Quantum Entanglements, Part 1 | Lecture 7

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Quantum Entanglements, Part 1 | Lecture 7
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Lecture 7 of Leonard Susskind's course concentrating on Quantum Entanglements (Part 1, Fall 2006). Recorded November 6, 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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experiment again I don't think we ever quite finished so let me finish that discussions were experimentally made a very simple are version of a truce with experiment was source we I called an electronic could be a photon could be any kind of particle whatever but let Corbin electrons is used to it electron is emitted from the source not feel like you go imagine that electron community from anyone are several places 5 areas a barrier for me electron or he with some openings again the openings could be in anyone of any of us several places but for simplicity let's just suppose that there are only 2 openings are on column last time I think about called plus 1 like this 1 I hate that that was a good it's less common name be this time any errors P I think this . 0 and the rest of the electron can grow frugal Hall at 18 growth will hold B and Justin Nagin arrest was blocked off so from electron its anyone the other side wasn't was to be you could just pretend that it gets reflect back if it arrives at a candle Bush led arrived from where a final and over here and I think our label lows points by integers solve this could be an site over here at 0 site revealed and equals 1 2 3 murders 1 rooms tumors himself what was on electron over here initially we haven't discussed where remarked house change and we're going to to do that much tonight effect but the but a arrange you we well but they most important thing is that states change in a linear way that means if you start with a linear superposition of 2 states with coefficients where the evolves into was the scene linear superposition of what I have at any 1 of them Robert through source for example are 5 work we started with an electron state 0 incidentally president purposes be electron has no fear we don't have to worry about electron spin etc. relevant for moment it just is add up place and the states of the electron or simply characterized by which 1 of these vertical positions that so all we saw electron right center at . 0 romp it involves end it evolves into a linear superposition states and a and erect are no particular prefer reasons the Garvey and supposedly rises that that so it has a goal epochal amplitude To get Sosa crossed through a posture bait I was a course they mean means state off in which the electron could be a very you could find it were probability half of a technical problem but McGriff OTTAWA probability be probably would have beat them just the quantum superposition of 2 configuration that was if they can if it gets very move on move on screen here so we cursory AD evolves into some supper position or arbitrary superposition nothing special Oussama armed and are some coefficients which represent the probability and withdrew its size of the end of charm now being at and place as happens if you want to start a wave think about it is to get the whole be altogether start an electronic AT and wait a while and I will arrive at a superposition states with different amplitudes size of alright if you start to be your also of all think I call we coefficients here fight cars OK so let's but 1st begins with 1 Hall completely closed electron can only go through Kinkel through be altogether we can forget this requires about and probability of electron arrives at the air strike he ever is just going to be sigh start him and With blunt focused on me and I don't worry about
any of the others the amplitude of the electron gets the point and his side and the probability that it gets there would be called P & sigh star I'm sorry that's a storming a because road another words as if we have blocked off Colby they should give remain stricken aII over here that indicate that went through the opening likewise thereby close up from 0 Haiti and open up a whole beat then the probability five-star em fire Our classical logic classical statistics classical uh probability theory statistical mechanics if you have a beam of particles Our which was Our Mark made with enough precision of some hours on certainty in some other particles Michael Rose a summer may go to beat but our expectation would be that the probability forget and would be the sum of the probabilities of going through a or going to beat Sabres probably not start Quebec mechanics In quantum mechanics or would have the right is that if we started 80 we get year precise B. we get here but if we start at a Class B a restored would be state aid plus was the electron is so of equally between these 2 men or we have to get a Class B will grow to summation of end of and friends and what size of Empire and my guess at and class Vice Obama plans Americans as represent that by giving site the fight so it just becomes From sigh and plus Y M Prime Air Our let's let's forget for the moment the summer just focus on a particular point but images of some here just focus on a particular point and what is the probability the electron gets and the answer is the squalor of this amplitude of square it times its complex conjugate areas he and the probability forget the M 1 a and B are both all them 5 of just