Quantum Entanglements, Part 1 | Lecture 6

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Quantum Entanglements, Part 1 | Lecture 6
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Lecture 6 of Leonard Susskind's course concentrating on Quantum Entanglements (Part 1, Fall 2006). Recorded October 30, 2006 at Stanford University. This Stanford Continuing Studies course is the first of a three-quarter sequence of classes exploring the "quantum entanglements" in modern theoretical physics. Leonard Susskind is the Felix Bloch Professor of Physics at Stanford University.
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of the world the U.S. Paris was worried over parts of his era of flow over a war games raw wounds were murdered was well as a form are ROC say nothing goes out but terrible the question is how long we and the degree of a triangle remain whatever you do with the other and you may change things at the other end 1 way or another but that electron which is over there will stay entangled with the system over here to stay entangled even know the system change was still stay entangled made wickerwork example maybe at some point a work out exactly the examples supposing you walk rotated the spin on the right here by putting in a magnetic field houses the state vector change for the 2 particle system and users some which the measure of entanglement remains the same the answers yes there is we haven't talked about measures of entanglement quantitative how much mentors between systems but as long as those systems don't stop talking to each other again as a measure of entanglement which never changes between them and even know a particular form of handshake somebody asked me about what happens if you take a tangled pale and you 1 of the particles To a black hole of what happens to the entanglement well it becomes entangled with degrees of freedom of the black called black hole evaporates that electron over there still entangled with all the photons command of the Out of the year out of these black hole and so forth and thank them even think of much simpler examples of black holes of much the whole story sounds very crazy but nevertheless forget black just in a bathtub bathtub pair of electrons their entangle the missing with state and the better the pool water and you take free-electron Adam Kobe wore out of the water and the often came to untangle what could go water molecules and any job 1 of the bigger the backed up and it gets all mixed up terribly mixed up with all of the stuff in the better here this electoral is still entangled not so much with the single electron which is there but when the entire state Of the ban you have now treat the bath tub as a system and talk about the entanglement of the bath tub with electron thistlelike its to mixed up with the others to have a clear identity but nevertheless the state of the entire backed up here stays entangled with electron and then the bet that can evaporate there's no molecules left we've all does nothing left to be entangled with will of course is what's left to be entangled with because all of the year at the same the vapor that escaped the death of the a man entanglement will remain there closing in on a less something from the side actually comes over to the side and interact with each other and then they can become disentangled is quite possible to take an entangled state and this let anything we can go 1 way or the other way so you would take to untangle electrons let them interact for would become entangled there are anything can happen can the opposite directions with distant can only take place but only if the systems are close enough that actually interact with each other so if something from the here comes over hear it can you Kendall Bermudan yacht thought that was they z-axis we will make all those they still fragile unity of of the will although can they can live with talk about that those measurements are called measurements the core of observations and talk tonight about how observations on measurements have the interplay between measurement concept the collapse of the wave packet and the concept of entangled before I do I wanted just I'll spend a little bit time because I don't think I didn't Burwell talking about projection operators just just review a little bit of a mathematics just very briefly of discuss subspaces manner and end the mathematics of subspace isn't projection operators that is very elementary level might do anything for you use more heavy hammers necessary to explain the idea but 1st of all we have is that it's a space of the kind it with talking about where you have
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
lost right so you could always find a mutually orthogonal said of unit vectors they call bases there is not unique but want to find way to find the bases you have a basis and so let's imagine and we have a solid basis vectors of corn In a way in there's a numerical integer that can run from 1 3rd dimension of the space we have these such vectors and there orthogonal to each other mutually orthogonal so that means that M In the end 0 0 I'm West and is equal to and in which case it gives 1 of the symbols that the crime Dr. symbol chronic about the symbol simply means the right hand side is 1 of em is equal to end and 0 if there was not equal to let's be better that's basically the definition of a basis set Rosser basis vectors for a love for a space of of the kind of talking about now years in interesting theorem which is is hardly of pyramids more or less obvious that if you have a basis said you have a basis set which has as many vectors and there are dimensions in the space that follows it is that the whatever ever it can be expanded as some of the bases vectors With coefficients were a set of coefficients of course coefficients A sub and tyrants and some money in other words this is just saying that any vector anywhere as can be expanded in terms of over 3 dimensions the 3 basis vectors and now we can related on the expansion coefficients here the inner products of these societies with the basis vectors let's do that just 50 in a product of this equation with any sector on the left were bases vectors on left and we find it and sigh His equal to someone a sub end them this is a summary of but does terms because all the land in is equal to am do we get an answer do we get something on 0 when it is not equal to an orthogonal when it is equal to an hour and you get ants in the war just 1 receiver only 1 term in here and that terms just equal pay and some