to be in his explain Sowerby says spin 1 is up along with a 45 degree angle in this year explained seat says that 1 is up longer than 90 degree direction in exwife and the IOC 1 c explained cancel his are pictured says of the spinners up along the 3rd axis beat lies 45 degrees along with our along 44 or actors and she's the wise and 90 degrees along racks a B C OK now about not be stake our 1st ignored a winning lottery for anything we will lead not be be not be says that 1 does not lie on the street is not up along the 45 reactors is value along 45 degree Texas but this is its bid to what is along with 45 degrees items not be it is that Spain true what is up along 45
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degrees not beaten he is that what is up along a 45 degree not fee is a killers up along a 45 degree of triggered the idea saying thing for Nazi is that up along 90 degrees OK so now we can plug in a year what the various statements site and them Nazi with see says that war was Opel on Monday Nazi says 1 down along with them tool is up OK so what I'm a statement is not Of the statement is original statement except near the spin of that Quinn villas in the opposite direction of solar says so whenever I'm not statement is that becomes an yes statement about the others so let's write down what this says this says the number of pairs entities now appears of electron pairs of electrons this becomes and the number of love Perez with Spain up but call it Spain 1 up alone up at 0 degrees up 0 degrees and understand what do we want not be not be the other spinners along with 45 degree angle shots of the 2nd spin the 1st been here writing in the 2nd stand up Hong 45 degrees 8 dessertspoon 1 was along the 3rd axis and not be as it's been 2 0 is up all 45 degrees and got a sense of the fans electron ones electron to paint the 1st entry here is electron want the 2nd entry views or to next not say that I don't have to worry about being white it's got to be exactly the same as they not be wider set on B a not be and being Nazi simply related by rotating by a 45 anything I could say about 80 and not be will be exactly the same as being in Nazi biggest identical to each other except for a rotation 45 degrees and laugh the singlet state doesn't change a few rotated 45 degrees that's the that's the main thing about it doesn't change when you rotate we saw that by saying that the spin fills the same in every direction so in fact off we can just take this to be twice the sum of listeners I just twice the number of up at 0 degrees and down 45 degrees and bells fear a would tell us that this is great event necessarily greater greater than or equal to the number of Paris with the property that up but I course this should be C should be merits to correctly now yes C is up 90 degrees or age 1 0 that along Yafo case right 8 years 1 is up and 0 degrees and Nazi means To was up along 90 degrees is up at 90 degrees well now we have something incidentally counting the number of missing counting the probability these things to be true a large number of identical thing an ensemble measuring the number of is just measuring the profitability for have either thousands of Paris number which has a certain property is just a certain percentage of a 100 thousand which is given percentages given by proverb OK so here's a question and it is single state what is the probability varies probabilities of probability for a after a 1st electron yet 0 degrees in the 2nd electron be up at 45 degrees how we calculate that I want to move is a question tell the answer right now the answer right now is this is less us when you calculate when a calculated but this is less than this simply violates the very very elementary theorem that the dealt hardly appearance is just a very simple observation about the probabilities of arms black let's see if we can calculate these things eliminated Illinois to calculate and efficiently the teacher about projection operators they were projection operators and let me give you an alternative definition of the probability of postulated choir mechanics a projection operator labor like that projection operator P projection operator has a number of properties but 1st and tell you would to give you a picture work where would do little on ordinary projection operator threedimensional space my vector space happened to be just that 3 dimensions then subspaces was subspace of threedimensional space could be 0 dimensional point to stay origin nothing wrong it could be a onedimensional subspace which is the lines sticking out of the origin twodimensional subspace which is a plane passing through the origin on if you like it's the space which is well you get it you do if you have a pair of the twodimensional vector space would be take any pair of actors and think Of all the vectors it you can make bye a adding tool With arbitrary coefficients all those vectors will lie in the plane of those 2 vectors that's the subspace which is Spain's by 2 vectors of I have only 1 that I want that's a onedimensional subspaces all the sectors that lie along line if you have a higher dimensional subspace band higher dimensional space that you could have some faces up to the dimensionality 1 less than a year of space itself I wasn't projection