square about which for terms star M side and five-star and fired him class stars and fired him last society 5 I yesterday the start of the tour until probabilities plus an extra turned extra terms Colby interference interference between the reject tragic theories 1 of em going we are all aware of the north law not dialog in any kind of classical statistics the sector and has now where it is for example if it ever happened Hawaii right the symmetry point here then you'd expect sign a N fight to be equal by symmetry and this will give you support times the probability of going through Idaho it's not hard to arrange by going off center here you can find a way out of off-center or you could do other little tricks are being particles here to make sigh sign might equal the minus 5 and then you just get 0 probability of despite the fact that electron get him through a or through if you open a N if you open a and B in that particular circumstances electron Falkirk SMS weird as anything I nor quite mechanics serves writer simple Arthur but Malcolm's a question which we started to talk about last time we reviewed finished our review again what happens if somebody actually president and actually watch is whether electron goes the 1 or the other and even recall whether it goes through 1 Hall Benoit half surely then you must know what electron went through a or B so we made up a yard little system we ended on other element here we enter the spin the spin can the br poured down it could be electrons Spain itself is not always way could be electrons by simply want the think of it as an external measuring apparatus that recall the the presence of electron sorter more think of it as part of your charm of Suffolk as part of electron we and another degree of freedom over here which is a spin which be off for Darryl upward on this what's not ask exactly how it happens but when electron passes through the whole 80 was most be when
the electron passes through the hole at 80 buyers Is it gives a kick the spin and reflects that assumption assumption about the latest spin located 1 how it interacts with the electron we measure all over and over again that when the electron does passed by the spring bus to ticket so we know what happens electron passing by the spring of over picture from receipt from Babcock right now listeria OIC experiment will just have an electron we have a electron and the mother spirit story have AS something state vector here we can't just described it as an electron . 0 what we have described it as an electronic . 0 with the speed down or are apparatus on activated we configure rivers the apparatus being on aggravated are Erin straw mats or initial state if he elects on most of the opera hall begins it becomes state AD electron . 80 by the spend up if it goes through a whole be the spend a electron and stays pointing Barrow so now we have something new ripped through states a are collated and entangled with a spare their entangled with a spear such backed by measuring the spare you can go from through overhaul of the law well what happens so ATR once once the the electron departs from here longer interacts with spare us so I assume that it will market the spin all over again although happened is it will evolve exactly as a while before except the brand will remain Bob not measure this is a while your way class aware practice with strong the I thought had a year we're changing them fast 1 your focus yeah that constitutes a measurement of York truck and constitutes a collapse of the wave packet of Elektra with fathers a compounds its system of electronic against no doesn't collapse forward aware co-ops away packet of a combined systems for something else to book ever vehicle collapse closer so if you could call it created plus that you will never be you plus Beattyville for an hour collapsing the wave packet is a language to you if you don't want to remember that this others then you forget the others and you just say the wave Capac collapsed fever a or B by the presence of spent by the presence of the spent hours you keep this didn't present to keep been present them over which if you include the apparatus including the president of a description and you don't talk about collapse of a wave packet you talk about establishment of entanglement but there system with respect could be anything it could be a pointer that has states it could be the emission of ballpark so that up a server ban could stand no photons and up could stand for a photon beam present so it could be anything which distinguishes which of the 2 homework even if it was a very good job market at was lost mobile at the polls young age more questions whether left on mark for the euro area now answers yes she could leave a mark it they could leave a mark by that depends that depends on details on the metal plate and particular if a metal plate has an energy gap pervert pixel on energy toe excited electrons low-energy you may not have enough energy to excite video hero of anything interesting system here so it will not leave a mark the bus separate reversing we will mark a Of course they Brewer as if us by but it will be at your best friend for a moment studies at site or are you asking whether we were were resource are you were asking whether you couldn't tell which wholly electron went through overlooking the recoil R yes got there but said that the electron goes to the upper hole and obviously because it will kick upward to be grow