money and we sort picked that we select them out In the project on the side of the howitzer is just right so that's a nice simple but that the expansion coefficients here for a given vector on nothing but in a product of these sigh vector with the bases factors that tells us what they if we take these Nikon rewrite the equation 3 this equation we now don't know what days ah let's Friday the following 4 signed physical force on Monday and announced that a righty a AD I'm going to write this in a product in a product is just and sigh that area firms I've taken the liberty of writing the coefficient on the right side of the vector instead of the left side of actor that doesn't you are allowed to write 3 times sigh as equal to sigh times 3 objections that so this is a member of the city in a product this is the end they suspected sustained formula except that I've used for the coefficient here begin in a product Niven interesting equation I'm lead this the following ways I read this summer separated following invent the new symbol no symbol is the sum of in the current what on earth does that mean we never defining putting vectors back-to-back like that but don't tell you exactly what such an object to when you put the vectors back-to-back like that any 2 vectors it doesn't have to be basis vectors think any correct scored a and B but the back-to-back what you get is a linear operators get operator and the way that operates is multiplies of vector you just take in product b with Woodside and by 8 cents of this is an object which when you will it apply until actor with the Royal but you just take the inner product be side would you get back is another vectors it's a linear operator the school the die had a pair of vectors back-to-back like that all Diane's like that are linear operators so we can now read off From this equation this operator is it's an opera later which does nothing when move this all over a little bit but sigh of think of this as an operate which had society however extra redundant vertical signed here but I wanted to win this construction hits it just gives side back so this tells you immediately what the meaning of this object if it's just a unit operator How you read it when this object over here it's any vector at all because that the same vector and that's call called the resolution of the identity but any time you see this symbol some over the vectors of aliens back-to-back like that it is just too you move up the and hand them in the head In this summer it is a somewhat basis yes has to be a basis has to be a set of orthogonal we use that year what we calculated that coefficients were equal to times and we use the orthogonality and the unique character of each 1 of these vectors Oh yes we did use that these must be a basis diary called for my guess is that yes the unit operator every vectors and eigenvectors operator with a W 1 human operators special operator that has all what every that is an eigenvectors of it in every sector has ID value plus 1 so it's it just says that 1 nominee vector gives you back the same I use the identity of called the identity the fiber now we might be interested in subspace with find some subspaces some interesting subspaces some spaces correspond to certain observable questions that you could escalate to my answer suffer example suppose that we have some operators some observable from permission operate a Coke 80 something that we can measure that has an eigenvalue spectrum mission all the properties at we identify with observable quantities of supposedly have a eigenvectors any of it with eigenvalue Amber I just says if you measure K 1 system is in the state aid you'll get cancel and the officer let's opposes another eigenvectors with exactly the same Eigen value not different ideas are you the saying 1 different bacteria in fact the newly independent of these not pointing in the same direction as are multiple very with exactly the same been any linear combination of Iranian B the after a post David B. any combination more bad news also in eigenvectors K with exactly the the same Eigen it's easy to check he just apply but each just gets multiplied by 1 member of the land the comes on the outside reply the cages
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
expectation value of this projection operator see can make any sense out of that I clothing for the projection operator its expression here what I find sigh Henry sigh some does is make sense yes it says that the probability of finding a particular property is the son of the probabilities Of the particular way use but you could find that all the orthogonal ways that you could find out that particular property so for example of the property correspondent the 1st thing being In other words corresponded to the projection operator on the 1st day being up that's right them with a projection operator would be it would be up up up up plus up down up there 2 basis states in which 1st spinners family up up and up down and reconstruct the we would leaving out some things here found bound so for but this is the projection operator subspace were the 1st spin Is up on if I take the expectation value of me on this operator any state the probability that the 1st up but comes from the places a comes from the probability that the 1st thing is up and the 2nd spinners op plus the probability that the 1st is up and the 2nd back so that's the basic past past Europe probabilities what I really Burford water it now is is is little more observables Army Corps of the operators which have a collection of modern a property of having a particular value Mr. property that were new measure you will definitely get so the property is characterized by an operator and I get value a projection operator it is a tool for calculating the probability that the property is really there are no words in this case it would be probability the case equals the projection operators just pool and it's a thing whose expectation value is due give that probability most of us really have me has said that there might those are always all of this is for DVD Ryan yet you could find orthogonal basis do wanted done so you Malkin just think of the other part of the other directions in the spaces being incidental you now have a space where several factors will be in the form of a vote on the use of bases suffer if all only the colors of the received Drexel was part of a vast array of 1 yeah sigh so I could be any better the and then then this case is a 3rd sector which does not won the subspace you could only in as