operator the projection operator in particular a projection operator on a onedimensional space in particular its operation which takes any vectors takes its projections on that subspace and give you the vector which is the projection on warned the subspace Our Of the original but that's not really the source who operate on a vector Alpha To give you the projections scored alphasub P E onto the appropriate subspace before subspaces this line here would act on any vector begins the projection onto a subspace that's what that's it takes away ring molds the components of the vector in directions perpendicular the subspace just Thompson amount remote but state about projection onto a twodimensional such a simple twodimensional subspace would just be a vicinity exwife plane just EXY playing here would be a twodimensional subspace I have an arbitrary vector then it just projects on the plane and use you projection or place that's the notion of a projection operator that I suppose we have
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some property they were Intourist survey the property is what's take cover property could be the spill is up along the 3rd axis is eigenvectors goes with the up eigenvectors along the 3rd axis therefore there's a projection operator that goes with that statement statement spinners up along the xaxis corresponds to a projection operator is the projection operator which projects on 2 0 eigenvectors which the spinners up along the 3rd axis so was it do it takes any state the projection operator projects on the spin be opt for 1 Spain vs. talk about once what is that it's an operator which takes sector but court Alpha Beta was it gives its projection onto 0 Sigma 3 equals 1 projection onto sit with 3 8 was war was again at half time for this sets the life of the source of this data His orthogonal mistake with the low entry here is all part of the state in which 2 spinners I think so what does Is it picks out the component of the vector or longer the particular property that you're interested in such as gives you offer 0 it throws away the preselected has to do with spin being now what does what does P3 Eagle minus 3 piece in a freak was minus 1 storms saying that but over well for but here's an alternative formulation 4 set see was was minus 1 excuse me incidentally P is an operator it operates on a vector forgiven Novak there must be a matrix that goes with Mateo Matrix the kid but at the projection operator a signatory equals plus 1 that's the matrix 1 0 0 0 0 what's check that suppose that that somehow from beta what is it gives gives 1 times out front 0 0 times beta 0 found later OK Webster was another later right it which is gonna be very efficient by very useful for writers operator in the form of sigma 3 plus 1 divided by 2 bucks check that signatory it is 1 minus 1 0 0 plus 1 which is 1 1 0 0 1 place the L'Oreal elements here amateurs might as well as 1 and then divide by 2 Sony upper entry he'll want us want to a divided by 2 that's 1 every place else to 0 this is a useful factor that the projection operator of onto a configuration with a 3rd component spinners plus is just Sigma 3 plus 1 divided by a wall of that OK so now it's says that question what would you expect me into the B of ISU what's the projection operator projects on the eigenvectors signal 1 equals plus 1 projection operator for let's just call a projection operator for Sidman ferine his projection operator for signal 1 and by that I mean that single 1 is equal to plus 1 that's just equal the 1 plus 1 or 2 you could check that easy checked to check that's a projection operator so early give you know an alternative definition of the probability bomber posturer part mechanics if you have a state and arbitrary state call sign and you're interested in the probability that a certain thing is for a war where you measure it at a certain thing is you construct the projection operator for that certain things you call a the expectation value of the projection operator is the probability that the probability let's let's just check that prepare for simple case supposing again is just Alpha Beta is just a vector Alpha Beta there P 3 it's in the 3 plus 1 Hillier Alpha Beta we have seen the 3 plus 1 divided by tool and then we have Alpha star I thought this is the expectation value of the projection operator where we get lower in the 3 due on this sector multiplies the Alpha by 1 and the beta by minus 1 so it gives when sigma 3 plus a hair from symmetry plus 1 of either by 2 acts on the specter here just get something simple signatory 3 plus 1 either by 2 0 gives plus 1 on Alpha and wears given due 0 on data is give 0 just wearing just business Times Alpha star status this is just tickled Alpha star offer so an example where calculating the expectation value of a projection operator gives you the probability in this case for the sprint to be out so this is not only a definition of probability assumption your interest in the certain proposition profit is draw you construct the projection operator for Alpha In the way that I shall you it's a projection operator the direction which is the idea that the corresponding to the proposition and then you just calculate its expectation value that gives you the probability that a simple alternative alternative route to getting the public probabilities arts and now we have Everything we need to look calculate the various probabilities never in this expression here show a staff from London with stuff from With suffer from getting a headache OK let let me
