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# Quantum Entanglements, Part 1 | Lecture 6

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a concept of vectors and they're complex conjugate formerly called do a broad sectors in the vectors recalled rules of each other world we're inclined to sort of think of them as like complex conjugate whenever we have a concept of orthogonality between those factors as long as we stay in vector spaces which have finitely many dimensions and my personal what the quest what's the definition of the mention of the space definition of any of the 3 dimensional vector space we ought we ought to get it abstractly forget for the moment using column vectors just counting the number of entries are give a collection of abstract sectors with some orthogonality relations and particular somewhat relations of linear dependence linear dependence means that a vector may be linearly dependent on some other factors that would mean that the 1st sector can be expanded in terms of the other vectors of linear dependence means that people write a particular vector In terms of a subset of vectors but it said to be linearly dependent on dimensional of vector space a very simple it's the maximum number of linearly independent vectors confined in the space of words in three-dimensional space you confined mid middlemen Mideast triplets of vectors which a linearly independent linearly independent means that you cannot write any 1 of them as a some of the other and minus 1 of them or an equivalent waiters is that you can't find any linear combination of the vectors which adds up to 0 out if you can't than those sectors is said to be linearly independent obviously there is no way that these sectors among all the black blackboard incidentally orders were sticking out the blackboard boastfully vectors are not the Neely deep candidates they are linearly independent because there's no linear combination of them the heads of 0 0 motorway Norway To Expand the 3rd 1 in terms of these other they don't have to be off on them Toby linearly independent take 3 vectors memory withdrawn these things take me vectors which happened not to be off Arnold which just 2 plus this 1 over here with this 1 is not lying in the plane of the other 2 were there is no way to write the 3rd 1 in terms of the truce or get a linearly deep linearly independent known a three-dimensional let's start with two-dimensional the space what's the maximum number of linearly independent vectors are complying he answers to a fire and a 3rd 1 In the plane In the playing in any 3rd 1 and now destroy exactly the same thing but think of it now as a victory in the plane and finished 3rd 1 can be written in terms of the 1st 2 for example we can still get a vector he and then with a might signed up that minus signs about the multiply by Constance about Ed vectors multiply them by constant we could certainly builders vector from the other killed so these 3 vectors drunk blackboard are not legally independent it would really be packed the maximum number of linearly independent actors in two-dimensional spaces to the maximum number of three-dimensional space history and so forth and that's that the basic definition of please but mention of that space the maximum number of Earth that's a theorem which were not moved by it straightforward to pull that if you could be they mentioned vector space be dimensional vector space for a concept orthogonality that you could always find d basis vectors D basis vectors of basis factors mean that the base the vectors of 1st of all normalized means the unit vectors and they all orthogonal to each other you could find so a two-dimensional spend in anyway it's not just in 1 way in many ways but in particular there exist families of vectors which are orthogonal mutually orthogonal unit vectors to within a two-dimensional space 3 of them a three-dimensional space so for us hand goes back or set the of virus will have a real real real if we want to be part of a violent that is a group of men longer offers who movies face an art works definition of reason for your job that's out there there were gaps especially the various the space of states are to be bees what they are moving in the wrong place a huge number of possible what bottles said OK and now that each of
goes right through air and multiplies each 1 of these by land already is just a multiplication by land and so any linear combination of a pair of eigenvectors with the scene I get value is also eigenvectors without Eigen Lewis well would this will begin to discover is that there's a subspace subspace subspace a subspace means a set of factors our which are closed under addition richer which have the property that they themselves were vector space but might be lower dimensional a horizontal plane here is a subspace is also a vector space consists of all the vectors and playing here and basically it's all the sectors that can be built up over 2 vectors of applying the same thing here a linear combination of these 2 vectors will be and eigenvectors and so you could talk now about Eigen space subspace vectors subspace of original space which corresponds to a particular Eigen value case got that could state 2 states which happened this share the same property of having a particular value of some security not could example you might be an example of up down the door particle system up down and go up up both of these have the same Eigen value for sigma 3 different values the membership same they are part of that both have equals pop singer 3 eagles plus 1 they have different ideas of how 3 of the same value of sigma 3 really a combination of these 2 also has the same value for the openness as of the 1st century here so this they of course there's other vectors which down up and down down which have different values for sigma 3 of these show Ban a subspace subspace In which sigma 3 years opt so that's the idea of a subspace you could have a collection of vectors it doesn't have to be two-dimensional there could be 20 find that sectors all with the same item value and 15 others with different ideas values but of you pick out all the sectors that correspond to a given I value of mission operator a form subspace linear subspace that if fuel a vector subtract vectors multiplied by constant they are still in the same In the same subspace that tells us is that properties such as equals Lambda that's whatever K happens to be the property became equals land