symbolic as a little kid downward so it changes be momentum of the screen here right now because I'm you with getting a little bit ahead of ourselves in tour drew more complex situation opera story but let me tell you what the answer is yes said is that we are being screener itself has a concert pianist momentum there be costs it's localized space through the uncertainty principle which we haven't talked about 5 screen hero what we call a barrier the barrier is well localized because it's well localized that has spread its moment that's now that's be momentum space so distribution that's a problem brewing distribution and it's pretty well spread the reasons pretty well spread is because location is very very nicely definite of pretty well spread bid electron the all of also the alcoholic will kick and shifts the distribution a little bit the point of the law hold which shifted the other way but you can't tell uniquely facts as long as the as long as they can a small by comparison would be spread in the momentum space wave function you basically a it is almost completely overlaps with yeah you don't get kick for North fog state of the year of of most of us ever had better there that's a very very good question but that was a very very good but they were locked up Juliet of right right so that's it's exactly the same thing as saying when the electron goes through here it takes down top instead it takes through all almost down plus a little bit about for the overweight wherever there you command that will allow you have different degrees of the structure of the interference pattern by what they extreme example will you would do with a complete measures for you do an absolute measurement which means that you establish a complete correlation between a and be on the 1 hand and and Barry other Harris server are looking at the speed you could be turning completely in unique which of the 2 holes your charts for Fatah leader that's the notion if you like of 80 express ovary over here a measure of a measure establishing an entanglement between system apparatus apparatus just beings been OK so let's just let's just to do these and that not all the i's and cross all the fees and now In this situation to be with BE did not kick me electron optional BE of electron goes beat the Spain's stays down and they just continues state bound for for now we have a state of Israel were very are workers and Pangle between the wave function of the electron foggy on the 1 hand and but let's just go through formal the very formal motion of calculating the probability the electron was found at point and stopped there are 2 ways of electron to be found again it could be found with with electron and end with spin or electron at em with a Witherspoon back so we have to do is build a projection operator
onto the states with electron is that point in more the Ruiz you have some property that you want to represent by a projection operator you take the subspace corresponding that property you find a basis of orthogonal states have space and then you build the projection operator as a somber have our some or all of the people all the other they suspect in the subspace our duty I had composer while the states in which the electron em What do states them pop and off 1st and Barak and that on budget those are the only 2 states in which the electron is found at him so this is a projection operator onto a stay on the configurations with electron is at so now we have to do is we have to take a final state was the final state of final status of some of these 2 these together now we cannot factor out em up and down right in this form here because each 1 multiplies different get Siam multiplies 1 can affect the sorry of fire and more bars America Violette's let's acts such as jewelry so bite the bullet and do the whole thing and not any more room em and bound and down that's a projection operator of the army now we take state back there the state vector Our ears Romney 5 some are end yet at a summation Rex there some money and came up sigh and crossfire and there you down likewise Algeciras here we have the same thing star star errors of sigh star and end up star in and backups of but now 1st of all there overnight across the non interference and life-size starts I was just look at that hour so I am here but this he's here as an Opel lecture Have the cops and it can only combined would be up spin-off here it's on final the dams so we a product of an off for the value 0 you only get a contribution From up with and furthermore you only get a contribution when it is equal pay him in a product of art and is only nonzero with em up when a musical to air their musical and then he'll abroad as water OK so out of here you pick out the single where M is equal to enter likewise from this side area has the equal in an are past equal out otherwise you don't get a contribution song from the Barak terms you get sigh star and and also was similarly star can fight that 1 comes from this chairman projection after a while you get nothing from across a pig crossed our colleagues when you try the combined down with enough RPM kind goal up here but then operate here will kill them down over here the projection operator was not allow you to combine a fine costs bearings Oral farmer Maryland the various I so he is the result is the result theirs if you do a measure the result is the sum of probabilities exactly as you would imagine classical physics Soviet or look at the