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LME simplify what I said Gunter bare essentials projection operators correspond to statements well let me go back a step classical physics statements are a subset properties are subset of radio radio set the set of all things we saw the set of all things set of all things such back that's a subset of basic ideas intersection and union and all those kind of things about subset are classical logic that's what classical logic is about the classical logic of propositions propositions properties properties in physics off in quantum mechanics a very very different set up propositions do not correspond to subsets they correspond subspaces however a vector space subspaces of a vector space are a very different things and subsets of very set and therein lies the very big differences and therein with oversee allows us to calculate the use the fact that properties correspond projection operators uproar you could say that because fund the subspaces subspaces states which has this or that property manager for our purposes now the only projection operated be interested in other ones which correspond to speak in a given direction being part of OK the projection operator for life start with wares anybody remember what a B and C Warren are AU was the 1st spin is up along the 3rd axis what's the projection operator forebears 3 plus 1 of either by that's only half the statement fell the 2nd space is up along a 45 degree axis I went there was a projection operator for that the 45 degree axes what's the Spain what Steve component of Spain along with 45 reacts personal something like how 1 Postell 3 right minutes longer annexes halfway between 1 and 3 that's how 1 plus 3 but that's not quite right was was squared it too it was a widower were taking the and the unit vector along 45 degrees and diving it into what's a with the components of Ian along 45 degrees they won over the square root of 2 services our town 1 over the square root of 2 must have real square to that's the component of towel along 45 deg axes swords the projection operator for about 4 the 2nd statement here is how 1 plus Tower 3 plus although the squares 2 0 plus 1 where using his the projection operator for similar not who's just seem about and the projection operator assume about end plus 1 over 2 that project your subspace in which Seidman not in this plus 1 None yes whole thing is driver too thank you good now if you apply both projection operators who state this 1 acts on the 1st day and and projects the state onto the component of the index only the 1st day and takes away anything it which Sigmund . 3 is in the wrong direction and they if you like this 2nd it takes away anything on the 2nd spin which is in the wrong direction so the whole projection operator that corresponds to the combined statement that on that the 1st up along 0 degrees 2nd spinners up along 45 degrees is just the product of these to your act 1st was 1 of them that throws away everything in the state sector that's in the wrong guy in the wrong direction in these space of states and radioactive via the 1 that throws away everything that's in the wrong direction corresponding to the others so the product of these 2 is the projection operator the projector where projects for you up along 0 degrees 1st In an Opel all 45 degrees for the 2nd number about 80 and not did not be because beat you remember be corresponds to pop longer up along be 45 degree access not be corresponds to down along with 45 to pay taxes but that we look at particle is that a particle 1 and particle 1 always has the opposite spin becomes up along a 45 degree Texas was the No. 2 this is a projection operator of this object for this property over here for the property that AT and not b its expectation value in this English state corresponds to probability for AT and not I solely can do do is only 1 way to do this and that's just hold your breath and start writing and working out the details 1 by 1 under the door this is tedious bore you can close your eyes and I'll tell you what the answer is but you might as well see how it's done for those who are really interested foes wanted terrorist assume home OK so that's right down the singlet state here is saying what state is up down the minor stand up and now we better put in square of nolayoff would take me in a product of what we have and projected back onto the same vector so later on the project back on a single actor Fernando I just want to calculate what this operated does when it hits the state OK it's not too bad column 1st of all what the signatory plus 1 due to this Over here and I just throws a story white because in this state he the 1st and is now but this is the projection operator onto the 1st been beat up so symmetry 3 plus 1 or 2 acts on a state with the 1st now it just kills it so as that OK and now we don't review a signatory plus 1 over so we can we cannot rewrite because someone erased all right if you get rid of this we acted with it and it just kills a 2nd the 2nd entry the best of it 0 1 plus Pelfrey over tool route plus onehalf all acting on 1 or the square root of 2 0 World 1 of the Worlds to our fewer that this 1 and that acts on OK residue 1st of all what this town 1 due to warm flips remembers how was act on the 2nd entry 1001 flip stand up so we 1 acts it gives up here Watson here is a anything without stopping here know nothing without the been here all areas abandoned them up so that means 1001 acts it give