that is not necessarily identified with a single vector but identified with a whole subspace question they want seem identified as cyberspace like you got a little bit further and find a basis With Eden and that subspace so his here some subspace down here and might actually got some subspace Baron subs based on the floor once we find that space we can find bases vectors they suspected that don't spend the whole space but just stand the subs that so just think of the subspaces faces a space by itself and we have a set of car Arthur basis vectors Bundesliga Cohen a isn't being used for this particular example over here what's called the bases vectors a bees themselves assumed that we found ABC however many we need will be led in general would be be lessened D B is the dimension of the whole space but a set of bases vectors for the subspace only for the substrate and both by definition have a property that any 2 of them are orthogonal on must have the same otherwise Ian a product is 1 about a by now don't run from 1 could be they run from 1 to whatever the dimension of the subspaces so we now have the concept of a basis for a subspace let's take that basis and construct that added something which looks like this let's take the basis in the subspace right the expression some already some of all all the bases vectors of what is that object never forsaken tell value is it is it would give you back saying if you had any vector with it dealers sector as long as the victories in the subspace so that there is some space for all practical purposes the other directions don't matter so we just 3 writing this kind of formula role he of size in the subspace then we just get back side again is not in subspace when you get a size not then the subspace then what you get is the projection of sign onto a subspace stores at work like proposed vector over here someplace sticking out of a blackboard from mentioned broader properly the projection I the subspace is the rejection of the vector I want subspace aren't we just discovered what this object 80 80 is it's the projection operate its projection operator punter the subspace which in this case corresponds to a property the cables slammed sold this subject over here is a projection operator onto the subspace in this case where Keyes Eagle where that's how you construct a projection operator if you have a property that you're interested in you find the subspace which corresponds to you find a basis in that space but this is the Ismail may not be hard to do it in a particularly easy or hard but you find that bases and a new construct what 1 of bin be identity operator accepted not really anymore because it doesn't have those sectors which stick out of the plane which together the subspace the projection operator the Serbs yes yes disappeared operator is projection operates a projection operator on the whole space on the whole of birds that you only have 1 vector it would be the projection onto some particular accidents that you have to vectors two-dimensional subspace would be a projection onto a plane after Favre and other sectors factory corresponds just to a projection back onto the whole space but that declared corresponds to a property our properties but the property that's associated with the identity operator a trivial property just to the system exists no information that is contained West that part but Kate yeah projector operator carrot characterizes or property it's a property which is truly matchups based solely on an area is just a little bit rise finals thing about projection operators is the probability posture the probability pustulate quandary characters most nicely expressed in terms of these projection operators normally can say that it is if I give you an arbitrary stateside which is neither of which was made only not be In the subspace mainly now appears subspace that I give you an arbitrary normalized State Society and the probability of has property cables land there is just the
end what is the projection operator for the end statement for the property Katie Eagles land that and l equals meal while the answer is a product of the 2 of them if they commute if they commute the answer is the product of the 2 of them Our but seriously wiretaps but it is cos PKM market Court PK equals land that that's too much notation just he K a P L now if they commute but does not a witch or you write them down let's take any vector all all let's go back step commingle backers that some of the factors that if you take the projection operator P K and you apply it in the world 1 of our plight any vector you will get a victory in the subspace right so it stands to reason that if you take this whole vector here it must be an eigenvectors of with Eigen value Lambda This is a back there In these subspace cables and so it must be the case that there's just gives you land there fires PK I'm sorry wasn't very deep it just shows that PK onside gives you a vector In knee subspace where Katie is definitely equal Calamba OK now let's take the product PK times P L applies to any state whatever I say this sector will have both the the property the case is equal to Lander and they'll lose equal from you and we check that well 1st of all I told you that you to take any state and multiply by candy will have the property can but property carrying case equals warm Alicia scored a property case vector if you multiply by PKB has a property case will these that there are multiplied by PKT from OF therefore has a property can but it's also true that PKM commute with each other I can also write as P of times P. Kennedy side and now exhibit a fact that has the property L so if I have to look commuting projection operators to projection operators which commute with each other that robustly means that the 2 observables compatible with each other the 2 could measure both of them independently measurable simultaneously they and the product of the 2 projection operators corresponds to be and state another words it corresponds to a finding that corresponds to the vectors or a corresponds Excuse me to world the projection onto the simultaneous property cake was when the cable Ehrlich was merely for example if I want to know the probability that Katie equals land that end illegals new it's just the expectation value of the product so I became known side is the property is it probability not just a K E equals land that Ehrlich was new but that K equals land that handle equals me that's the end