root of electron cakes nothing works not just you'll known what is left you can ever or anybody else can ever determined after Woods with electron went up or down approved that Loroupe failure is interference pattern but probably where yet know about this probably barred from robbery the hours of Korea yet Vice yah yah but the main point is the main point is you've got that Mayor not just what f the was are you that the Mets not only the and but down a these state vectors and that will cancel or kill or destroyed the interference patterns between size of part of while thought think that laughter at exactly the same as it was classical classical bricks not to others and get his years at Eurostat and if there's this yes that OK I said if you'll start to review process be of a few hours spare Yap got OK let's suppose love right let's suppose the spin is located in some symmetrical situation here where gets equally Our whether goes through a or B they will come out let's see where come over here then then will just have a payoff past of jack up be bright and they will be exactly as if and the Buick and they will be in prepared at the end you will not be able to tell from the fact of the spinners up which hold when thrilled so it if the Fund but and with that would leave us in this situation Qatar Hot Wheels violate crashes Botha do and I do not have the measures from electrons was measured electron has done the measured electron was on the measurement not of electron spin electrons from the measurement which Callaway I don't have the look at electron Ike so I don't have a look at spin the nor that the interference pattern will be destroyed don't actually have a look appear work from just a fact that the information is recorded by a system I could look at it but just a fact was recorded was recorded in and the orthogonality of up-and-down that's enough to destroy viewed as 1 of half a look at your chance I just look at electron Roger I wanna have a look at the spear I just look at electrons coming from rules have a beam of electrons coming through this over and over again I said electron experiment electron arrived somewhere and then I put these beamed back in its original position go the whole experiment again but this is back in its original position Delors experiment began and gathers the prospects what the probability is electron arrived in different places I never actually have the look of a spin Our all I have to do is look at electron Stewart came out of our will discover that the was north of Paris not it is sufficient to ask for a walls is yacht fathers rights are if iffy Laura another we are basically momentum wave functions of 3 screen years narrow enough so that when electron goes through
a dessert kicks and kicks it to a configuration which is orthogonal original configuration and they will destroy the interference right in fact is another way that interference patterns of astride yet the electron now Mrs. Sobel very but if electron goes the opera hall here then in me and slow electron Herzl electron pressed it will e-mail some radiation every time the electron accelerators has to be accelerated to go from here he ate it will a so all are need a four-part may amid after we were for me and that of the ghost the law Hall also may report on the probability of a meeting of all time considerably by electrons a lot what's about 1 part the hundreds low-energy electron so only 1 part of 100 over hour our because in electron so it does leave a little bit of information in the radiation field but with only in Haiti and a photon 1 part in a 100 you cannot be certain when you look at the U.S. system and discover there was no fault on you can be certain which went but on the other hand if you really would be electron through their fast bowlers a high energy electrons and you really are arranged it so that the electron went through 1 whole way up on top of hallway by Obama Bob so that really have the whip around accelerator lot and the likelihood of emitting a photon would be larger any interference pattern would be more destroyed so just ability to emit folk enough to partially destroyed interference pattern but those 1st as you go France probably herbs that SO what you say it is a case of a truck full probability of VAT at source guess what while sources here so source of distraught parents who are both of which prior to to her says those sectors that are areas that are old goat but in fact our duty to about 3 years researchers usual number show area so far vote a bus a group you have wars you pass it says that for Tuesday's Kate Moss she will turn back crucial that yes yes but you may also have such up partial situations where you by looking at something you may be able to say it was more probable with you when you have a horse or yeah that is no for example of electron passes through here instead of kicking back prop it just kicks the year just 60 of directs spin a little so that's mostly still down just a little bit off that will partially that will destroyed the interference pattern a little bit Rideau guy a lot of time at the half what can I tell him that we won we got got lot about what what you could do was you could collect statistics only for those events in which a was injured at OK and then you would find the interference pattern worse now it's not quite completely clear because of a both that the photons emitted from here it may not be completely orthogonal from of photon emitted from