1:05:43
something which is completely orthogonal toward on left inside it won't contribute whom we take me in a product of the city where it went how 1 occurs and Exxon down to give up you give something on the right hand side just doesn't appear in corresponding Robert we can be quite sure that the tell 1 will have no expectations among just not contribute to this To this in a brother don't throw away what this town 3 Due to Aris I them by shut yeah that is there are 3 tenor acts on 2nd entreaty a cow 3 measures the component of the 2nd Spain and the sentence is now so this is first 1 where ballot disk is minus 1 it is multiplies by 1 from minus warned as it's the numbers the whole effect of all of this stuff here Mr. throwing the 2nd state that signatory plus 1 and how operation here just gives this number not quite true there's another pieces please but up peace will not have anything to work to take in a proper were either strike there say would stay on the side missing would stay on this side of town mind is paramount the square root of 2 well down not configuration will contribute its orthogonal Of them our Belmar operas or far brought their own so this warm Hallmark contribute we're Everett you don't try but the rock down with itself is just 1 of our Hall of Fame His just given by these numbers 1 over square 1 squared to win this beast over here this is probability for AT and not be for a square to 1 half 1 half minus 1 over square too anybody can do numbers quicken their hand show on board were not yet born for Ava we have 1 might when I was worded to direct I wanna square toes . 7 . 7 1 form a square 1 square toes 1 . 4 1 0 1 . 4 was about . 7 1 minus points 7 . 3 services . 3 or 4 because that set . 0 7 5 . 0 7 5 right but remember that we have twice that this 1 is for equal and so we have twice that of this number here is White . 0 75 which is what pointer 75 I a 150 . 0 . 1 5 . 1 5 Rice a small for now let's calculate disorder cake this is will do what is due on the same saying construction here but we still have 1 we still have electron 1 up along with 0 axes that there's no we have electron tool is up only 90 degrees 90 degrees means the 1 axis be projection operator for the 1 axis is just how 1 over how 1 plus 1 over 2 number Michael erase this I'm just gonna and just going to war doctored up so that corresponds to the 2nd stage door here so that's so get rid of have 3 just how 1 over to Class a half say OK let's say OK again said 3 plus 1 2 0 game gets rid of his term he picks out the component of the state in which the 1st so nothing was for that concerns and now let's see what tower 1 tower 1 takes up down top so how 1 acted makes up up but there's no woke up on this side here so this fund doesn't contribute all received my plus plus an excellent how 1 doesn't give anything either How 1 acting on Dan gives up your up up here but you have no up up over here dead and all you get is 1 half is the triples wanted nothing go with 1 hand and now you have some squares To your square is 2 from here and you have a squirrel 2 from here so all thing is quarter . 2 5 thing years . 2 5 1 0 1 4 . 1 5 is not begin point higher we went this whole exercise is to see that the right inside here is bigger than the left hand side This is what John Bell did Father Knows don't forget physics but 3 brilliant will exercise in quantum mechanics years when you could go home and read do for yourself but the upshot is that there may be quantum singlet state of a pair of electrons violates Bell's theorem those quality was a that means that send them there is no possibility that isn't underlying classical underlined classical some way of thinking about quantum mechanics where properties are somehow were governed by ordinary set theory is no possible underlying a description of classical on them underlined description required mechanics it cannot be the statistical theory of some complicated chaotic jumbled classical system you know how to do this unless there is of about the question is whether not Meacher full autonomy Hanukkah everybody know the nature of course can tested convert highprecision let's put it this way well but it's a beautiful experiment and show it was fun to do it it teaches anything we didn't know well it might oppose something about the technology of manipulating electrons and I suspect it and I suspect it was a hard experiment was very delicate experiment and so it probably is part Our march toward being able to manipulate small thing I think in that sense it was a piece of technology was the result certain before he absolutely Irene it now the Apple yet I mean be expended expanded could not have occurred on Canucks check so many ways before that nobody expected not Taylor desperate bid to which the hard thing to do with sampling to do what it checked yes that is what you want OK it is there now is in an hour is any way you classical logic could somehow stimulate Kwan the answer is yes but only