statement you multiply projection operators is also on or statement be always and they also known as in the classical logic would be intersection on natural Corby intersection vector spaces but it is a busy intersection in classical logical would correspond exactly that we do have a property described by a by a subset the point In a reset and you have another property that it's the intersection which corresponds to the or statement sorry end statement the overstatement corresponds to the sum of the 2 projection operators Quality Charter prove that yourself ended his product or is so if you're interested in what sectors I have the property that K equals L. K equals war L equals new or statement than the projection operator happens to be the sum of over there be no interest that I think in the proper it is not compatible you just get Hayashi you can measure both things together at a patrol I think is clearest of compatible not compatible still defined it but them but I'm not a very good definition what's set that OK the spinner 1 every electron system missed any component of 1 spin is compatible with any other component of the others that we could check that in charges could be another 1 young let's Tekoa Coetzer let's take a case of 2 spin have a budget of the a Farm yup yup class this spin operate act completely independently on the 2 sets of degrees of freedom they always knew with each other a few 1st multiplied is back there by a sum and no and then multiplied by how get exactly the same inches was but by now so usually often our compatible operators are simply operators for 2 separate subsystems when you build subsystems nobody that bring systems together a quantity associated with 1 subsystem is automatically compatible with quantities from of Osub vessel of you to vote higher of them yes vomit and happened to be compatible yet yup but they could belong to the same electron our view at 2 distinct electrons and anything associate 1 electron was automatically compatible with the other lecture on the other hand there can be things associated with a single electron compatible for example of electrons moving three-dimensional space be components of a position x y and z e also compatible with each other we haven't spoken about that because we haven't done that can require mechanics doing bit quantum mechanics of particles the Canucks but a similar situation for compatibility is when wind 2 things belonged to different but called QBX in the case of cube bits in the case of simple spins b criteria would be that they belong to different spent so different the the compatible now None on something but something which is not compatible with the x component of 1 Spain and the white component of things you can measure the x component of a spin and the Y component the same kind of our they're not compatible you only get 1 vote when they met 1 led to medical but the magnetic field and the measure the spin of a particular electronic can put the magnetic fields simultaneously in 2 directions and travel and try measure to different components of the same spin annealing interview have tool electrons and simple situation just pick them apart but 1 magnetic field or here in 1 direction another magnetic field over here another direction and you committed to different components of 2 different spin so they become compatible but 2 components of the saying spin and not compatible this is a just as they did in the 1st year of the story is located positional mobile phone division of all woes issue a matter of fact sewing have a concept
1 post turnout because palaces been operator for the 2nd electron but not however 3rd axis but Powell 45 degree was that that's how 1 plus have 3 Over the square root of 2 that's a projection of wanted a 45 degree axis and then we should arrive by this is a projection operator for 1 statement the No. 1 was up the projection operator for the other statements No. 2 was a long 45 degrees as this 1 what is the projection operator statement that electron number 1 up and electron No. 2 was long 45 degrees it's a product it's the products so we multiply these 2 operators together not just means multiplication we do the same for this 1 the same kind of thing for this war moves over a little bit of a 2nd projection operator over here would be the 1st spin long 45 degrees bats 1 plus 1 plus 3 Over square tool all divided by tool but then the 2nd spin along the
horizontal axis so that 1 plus Powell 1 divided by 2 that's a projection operator for the 2nd configuration and the projection operator for the 3rd configuration is 1 plus signatory of 2 kinds 1 plus How we won over well you know how to work with these Sigmund operators and if I give you any state any state of the electron system you can calculate the expectation value of these 3 projection operators and you could simply check is the expectation value of this 1 plus the expectation value of this one's bigger or is it not bigger than the expectation value of this 1 over here this is the exercise that we did last time for the singlet state in what state down miners down up and we found the be inequality was violated our there is no reason why it should not have been violated were not doing classical set theory the derivation of this formula he came from drawing various classical sets and looking at the just noticed camping not a witch what things corresponded steady and not be what things cars for regions co-sponsored a Beeler regions correspond be in Nazi or regions correspondent see it didn't have to do with these intersections of projection operators it had to do with the intersection of sex so there was no reason why the inequality should have been satisfied the quantum context and not we know that work the belt either by or what they think this is the end write them if there are more than they love the war will not have the only thing you could say about it this is a very clear demonstration that quantum mechanics can cannot beat the statistical theory of some kind of system with classical but that's really governed by classical Archer a classical set theory that all the it cannot be there are a little below what they say is the size of the with of the there's more class will also they remember there are many many variables that you could check in this case well Bell's inequality McCollum you may be butcher very very many different ways inequalities for the same system sold by choosing properties by choosing a particular set of 3 properties you might find for certain system state inequality is not violated but some other properties for the same system you might find it is suspect that would generally be true think that would probably always be true that you could probably always find properties a B and C that the quarry would be a review of Iowa and having difficulty furniture or carry firm of the summit magnetic population of electron last I had a B articles of the particles command payers becoming payers Our view that we might start might start with a pair of electrons a pair of electrons and knew each other because they knew each other they interact with each other and the lowest energy state happens to be the singlet state so after a while you could do without the need on the beginning of this Singh state and electrons are taken apart and that's not what to do