a basis for the whole space but it's not part of the the men were new projects sigh aren't a and then project sign on the B E and they had the answers you get the projections are 2 of the major Jedi misfit a lot of unity of where we're dead a yes vote for 1st creation was for a year yet yacht life this is case in this equation here really talking about a particular back but in this equation I'm talking about some of that I really should I really should use a different location services is a summation variable here now we can quote some are a prime or something like that what kind we can we can decorate with as many indices as we want just very confusing but when you see a son like us don't think of a particular vector being picked think of something over of actors wears an equation like this without a summation in some specific some specific sectors some AM that's right this could be a sum already and be right write a prime could stand a or B or C or D depending on how many vectors exactly that in 1 in 1 communications and worked with subsequent subscripts on rotations yes you got the idea summation index like this year's just some over something period picked out the a victory would be vector God you could call it wanting to that work I would make more sense to call 1 and 2 of Vogt and Vallejo have version of the advice the advert vote probability yup these a probability amplitudes and there is a probability amplitudes multiplying them by the complex conjugate give you probabilities and then something all the different ways that you could get you see the different ways that you equals 1 that you have 1 vector or the other victory cause summing all the various ways gives you the total probability the Katie cocoa and there's a question that broke a good kid just anymore right so that's that's another way of stating fundamental love Our probability boss to it now if we Hill tool commuting a projection operators to commuting projection operators meaning Our well exactly what I said Phil commuting projection operators give you some examples of of score key power k Eagles land there and another projection operator for another property let's say L equals is still distinct projection operators for 2 distinct properties if they have the property that become commute then the some simple definitions if you could make of what you mean by or a all about and what is the projection operator Katie equals of land and illegals knew what I think what is the projection on the simultaneous conditions that both Katie Eagles land that an l equals in order for there to be a man and begin insert there have to be vectors in common in subspaces unit to subspaces have intersect with each other after winter sacked a you had till about spaces which didn't intersect at all except at the origin of all subspaces always intersect the origin of as subspaces which don't intersect except the origin and then but what's places 1 vertically upward simplicity then clearly there are no states which share the property associated with this projection operator with this projection was nothing in common between on the other hand if we have Our subspaces which shared another subspace along with dimensional subspace then knew actors in the space which share both properties which have the property associated with subspace No. 1 and the property associated with subspace No. 2 we know we can ask which sectors which states of the system have both property 1 and property to
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
of parents that that we just remind you I I just want to go through this again make the logic I hope clear what we did last time ii implicitly pretty much assumed everything that I've done tonight when I describe it end it may have left you less than comfortable With the procedures but it is remarkable no matter before the calculation again is to remind you we did that this was the year of the Bell equality oversold pulled the completely classical completely classical theory had nothing to to do with quantum mechanics and value to inquire mechanics it's as if we have some properties a B and C The number of quite a number of objects which have aid but not and not be plus the number of objects in a number of objects closest in probabilities number of objects will be but not me there's greater than our equal to the number of objects with Eddie and Nazi now we can apply this facility if you if electrons was strictly classical objects we could apply it just electrons and take a B and C to be the following quantities a could stand for electron busy component of the electron circumpolar electron eldest indicated but point think off a beat could be mad at me electron was pointing along 45 deg accidents in the 1 3 Playing be simple permissible and see could be trapped electron sorter warmer for example axis that measure along the horizontal axis you definitely not a B and C or a particular announced they would correspond to electron pointing in the opposite direction would be corresponds electron pointing in this direction but not be corresponds with pointing in the office of a sale of 1 of our most of his love van is they as letters then that it was a lot of time version is his goal sometimes it sometimes full the victory 8 plus B might be quite different from the correct my speed and I give you an example up down like stand up is quite different made up a plot sank to make a difference for the overall the view violated voters need all of them were wounded 1 of them a column I don't I don't honestly know the complete answer questions it's certainly not violated for any state for example I can find of the 2 particle electron system for the belly quality will be satisfied short so the answer is but there certainly states for which the belly inequality is satisfied that the such a little bit to try find situations where the virus but not very hard to just throw artistic around the guests or 60 40 60 % times you've satisfied equality 40 % you want just made that up idea really but the most likely I would guess that it's more likely that you randomly selected state that inequality would be satisfied and that's a sort of a smaller percentage of basis states that doesn't satisfy quality never check that something is then via video how can they made head but you could could set Yucatec a punch now that yet Yao yup yup by new could also have CPU Caribbean angle analogous of the angles a close together that but you don't violate inequality but we can check that that's an interesting thing to check at what angles so