1:14:44
if the system that keep talking about has wires connected but left the Of the right electron In some ways it makes these experiment on the left nut and not independent of the experiment arise if you had somehow that when you did the experiment tho he some signal when all the here and told it rearrange and do something complicated or do you have a whole bunch of wires coming from the left and the right to join together in a processing the center you can simulate corner but it's at the course that signals have to be able to go faster than the speed of light that if you have any see any classical system that could somehow detected you did over here and send the message quickly over to their instantly before the 2nd experiment another words of Bob could send the message that our instantly and uh and tell her what's going on yes there are ways to stimulate a corn carrots classically so the 2nd element of this is that the experiments on the left and the right measures the 2 components spin really are independent that far from each other and could interfere with each other now 1 way of making absolutely certain that they could interfered with each other is to make all of Alice's experiments take place right at the basically at the same time or very very shortly after Bob Tewksbury Alice's experiment takes place time for which light could not have traveled from bomb then Alice's experiment must be independent nor wire could send a piece of information will be no way that any classical classical signal could have been sent from 1 of the soul 1 of the things it was in this fact experiment was not just that taking these single electrons and measuring himself forth the left 1 the right 1 and checking whether I would be inequality was set by but he managed to do in such a way that it was no ambiguity Alice's electron was measured sufficiently soon and the barbs electron the light could not have propagated between that was the hard part of the experiment as far as I know To be able to to be able to run do both experiments sufficiently simultaneously that there was no possibility of information flow classical information flow from 1 to the other so the 2 experiments of Allison worrying that our independent of each other under the assumption that no signals could travel faster than the speed of light arms all attempts to simulate quantum mechanics From classical underlying logic always make use of some nonlocal assumptions some kind of thing which can process so signals at 1 end and send them to the other and were essentially instantaneously so that I think that was the hard part of our aspects experiment was the kind issue I think and I think that was the technological triumph that they were able to be quite certain that the 2nd experiment was done within the transit time of a light beam from the 1st express no interference between young this after I signed down died but like so that I send it he he came away when Valdez clarified his sister just made a very quantitative statement a very quantitative statement Einstein had this idea that somehow this is this fun will assist Smokey Wallace bookie about a missile was book about that violated the classical logic proposition that Ed not surprisingly given violate classical logic proposition quantum mechanics is not classical logic but he just pin 1 man just Finland and very very solidly and a rosary quantitative about the fact that the experiment was done was probably less important than he put his finger on a full experiment that could be done which would if it done would roll out a classical underlying basis to do the experiment now this Is enough is enough I know the car mechanics who work and therefore I know the quantum mechanics cannot have an underlying classical basis you have very ambivalent about all of this having some of the times it but this would brilliant summer and but that trivial and toward the end of his life he started the quantities tho was so brilliant that began the question Kwan mechanics and began to question where require mechanics was the right theory but this is the best gonna be against here that to lecture as defined by New York Times here as well it small things far apart after you could do it do bowling balls of food that worry but the appointed look taken but the but there is still have no it's still about small things but a small things you have to be careful to keep the quantum wave function intact and you take things apart when you take things apart disturbing influences can run away from for example just the atmosphere borrowing interacting with the atmosphere do things the wave function that ruined the experiment so when you take off part it's very delicate debate to it in a vacuum you better do it cold you better make sure that there's no photons around the interact with electrons and an care is required and for the you take them apart more care repair required because quantum wave function tends to get out of sync not by itself but by interaction with other things interaction with the environment will tour it which I talk a little about interaction with the