and you could send them through some sort of system which separates them it could be electron positron you put on the field the field sends electron 1 way positron the other way as hell well separated them allow separated the much as 1 of them many of them many such shares for each pair committee they are you measure are take divided up into groups bottom up into groups so that we can do 3 separate experiments are so we have a 100 billion Perez we take a 3rd of them on 3rd of them within a measure of the number of AT and not be colleges assertive on billions 100 billion the by by 3 within a 100 billion divided by 3 of the parents and the 1st said we measure of up component of electrons vertical component of electrons and a 45 degree component of the positron became away can't up altogether any pairs are rare in which the 1st compiled lecture the 1st electron was up along the 3 axes and the 2nd electron was up along review of the 45 degree Texas again and that a probability that gives us a probability forget it just use probabilities here that gives us the probability that the 1st electron is up 2nd electrons along 45 degrees than we take the 2nd group and do exactly the same thing except the 2nd world we don't measure the 1st electron up in the 2nd electron 45 degrees we make this 2nd electron 45 degrees 1st electron 45 degrees the 2nd electron 90 degrees we can't up the probability that we and then finally we take the last 3rd of electron pairs of electron-positron pairs and measure the 1st electron up easy access and the 2nd electron along the horizontal axis and count how many of them have the property that 1st electrons up what what percentage of the probability the 1st electron was up in the 2nd electrons horizontal clearance says classical classic wafer were true it would say that this was the should be driven is Cher voters PARIS July the population of the different about that day from Frank is dealing with probabilities we take it is doing work probabilities and we have our identically selected set of cloth 3 populations we can measure the probability distributions by looking at just the separate there the film would be truly if we provided the system identically prepared assuming their identically prepared on all 3 populations and every single payer Our the would be true about some populations is always a prepare them in the same way that prepared typical were each store that a doubt that were vital writing on the wall here is very diverse videos based their last verses get on an arm and shows how falls some which are outside that Peru decided that there must be a is that a of their so they got a hell of a lot of work to a view that is years schlep that was that we're a but was stare at tips burners it was done in such a way that the 2nd of the tool that the 2 measurements could not bell-like signal could not have been sent between now and I don't know if it was time for light signal Co from 1 to the other but it would not be a contradiction you might wonder how the hell that the signal get across there were what was the mechanism whose the light beam and so forth but you wouldn't have a contradiction with causality over locality Europe Clark if the 2nd electron was measured after a time during which alike been could've gone from 1 place to another you might say all that I didn't see a light beam gold I must have discovered some no force of nature that I didn't notice before but at least it doesn't exceed the speed of light arm but the trick is to do the experiments in such a way that Allied-Signal could not have gone for the 2nd electron when you measure it from the 1st electronic companies electron these photons sleeper there no vying with a bat as well as we're classical and a man on his but right but with yet right and that's right that's exactly right so part of this whole queerness of this entanglement is connected with signals not going faster than the speed of light when people say that an audit that hidden variable theories and that means classical logic systems which could somehow mop up the effects of quantum mechanics when they say they have to be nonlocal what they mean Is it signals have to be able will faster than the speed of light work to send messages between the 2 electron I ghetto another thing and that can get to it but if we are academic giving you any freedom removed In the hallways Is nothing sacred about the years now know our house that's right is there's nothing guns sacred about conserved quantities context here it's just that Nice charitable work with and I is don't even really have to be Spain's they could be any ecosystem any state any system that has to stay and upper-level a comic areas that are young young not now but it was not important this was an spins a angular momenta or anything like that all right where I want to do next was talk about the connection between 3 things Was interference phenomenon measurement phenomenon how you were measurement and its connection with entanglement Howell measurement is really the establishment of entanglement between systems measurement is and how measurements destroy interference how they call the collapse of a wave packet or thought we would get to that next North Sea what see how far we can go our when they wanna back later fled or were so are actually do be speaking about Is it to slit experiment but under simplified to slit experiment so that we can think about is just having a finite number of states I will support expanded that is toast experiment but just imagine that photon or electron whatever happens to be
I wore blue Roy the
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