that his two-year inquiry satisfy the angles of Iowa and advocated using his own advice and vivid and checked it but he took about mouthful photons I think photons electrons OK so that others will be different but yacht other certainly situations where these inequality is not violated arms but but let me just remind you of what we did with all we did was take a tool electron system imagine doing many of many checks to electrons are missing with state and we do many many checks on us and every single time we find out that way and our that these if we measure the spin of 1 of them along any axis the spin of the art the other 1 is along opposite axis that would suggest that the property be the property and not be the property not be for electron is the same as the property be for electron 2 would suggest that the property not be which says that electrons 1 is His equivalent of a property be for electron number 2 start we could rewrite this inequality as if we work in terms of hands of electrons with pairs of electrons which Rabin created created way that spins were always end by aligned we can substitute for the property b we could restore could still be associated with electron 1 a and not be would then become the number of Perez where the 1st electron is but the 2nd electron is a 45 degree the 2nd electron being a 45 degrees would imply that the 1st electron was in not if it made sense to talk about 80 and not be for electron No. 1 that's Osorno rear end is not be von not be means the electrons pointing this way but if electron 1 is pointing this way electron too was pointing the Long Beach so not be for electron 1 is B for electron to good I'm not something for electron once instead opposite since their opposite not statement for 1 becomes the leader of the positive statement for the other guy right so we can now we rewrite this inequality as our the number of our pairs in which the 1st spinners up but a 2nd spinners along 45 degrees plus the number of Paris in which we 1st band at 45 degrees but the 2nd Spain aims at 90 degrees is greater than eatery called for a number of spare of Paris where the 1st thing is up on the 2nd day in a 90 degrees but all we have to do to check the quality for any given state used to construct the projection operators for these property services toll electron system that will replace the 1 electron system by 2 electrons system if we would in classical physics with classical logic these opposition of the 2 Spain would tell us that not be the beat the 2nd spent so about the logic now OK so what are the projection operators we take any state whatever and we can ask what's the probability for this property what the probability for this property what probability for this property go back now we want to construct the projection operator for this property this property has the No. 1 pointing out the projection operator for that is 1 Sigma 3 over 2 if the 1st is up Sigma 3 gives plus 1 plus 1 is to a divided by 2 that's 1 thing so yes that's uh if his 1st is now the signatories is minus 1 1 plus minus 1 0 0 and we gives you all of this is a projection operator on the 1st not the projection operator of the 2nd spade pointing it along its axis is also on 1 plus the amount 1 plus but
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
lesbians forgets spin now a real particles it really move from 1 place to another but let's suppose that we the particle could be in any 1 of several vertical location just to make life simpler some set of course 0 1 2 0 3 minus 1 minus 2 0 minus 3 so the electrons what are called electron but forget it suspended its starts down 0 and this is where this is position of the electron along the vertical axis electrons actually moved the right might also move up or down but just simplifying by giving it only a finite number of places where we're just so that we can use the pools we've been talking about finite dimensional spaces so the number possibilities with electron be finite in this case like electron move over to the right off but there are also several places where it can be found over here Toledo From this side of the he nonetheless like that I would assume the electron the point he felt electron actually started 0 now it because I here when it gets all the years it's in 1 of these 7 places depending on which place it's in which places and it continues to go out and continues to go over to a 3rd place which also has say instead of 1 2 3 4 5 6 the same set of possibilities were really talking about the evolution of a system from 1 hastened the time for another instance of time but he is a game of their plight visitors Tuzla experiment on assumed that if electron arrives at this point that it just gets send but right so we ignore the possibility that the electrons which go backward because to this point only here it also get goes back in only goes the rule hits this point on that so from the point of view of a to slit experiment I've opened this slid and I'm open that's led and I've sent an electron here hero walking here 1 electron kind wasn't 1 electron time it either goes the upper of the lower sled there's also the upper so that it can go our novel use slipped and I arrived at any 1 of these places over here so I'm interested and do women make up some rules rules a consistent with quantum mechanics by a simple example I'm interested in in the probability of electron arrives here here here here here here what's apart at the various circumstances his assumption the evolution of systems for quantum mechanical evolution of system is linear that means that the system starts in the state aid and evolved to a state aid primarily in this case it could be that it starts at any 1 of these places here gets any 1 of these places here and the state B we've often beat prior benefits start the system in the state aid plus be below evolve into a prime source pride this is the assumption of a linear arity in fact we can do this with arbitrary coefficients are forbade that's the statement that of subsystems involved in a linear way linear superposition evolve in the linear superposition this is an experimental fact as it said the last time it's very very hard may be quite impossible nobody's ever succeeded in changing quantum mechanics a little bit too about coming to really silly you're conclusions so will take it as a pasture where the evolution of state vectors is linear linear superposition evolved a same linear superposition with the same set of coefficient let's that's that's about all going to assume ever-growing well