environment our prevented them a 2nd this level reforms in its 4 thousand words as she has a said would well no I mean this is it a trick which makes it look like give violating Eisenberg trendy principle but it is still true that you can directly measures are certain simultaneous properties about 1 Electra you really haven't done here where do you know who this is 1 of the I was a young always nonstop from doing that you could do that but it's not quite the same thing is measuring the position velocity Electra and factory is fear that you Greeley can't make a machine that will replicate that will replicate over her car quantum sisters with another 1 just like it we could prove that right now reward from do no cloning fears you use a lot of possession and says but not yet I'm OK a right I want to come to that I forgot about that APEC army point till propositions are independent of each other was a that means you committed both of them simultaneously it means that the means but the projection operators commute with each other if he'll have to look commuting projection operators In the end statement means a projection for example what doesn't mean something would be to say if you only have 1 electron if you have 1 electron you could ask What's the probability that spin both at along these along the 3rd axis end up the longest the 2nd act that would be meaningless question can you can't do that because a 3rd component that took a bone to spend on commute with each other if you multiply the take the projection operator is 1 of them is 1
1:23:44
plus plus 3 all the tools and the other is 1 plus signal won over to arrive Is that the way the the food and the other a different answer so arm is no meaningful and for the spin is up along the 3rd axis and the spring His up along the 1st axis that better that's safer bid and kind of but few would talking about 2 different electron and all the apparatus 1 electron commute with all the operators of the electron does know which ordered you apply Sigmund as because Sigma act on 1 of them and power acts and the other 1 and they completely Camille completely independent measurements they don't interfere with each other measuring 1 Spain doesn't really do anything want so particularly if the far part sold yes that something I should have stretched yes the part is the end operation but only makes sense if it the 2 projection operators the 2 properties are compatible if the proper it's a compatible which means a commute with each other and the product is the heirs John W. question yacht good good port yup where measure 1 4 poses for Tokyo the mutualist but on other particularly helpful way of looking at Bell's theorem and so forth but durable the column that's what I want to come to our want Cami the took no cloning for the measurement process in the collapse of a wave packet of where collapse of the wave function as being a special case of entanglement which 1 we don't think it I think we are I think we better off today doing no cloning and the next time will talk about the the measurement process as a process of entanglement entanglement of what with what the apparatus getting entangled with a system that you measure that's what happens when you do an experiment to experiment you have to hire you may have a dialog about point there that tells you if goes this way the spin was up because at Winston was Maryland it Victor electron and you put into the apparatus and if freeelectron is up then the pointer points this way of electrons down the point the point of this way sound like entanglements like entanglement between apparatus antielectron starts work measurement has broken the law L I want approval feared for you where after introduced some additional things we have not talked about how state vectors change would climb their talk about that next time I think but tell you 1 fact about about the time evolution of wave functions namely it's lead resent me that means if you start with a wave function Eddie then under time more under some process under some process transforms until a function of say a crime to end if we start with a wave function a different 1 and it goes to be prior to the end of you start with a wave function how flat AD possibly beat in trying new motives saying linear combination of a prime that's a posture quantum mechanics which is certainly experimentally extremely well tested North says Is that the wave function the authority of the state sector of the system changes and kind linearly with nearly means that super positions of state transforming super positions of state I'm in a linear way that's a postulated on mechanics which is experimentally known to be true to extremely high accuracy is linearity quite so only show you what it says are about trying to clone system is out of his goal of cloning supposing I have a system in any old quantum state whatever I want the machine that I could feed it into and that machine will spread out too copies tho identical copies each in the same way function that the original 1 system started where sort cloning operation and a particular for example effect whatever the initial state of the system it should produce 2 systems in the identical state that's the cloning goal the biblical Kabila clone quantum system now classical if possible Xerox machine classically clones of information on a piece of paper doesn't do exactly faithfully because Lueras happened so forth but the degree American be controlled and so you can clone classical system you could build machines that whatever state on that piece of paper you put into it it will produce 2 pieces of paper with identical information at her I actually we discovered the local and going called Xerox principal and strength you escorted no Xerox funds from No . 