here Our away from from the photons may not be orthogonal sorry could still have a little bit of glory 0 Rory situation but can you you can destroy interference in many many ways just the atmosphere and this is this is this is the biggest source ah interference destruction in particular why don't you see me as fast are wire I assume you answered back at that hearing you don't that you close your eyes call close your eyes are really close your eyes and now often you find in the same place you don't find my wave function having spread out over rights and thank you could wait for ever and ever and ever and you would not see or the needs wave packet thousand bird doesn't spread part of the reason is part of the reason is the was heavy but other reasons years where it's constantly bombarded by atmosphere by other things and so it's as if it's continuously measured every time measures at a few continuously keep measuring reference it will not spread terror your 1 of 5 I forgot while would not be returning to the remote that the leaders he again and not a lot of time alone I go well I'm try to give you an extremely precise example I give you extremely precise example and now we can extrapolate from that we could work out some other examples wickerwork work out some other examples of which partial experiments about over straightforward and give you crystal clear example our way no ambiguity so that you could see the mechanism by which they have destroyed any kind of work do you could do that don't it was not worth millions not a father collapse of a wave packet collapse of a wave packet of means collapse Of this wave packet which literally means it you throw away the off Barry be mixed terms of fact is you could describe it that way Mr. measurement a collapse away factor but you also want to be able to describe our composite system of electron and the detector In a way that doesn't involve any Yom you're your hanky panky with collapsing corresponding correct description of a composite system is an entanglement place so that's where you're that's the importance of entanglement in doing measurements of obviously it's a very important for the part OK if the will of a lot it is not a lot but does not both get because 5 4 in the period gave or the group based breath thought that a good faith color the like beat back from 0 2 8 of em her yard flow but it will of argued there is no well before hat but you Comoro electron accelerator or a corn bread there was a math see things that AT & T E or as a Messier that they make sure that there will be change it is I plan to get bigger 1 resource not be deflected but the other 1 would be about if there were we can think of a thousand million variations on Austrian and the story anything tags the route who leaves very Ma telling you which way the
electron where destroyers interference over stripe passed every gives you some probabilistic information about which were electron warned not certain information but were firm From buyers forward direction a real definition of her parents have reacted to learn to our that's all as a matter of visitors say they love of the same name as if are to be interference pattern here where there are many many electrons ordinary folk art from support when there are many many folk commute through what once the interference pattern is nothing but the interference pattern of avoid motion a waiver with but electromagnetic waves Our factor also happens for electrons cows you with some sort of weighty associated with electron Osaka work magnetic waves of 3 electron functions more but rather that are said told him that created said there is a wealth of the city about yah yah all we're which which would put between AT&T young but that if that well we're just assuming that way benefit arrives anywhere between bounces off repair the region's closed off rock just off I I I just want a doctor said on a Motorola can knock in a superposition of aligned their captors not a superposition alive in there that's exactly was not true wife what is sure is a cat is and tangled with another system was entangled where are the question of whether the bullet was fired on compounds assistant lets Australia because sorrow of bloodied and cat sought to calm posits is gone and cap and gun can be interstates Tuesday's part 1 fired and fired the cat captain me into states I think you start alive cat and Odin fired a gun you offer on fire and then something happens you wait awhile and after a while the system evolves into white cat was some coefficient alpha wife kept on fired a gun for some other coefficient big cat fired God is not really superposition why of course there it's and entangled state with regard to stop you cannot go any kind of interference experiment between life and death if you were to calculate the probability of life and did you would you have no crossed after its b b b gone itself the firing of the guns have left a mark of firing or not firing You don't point is you don't have a look at the cat fine them so I you could look at the guy so I you was tangled hair to much basically exactly the same kept destroying to experiment is not a superposition of did the calm posits system is a superposition alive on fired get fired if how at the start of a work of art I want a measurement that places that well says a I man by rural walk of life while reading a book about well that while whilst reviewed his or her breath Bob a if there is a probability posture would