makes some definitions From the start electron over here and after a certain time it will have arrived at 1 of these places I'm not interested I might as well throw away all places which just reflect electron backward because somebody back here Sony gets a 1 quarter minus 1 Mbyte symmetry by symmetry just op dams symmetry here but I really don't have enough dams symmetry of the length of this Pampas exactly the same as a way of despair and so forth and I stopped electrons symmetrically about the only state that I can get to over here is a linear combination of silicon right the 1st phase starting with 0 state will involve the state 1 plus minus 1 9 not terribly interested in normalizing the state I'm not interested in keeping track of cover for the probability is that I'm playing the role of groupings so we start with a lecture over here after a time by the time it gets to be over here it will be the superposition states would equal Tampa to equal likelihood of being here and here just because of the symmetry of the problems you expect equal coefficients here but now let's see what happens if an electron starts at 1 what happens to it when it goes over to this side over here so let's make a postulated assumption about that it started 0 0 initially it gets the 1 plus 1 part of my 1 1 plus minus 1 1 have visited starts at 1 over here all minus 1 over here what happens to the state vector over here so let's make the the following assumptions if you start at 1 of electron starts at 1 and by the time we get so we can cause screen here is what we measured electron screen panic at the screen all year the maker of arbitrary assumptions that it is in state summation sigh In area there has to be some linear combinations of these bases vectors they correspond orthogonal configurations is a different places don't get mixed up with each other the I over here it's not here it's here it's not here so we have these bases but this able and this is about the only thing can I am that warns equal warned goes that 1 becomes after of further interval of time it becomes so when your program but that it was not stretch area 1 of the leaders of a literary 1 slowly past fast start for it the rise of your yes I could by decorated with Romania symbols so this is good news to just the just 1 cent on separate out there I don't have to be so I'm not writing in is equal to anything else same state 1 over here will evolve into some states sign in and what happens if it starts at minus 1 mother starter minus 1 L . some other combinations and not in the trailer combination his other save some of other fine and carnage Air Force war use or a merger of the His are arbitrary linear
combinations over here they could be determined by experiment that might be determined by mathematics but right now the most general possibility that start at 1 over here you will get to some linear combinations characterized by coefficient stardom minus 1 who get to some other linear combination 130 with what happens if he starred in the initial state 1 plus my hope the combination of both of these the linear postulate tells you that became the final state will be sum end of the some of these which is fighting and plus IN times and other words by the time you get over here the coefficients will be the sum of the coefficients that you would get by going the opera route and going through the law if you go so the operator that even if you closed off the the lower route and the experiment would only involve 1 going to science to close lower to close out the upper routes the experiment would only involve going from minus 1 would involve this combination over here if you open them both and allow the electron could go either way be banned the rule areas the laniary of quantum mechanics tells you that you should they add the state vectors you start with a linear sum of 2 vectors you should edit who in the final state a get your job is to normalize does a because they don't coats while a way to get for us it is the last all the Riviera or if you cut off structure which only through your may be wise which has suggested that you have said they want to use but moved side by side another man this has led to the village which is vague about normalizing here at the moment they didn't want to worry about keeping track of total probability that sets summit will do next time it's called unit parity arrival worry about a right now our words normalized normalized muscle portend the important thing is that that you get some of these coefficients going through the opera route and the lower route and that's the postulate of linear arity that if you evolve from 1 you get signed a few evolved from minus 1 you get funny if you evolved from the state 1 last Mike this 1 would want to see work foot Hi this stateless this state that you get some of that past your minaret what's the probability B the daily that you measure the electron at and for the end and site while these are the coefficients that end up to give you the state vector pieces of auto normal basis it's quite clear that the probability the probability that you in the army and site over here is just and passed by and complex conjugated times sigh and cost 5 and we could do this rule projection operators were just use the simpler statement that the squares of the coefficients Whitacre coefficients year we complex conjugate square them and that gives us the probability that we arrive at the end site and could be 1 2 3 and 4 forgive us the various probabilities for arriving at different places we get we get terms were workers out we get sigh style and of and no 1 is that that's all we would have if we closed off the 2nd hole if we only had the 1st hole the answer would just be sigh star and am so what this is Is the probability that had we closed the 2nd hole we will arrive him and placed on and there's also start 5 that's a probability that a free close offered the and went to the bomb someone we will arrive place classically that's all it would be the probability that go from here here by 1 of 2 routes would just be the sum of the probabilities of going through the opera route and the lower of that's all you would have a classical physics if you have a random walking particle that ran the walked around you left up whole opened you would have a certain probability forget here if you let the law plan opened you have a different probability to get to the same place if you let best also opened the probability would just be the sum of the 2 probabilities but require mechanics we have 2 other sighs start a loose 5 and forced by Starr and and have a right yes we give you an example let's consider the probability that we get to the symmetric point over here for the symmetric point we might imagine that's 0 it's an equal 0 Nickel 0 we might imagine that signed fire equal to each other wife because 1 of them co-sponsored going to be up a wholly-owned co-sponsor go into the law hold up by symmetry we might just assume that the amplitude to go from here to here is the same as the amplitude to go from here 3 years would be that case enough I would be the same in other words if we weren't we Colombian 0 over here a probability to get right here central . 