0 principal actually is all you have to do is prove it for 1 system because if I controlled that I can't go prove that there but I can't cloned everything that I want to do a show I can't cloned 1 thing sorcery kick yes is my cloning apparatus it has poured over here to put stuff until has 2 points of India the text of our he goes 1 in having those you systems and out comes 2 identical copy that start with it up lecturer supposing we start with up electron so we start with 1 electron in upstate just call up up where we get out we get out to opt out of a nearby assumption by assumption but so we can say then met beat up the single electron state evolves in time into with 2 electron state how does this nobody knows incident that violates charge conservation so make it neutrons laugh and laugh None of them among on that means energy conservation of energy doesn't have to be considered Beckham borrow energy from from the machine Beckham Barney energy from issue but would violate charge conservation policy charge from machine then that's OK it is nothing wrong with that but assessing neutrons and make it simple and no electric charge conservation is zone he was a which is summary Caltech salt by itself goes to a state With 2 electrons which are on the rise as this electrons each any upstate towns go right between coops next to or will put down here what comes out To now Otis only that I was able to build a quantum Xerox machines and bears Commando Solo round will all eventually into Baron said OK now supposing I put electron in these this this was an electron was pointing up along 3rd axis again no put the electrons which happens not to be Oriental on their taxes but what happens to be worried that along the 1st axis X 1 axis which accords the downwards quote left right right up the up down left right In our focus another 1 opened
1:32:47
here a electron pointing for the right the words at 90 degrees along the xaxis for example and if I have a faithful calling machine library driver cloning machine faithful cloned machine should take right right right right right OK for what is right and what is the right state the right state is just up plus down the riven by square right so actually put him on the lefthand side is up plus down over the square root 2 was but I know I get I our put to it I put up I get up off if I put in it and if the evolution is linear writers than that already tells me when I get put him up was down Oregon while ever put up I get up up cover prim Bell source writers egg I did down square so they are ideal quantum mechanics affected B evolution of states as linear already tells me when I get if I put enough force them I don't have any choice but what is right right but see Seifert if agrees what is right right right right means up plus down for the 1st electron is kind up close down for the 2nd time and is juxtaposing them by radio to electrons the 1st factor corresponds 1 electrons 2nd factor corresponds to the other assured side always wary OK and multiply this will I have here I have a configuration with 2 up terms up have configuration with 2 dams but I also have up down and down not reliably if I take 2 electrons each with along the xaxis I will find if I measure the 3rd component and that some of the time I will find and up and down over here I have 2 options then homes but never knock them down so you see me on the assumption that you can build cloning machine which both clones optioned then arms and also cook clones left some rights is inclusive which inconsistent with of quality care 6 out to share you this season said got a much shorter would mean to clone things unless there were you will call me things to independent think what was sorry the assumption was that up produces 2 ups and down produces 2 Barrett said he's playing like a that's the way you build composite states that's the way below states firm of independent systems multiplied the state vectors fashion school taking we haven't had the time to every unfortunately have time to reading the the mathematical operations combined systems is called can surprise pick and product of world of state vectors and that's what this is This is the tensor product tool rights explained that out you see would you get what you get cannot be theirs but so you right degree of assuming that the 2 things are independent of each other but I'm not sure I would meet the clone cloning things if they independent of each other in new could do independent experiments that so it is not possible to take a object are giving in arbitrary wave function and put into machine and have it come out as a replica it because of a later so Seoul where the 1st during break down it was put out
1:37:36