be quantum mechanics you construct projection operator and take its expectation value and the probability know what they beyond that Our that deviated a it slower progress of while they were both food or you are girls we've not G 2 yeah I know less about G we're I about where you could talk about it you don't wanna talk about have a a B mustard measures if you measure it you can then go and ask the spin because so he wrote in the Book Review and you will have a probability for having 2 years ago now what about this at this end at some point you have to do any measurement as to where the electron you'll be electron has to do something it recorded the fact that has arrived at 1 of these positions we haven't talked about that Iger said want position but if it arrives at 1 of these positions are not that kind of measurement can happen kind of measurement would be the blackening cover or it could be recalled for example 1st in in every 1 of these sites if you're like electoral arrives that point arrives at this point the flips the spread out most of the end of the day I am imagining that we do some experiment here before that weary lecture arrived so suppressed or their part because they did area is much less a lot harder he then had to issue in some other way back anyway and struck Bob Bobby completely experiment you actually have the measure with electron with electronic means that some sort of Marquez left from the system by way electron arrives over here it could be because spins aligned along here or it could just be because when electrons arrive on photographic plates they blackened are spot on your plate are there we look at the Black Spot we are actually look electron were a black spot but black spot becalmed tangled with the position of electron troubles here this is because they had not yet is but process the process Our separating your system and that which is measured in that which measures is always ambiguous you could always our I look to expand really doesn't happen until I look at the space in that case you would add another element or you could either just say this combined wave function here collapses or you could add another ingredient in meat and meat also has 2 states mean having seen the spin off I'm having recourse to my brain and mean having seen spins down recording that my brain solve we go because once I look at 1st revolves something even more complicated a and he's having seen off fast be now and we haven't seen them but then you could be watching me the point is a degree of consistency is a degree of consistency that probability defined car a or B were often down is the same as the probability of 80 up BCE not be bound them sought some poor it is no port was no point in going further and haggling anybody
else but if you did you would still have consistent of version had earlier had told us that smoke about salsa but in in the end but ensuring goes on the outside of the box in these bond angle but any looks Ian any sees 1 or the other as trotting himself becomes entangled with box for guns and the cat the fastest was waiting outside the Astros what did you see he becomes and none of the whole world Riley ahead of holiday for the correlated the consistent way but yet Lloyd appearance not appearance if there are none busses be nothing to recorder which things happen 1 more aware of this will be the across reiterated that beach absolutely sure ought of 1 particular instance of it it was these you will strike about you might worry that the required all of this screen he might ruin your measurement by leaving a mark as to where we were to be of the whole of law is the uncertainty principle the uncertainty of the momentum of the of of screen here where charter doesn't allow you to work to determine unambiguously which wholly or from role certainly there are certain principles for his role here is really the most important thing that while not most support necessary but would you should keep in mind is that the difference between a classical experiments and and ADR and acquire a chemical classical experiment you always imagined a classical experiment can be done in a gentle enough away but you could find out what he once the find out without disturbing the system without disturbing go the state of the system here you concede that we have definitely start of the state of the system our measurement is not done aren't Phil b entanglement has been established in establishing entanglement is a significant change in the state of a system of measurement is not done on Tilak entanglement established once that entitlement is established it changes the answers it changes the answers whether for example the reserve isn't interference pattern so classical physics you always a magic that you could make your experiment gently enough that it doesn't influence anyway this is studying or perhaps a arbitrarily small amount over our changing research firm was quite mechanical you are forced to work in order to do an experiment to establish paranormal promotes a big change of Cecilia forgive but this is a day he says it is larger than lot of press said that while about the depends are rare prepares your assumptions arms are nothing particular wasn't it killed Of course when will build resume make the probability of glory either of the holes very small so both the size of the fires will be very very small holes we far away the overwhelming probability will be better hit the wall and bounced back