0 0 0 the Hall a answer it is for the war times the probability that you would just go through the opera hold if only hole was open and that's why you would have expected classically just twice the probability of your opened up both calls you expect me and so would be twice the probability to get here not 4 times a probability get to hear but would you get in quantum mechanics in this case is 4 times a probability and the reason is because you end up be amplitudes and then squalor of the amplitudes are equal and you square you get 4 times a probability that's a little bit odd but I think it's not that hard to do something order by putting in the right of the apparatus over here you could change you could make changes in the system basically just a little love potential or something that brought out ruled the effect on electron as away from here to here because I have the effect of changing the sign have the effect of changing the sign if they change the sign to lure also changes 1 in that case of sigh of Zero was equal 5 of 0 0 they would cancel and you would find Zero probability to get to this point over here this is destructive interference before it was constructive interference 0 is destructive interference and what is it saying it says that there are situations where with certain probability you will get here going through the upper route with an equal probability
she would get here going through the law route but a few at best routes you make it impossible to get to hear that is that that's actually for my money that is as far as the bell inequality backed by opening up you would think you make it easier to get to here and in fact it completely blacked out the possibility of getting to here if you start with a minus sign here but minus sign doesn't affect he never probabilities for the upper route of along a road it only affects what happens when both routes are open their throats open becomes completely impossible to get a year despite the fact that you only opened 1 road but this 1 off and get to hear about this off you get a year equal probabilities of no electrons will ever get to this point as destructive if you're so it's clear that probabilities in quantum mechanics and Rock doing the things they do classical worker or how he both those holes for the time unless they like the 1 that the law so that you we change from destructive but it is very very short of the electron source of the Middle Street might be it started just as it was being near east where did you achieve results that were too shaky experience so fast that it might have no had a bye bye bye thought is being case that the proper results said you that I was sick to change the sign you might wanna do something In the leg over here right right well right regular Watson flight but that would depend on whether you may be change before the electrons that you Ieft electron lasted just higher live but a smaller time travel yes but is that make that change there's nothing wrong with line the others were also saw how did you may change what we liked what you made but there at a point is that they should have their event over the air raid ways that would are somehow will is voice of along hold back there has been only that rarely violations just a probability disappear from over there were probability distributions right is not just the probability depends on whether was once Ciliberto slips of course it does it so the probabilities on a hat it's probabilities don't have me amplitudes heads the probabilities don't aired you get probability In either case depending on that sign what times bigger than than the probability to go through 1 opening with the other 1 closed all Tezu 0 violated their end love was that final results about issue here again for long as that Vibe them the thing is we have a rule is a shared Saturday head of there's lot of my eyesight and some of them villages as part of his game on of an Arabic I think figure after give more precise shut up over now I'm trying to do trying to to study reasonably precise setups and we will have to investigate each 1 separately but what I really want to get was every year to rely on the public to back it is what happens if somebody over here it is an apparatus or here which monitors which way the electron gross 3 which measures of reports which way the electrons grows this is the interesting question here I have assumed that nobody has recorded which way the electrons God's will so they can expand and you can't really say the electron went Trujillo through here you don't know which way went throw and there is no work no clear notion of being able to check afterward that electron went through the upper or the law Hall but suppose set the problem up away that afterward you can check the something which we call or whether the electron goes the upper hold of more then 1 happens the probability distribution well you might think OK as worthy of the body measured not careful measured in such a way that doesn't disturb anything very much very minimal gentle detection of electron whether goes through their not you shouldn't disturb the system's very much but a quantum mechanics it's not always possible to do minimal gentle experiments and if you do an experiment which does recorders which way the electron goes you will be the answers in a rather dramatic way and basically destroyed interference basically destroyed be positive interference of negative interference and show you how that works this is an issue of entanglement entanglement of the electrons with apparatus which measures so let's invent an apparatus very simple apparatus for electoral goes through war the upper just forgot a whole over here all over he electronic resist where goes over here and here's my assumption if electron passes by this whole well would open apparatus president simple could either up or down the spinners you up or downed along the 