player 0 5 ahead of saying you're going want to some relief they said later polarization not so maybe that 3rd Premier the initial it said that last you as well as all of that figure but the question D later I have the question to say you take time will be given polarization and stimulates the mission Quetta think about what Her a dead but it may be up to the information a tree for what direct have that talk about this week even if however that there is no controversy about the fact that you can't build a cloning machine I think there is quite a country not only know what's controversy think there's a question of how close you could get a close can you get all machine matching papers saying different amount different gives some quantitative measure of how close you could get some number like threequarters eclectic could Corning machine that threequarters accurate some things but if that is so much about you it in the basic idea that you can have a enemies simple arms shit multiplying 2 things together like taking anything and Square and it is not a linear operations to quadratic operation you take any state basically and replicating it is sort of Square a sort of operation which is quadratic Misik operations are not squaring something but sort multiplying it by itself Our are here for 4 years and are simply not linear operation that can never be linear it's always quadratic and since all the evolution of state vectors is always Lanier you can't quite all you can do it you can build the machine that will call ups and down but there won't cloned rights unless you Kabul machine Oklahoma right and left it won't grownups in that area here yes yes if that's a different issue that's a different issue that you can transport but you transfer some information from 1 thing or another yet but what you can't do is duplicated in 2 systems right so you might have some kind of information for example the electron was up and you might want to translate you might want to transfer that tore photons and send it turned out of the question is how well could you transfer toward of a system that that's a separate question and I can't really use if you know the state you could build a machine will clone but will be a different machine and freeze at if I noticed state is up I can build a machine that Oklahoma and the same machine will climb down if I know the status right I can build a machine that will call the same machine will not cloned ups and right the so yes if you not the state can build a thing that forms there was a question or answer was or not the fact is however that it is yet but let me just say a little a little differently than is a little differently you cannot build a machine that will clone anything will clone ops rights audience the same machine cannot clone more of you know what these state was somebody gave you a state said tho you don't look at it did you just want a clone you're machines may just happened upon it because it may be the other thing about chances are well it did wooden Kong things upset the rough times can't say again radio station in life but that's different than what the opera machinist to be require mechanics for that earlier this is a cloning result this is doing result gap different so 1 1 it be the same as village just 2 electrons both right have them peace with lot down down that's inconsistent that's what the linearity of quantum mechanics sorry no court less let None revenge data to an end Mr. received it as a way out of that chorea but but but but I am the last time I what I told you you have a single electron the state vectors are to parameter family if you'll have to independent electrons which had never been entangled or anything like that in the state vectors you would expect to be of 4 parameter family but in fact the state vectors of 2 electron or 6 per cent of families of the richer class of states now you how you if you want to keep quantify that by information theory of I'm sure you cannot I certainly know how to cover quantified the degree of entanglement of 2 systems is something which I hope we will get through which is called entanglement entropy and entanglement entropy is a measure of the degree of entanglement of 2 systems so we can certainly quantified the amount of entanglement between the systems and much of its rigorous but many case will try to get to the concept of hang on OK any other questions before I wore I am the
1:44:51
preceding program is copyrighted by Stanford University please visit us at stanford . EDU
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Metadaten
Formale Metadaten
Titel  Quantum Entanglements, Part 1  Lecture 5 
Serientitel  Course  Quantum Entanglements: Part 1 
Teil  5 
Anzahl der Teile  9 
Autor 
Susskind, Leonard

Lizenz 
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DOI  10.5446/15108 
Herausgeber  Stanford University 
Erscheinungsjahr  2008 
Sprache  Englisch 
Produktionsjahr  2006 
Inhaltliche Metadaten
Fachgebiet  Physik 
Abstract  Lecture 5 of Leonard Susskind's course concentrating on Quantum Entanglements (Part 1, Fall 2006). Recorded October 23, 2006 at Stanford University. This Stanford Continuing Studies course is the first of a threequarter sequence of classes exploring the "quantum entanglements" in modern theoretical physics. Leonard Susskind is the Felix Bloch Professor of Physics at Stanford University. 