3rd axis we'll start with this big down and see electron goes so the opera hole here it requires its presence by spinning by by flipping the spin Art if it goes to the law it does nothing for the electron This is the apparatus which recorded which way be electrons goes 1 extra spin it could be another electron or it could just be some abstract system with 2 states and if he electron goes to the upper here recorded presidents by flipping the spin go up let's duel but problem I think we can I think we will see if we can get the problem done but we will go through it again I think because that's probably the most interesting and vow have ahead of us is that weaker but a detective all too but we need a detective in the because it doesn't so here it does go through here so are we would have a funny situation a very funny situation if we could detect both places and both of them didn't recall ordered but electron velvety so presumably that's forbidden to really only needed detector in 1 place to tell you that you are tried to go to Europe didn't go through here the what visits electron which I go through both a big enough to go for broke he was suing the other right if you measure it will go through 1 of the
other but if you want you will find that goes to to 1 the other if you don't there's no way that you could know which 1 went through if you don't want security and you could go back and check that electron peekaboo another 1 over here too and analyze it but that that we would have to electrons and get more complicated apparatus would certainly do it be fun to do it and see what we get OK so let's then side we start with electron over here it starts in the state 0 0 but that's not enough information we also have the specified the state of the spin and now this is a composite system composed of 2 parts electoral which can be at any 1 of these sites have started 0 0 and the spin and spin starts back so let's put it in the course of the spin here that's the the initial state spitting then on electron of 0 the electron comes over a year no 1 happens if it goes the upper stage was close off this forget for for a moment ago so the operas upper configuration the electron comes out of the configuration the the whole system electron the whole system that comes out in the state or 1 because this His position number 1 position number minus 1 comes out 1 bowler hat electoral studio spin its up so if we go through the upper Hall the state of the system is 1 off what regards Alar I hate this notation 1 minus 1 I should not like this 1 and now for comment what happens if I opened both polls about urban bus calls by the linear Neil quantum mechanics state that comes out the other side here is some of these 2 1 last configuration electronic minus 1 thing now stop but I will actually comes out for electron arrives at this point arrives at this point the again we have a superposition states but not sets and singled out the position of the electron has become entangled with speed is very much like a down not we are entanglement between Lee recording apparatus and and these whether the electron went through the upper hold or the law that's what a measurement visited entanglement between the apparatus in the apparatus is just an electron spin here and the system the system being position of electrons that went through now what happens after that what happens by Ricketts over the screen or in your life who is on loan from each well let's assume and nothing happens the spin the Spillers up up stays up of electrode proceeds to do exactly where we're darned after it Pegasus through OK but assumed that just continues on its way leaving the spin pop remember previously won way tool summational and sigh and a vessel firm over here while if nothing happens to spin during the journey of electron over here we can just stick in the spring quarter and just say all excellently up configuration with spin-off evolves exactly as it did before without the spin doing anything spin just days after what happens if we start with electron lower stayed here I was this warned that evolved into some fine and but now we should put recorded of the spinners D 1 happens of both routes a of both groups are open then in the final state is some money here I can with it In Nevada all right and with in barrel again it's entangle state media coordinates of the position of the electron entangled with apparatus would be up this in the we cannot calculate and getting a little bit tired I think probably go 1 next week but it is a problem what is the probability that we arrived at a particular now we have a different situation we have orthogonal possibilities of spin the calculate for or murmur quit now because thank God I think are probably give you more than enough for tonight he is up problem to calculate the probability that we arrived at the embassy site In particular the probability that we've arrived at the origin of 0 labor nor the answers notably the ball just because some of the church and the reason is no cluster because the cost terms involve the inner product down with enough me in a product of down with 0 the cost terms won't be there anymore because when we go to calculate the probabilities ups and downs orthogonal to each other and the Ross terms will be on the go through its detailed views projection operators ever will would do a dozen different ways but what you will see it is that the presence of the recording device over here destroys the interference so if you actually did a measurement to determine which way he electron then probabilities would as they do in classical physics it's only if you don't report which way the electron grows so that nobody will ever know there's no record left in the system which way the electron when that P. interference takes place but the message for the moment is that making new measurement is establishing entanglement between a device and the system it's establishing entanglement between devices in the system and in this case wares part adding these up heading the state vectors has an entanglement between whether you're 1 in minus war however the electron spin or service Spain degree of freedom of backed off be destruction of the interference is what is called a year collapse of the wave function in particular it's the collapse of the interference part of a wave of public but which has to do with the cross terms between the size of the fire was drawn it it for you to play down there will also be few years they were saying there are also formula for a few days of world there is a exists for this sale is exactly say yes the side of the ball from 1 of my right off the bat Helen got my entanglement do have an entanglement with something but not as just a simple superposition state simple superposition of states it's not as only 1 system is only 1 system and electron
and it's in a superposition of states another case the to systems electron spin and 2 systems are entangled you learn something about 1 of them by looking at the other so you learn something about which orbit you went through by looking at these but looking at stake here and there was really are now they are here yes it introduces another subsystem direct introduces a system into which into which the electron with which he lectured gets entangled set establishment angle between system and apparatus which constitute a major advance in view of the was there a area which is also hold an herbal also was also so strong so the letter and used racial attitudes electoral while that where there was this the arresting arresting the offer is excessive errors is is all the more a Yom I think which arrest I think that's bodies a real the real thing sir can this electron get entangled with failure with a with an hat from Indiana in media yes answers it could if it does it destroys the interference pattern so that's why you call on your apparatuses that's for year you don't you don't want to have a lot of code of gas around the electrons and bump into photons are not so bad because both arms don't interact very strongly with anything that much more weakly interacting electrons if you really go with electrons from electron passing through here it's electric charge might disturb other might disturb us something over here and that disturbance of the disturbance was enough is of his quick care if the disturbance sufficient that when you look at the disturbance a afterward you could tell definitely be electron went through the upper select the new completely destroy the interference pattern they might have a situation like this you might have a situation where there's some degree of freedom in there which initially had some probability distribution which looks like some degree of freedom in here position of some Adam was something and it has some probability quantum probability distribution that looks like the electron does so here Murphy it shifts the probability distribution so that it's not overlapping then would you look after afterward at the system you know yes No whether went to be up along law 1 following me right under that circumstance that will destroy the interference pattern but there's a less severe situation with electron going up a whole he a major shift the probability distribution a little bit and then you can't tell uniquely weather by looking at the final situation with electron went through 1 hole hole then the be interference pattern will not be completely destroyed as a matter of degree yacht about how much of a disturbance and about how much that disturbance tells you that really things make a real cold and put everything interludes ground state was nothing that is easily disturbed then you can create pretty much the idealized situation where FPU gone rule this just nothing the recorded in particular the system has an energy gap takes some energy to make an expectation in the system works ideal to disturb it and the electron what time is very low energy it may not have enough energy to make the disturbance in that case you have the interference pattern so it can be a question of energetic whether there's an energy gap that it takes a certain minimum amount of energy to disturb the system and and a low energy photon low-energy electron just can't stay bit new word heard you would have to be perfect interference practice then what will you can't see the interfere you could never see a probability distribution with just 1 electron but what you can do was send 1 electron so at a time when electron time every 20 years use in the electron and if too many many electrons go through you look at the probability distribution here the probability distribution as the interference pattern 1 electronic just goes blip somewhere is number 2 Garrido of backed off now as always in experimental always really means any repetition of the same experiment over and over again not for many many identical experiment performs simultaneously but means the recording of statistics In the end after many events of the same car so if not to be clean about it that we don't have to worry about 2 electrons at the same time we send electrons through once every 20 years and make sure they don't talk to each other and don't give anything away to each other although it is I voted for it always has there is right rising not creates a home loan differ which will play at NatWest will write right in this but the 1st year but there can which corresponds with his former wants becomes the apparatus of talk of a fire but that would be a little the House by a very low level measurement over here with dawn of the measurement over here that measurement here would go further collapse away function and throw away some of the entitlement to write about this measurement tho the here is the establishment of entangled the 2nd measurement could be establishment of entanglement with yet another system if we put another electron over here it's a question of observers observing observers observing observers we could even have an electron which would most which monitor the spin-orbit here love you we will do all that we are little air you're changing that they had to do a little girls what those PG freezing about Yahoo could you can you could turn up the interaction between electrons spin Our so for example if you have a very weak interaction and most of the time the electron went through a would not flippers are 1 young the we are not could certainly that's right that's where we could make up some of the game a game that I made up is that we did a complete measures we did an accurate measurement told us whether we went through the upper slid over laws slick now we can talk about weaker measurements measurements were we only get some partial knowledge will you find out that it was more likely that it went we up 1 will do your way desire that they would not get that's right that's right of 1 being up there will be a peace with ban on another piece would not right among them might be the big 1 in which case you must we do have the interference a week Armstrong a measurement
I wore blue Roy the
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