chances of getting out very far distances much more right there but but but apart from that of his I grew up all the well lucky but there is planning a new Thursday this is fundamentally true Oh no no no and no well that's another question about our talk about this over and over again I keep telling you I don't know the answer except I know from experience how hard it is through a modified things a little bit of our signed that answered question because I honestly don't know the answer it's and purple fact and it's a basis of quantum mechanics it's not it's not easy to make a logically coherent story hour of a slight that of nonlinearity becomes a mess now relief sober the replacement of 1 kind of logic by numbers the replacement of logical logic of classical set logical subspaces wiII be and food wisdom decided on subspaces that something else about maybe the Rev. Haggard yours Royal T juicy bit of sorts for after a half hour fiery a set about it their which half of what I think what we could do was pose it was ever detectives over here and ask about interference over here what is up so we thought was 1st the a pack writer back added that at after which way they like forward Roque well all I can say is affairs is something laughter from allows you to determine that going the test here without leaving any mark on the system would get here are our off here priori but if there is a way could detect which way went through that will destroy a year and a half after he got a job as a secular have to do the detective as long as the detection Of course at four-year right over Jacques Greg mustn't by away course our course must proceed effect so that world theater primary task of started here at it dress for sure where I guess I guess we would have set up a very specific experimental situation and how because can lies in the abstract about having a very very definite concrete examples from above all chamber Riverside tell you which inside were now by Jack Aurora uh a day laughter Assyrian up a young all you need is the trajectory remembers which richer away electrons through bubble chamber over here plus a vote if he added that the where they both has a go back out of a raft of other that it all are it's a good ask some of what has assured us that is all I know is we're as we were last year just just to be precise about it we could oil Road we've wall of bubble chamber then afterward electron arrives
over here we look over here we see that we see track with years ago we see a track over the answer as anything anything that leaves track will certainly destroy apparent sure there it the area later if you are well either said each example the BOI separately but the rule the rule was very general if a mark Court promises from which you could do it Africa the degree of ambiguity will determine who will determine year of the magnitude of the effect of what say which could do some ambiguously which of the 2 holes Charlotte role then there will be no however the of sharing value because all sorts of examples and always trace it back ask whether there is there isn't anything left from a source from which remembers we back this year we are firm earlier hog Europe secures I'll crop offer bath sorority row some sort of World War it Jed caused Hi head on certain view that momentum Morris of Austria 1st reassessed do realizes it so that I have back or back with him Tom where are bad back from the worst case house as not that not me so I down as the boats or what's more war less move on a Richter and because we 1st now I understand why it's difficult to move on roses millions into a number of questions but I think it's a good idea having learned the rules go back and do the exercises yourself see if you could figure out yourself how are the of summaries questions because they're a good question worker questions were think we can go on and on and on ever ever that half of the year I say this is not of making a measure by making measures you actually do some disentangle for example making a measurements are br electoral overhear you crew you for all practical purposes you start all over again you start over again will be electron definitely over here that electronic mail from your perspective you roll away the other possibilities having electron and seen spin off you simply forget this 1 here you forget altogether if they are hot electrons over he let's start over again and now you could even say this Elektra our car this electron over here and would you start with it is a state aid the is not Pangle state you just 1 white USAir made measurement that look bad I saw was 1 of them was up so I know the initial starting point was throw away the get better faster so we could that's right so we cut our we could make of the expanded complicated we put the Our Paris here and spot with electrons this point and the spin replace No play a special role anymore it was simply we've been told you that electron over here or something code of electron start here and so it really was in some senses tangled with the euro electron gun or whatever was the sort of your from moving so it's it's sort a nested hierarchy of different levels of discussing problem has said the division division it just the here's your that era a somewhere walking had somebody looking up looking and so forth and so on and so on and way you draw the line between the system and action is ambiguous depending on the way you draw the line at some point you will have to stay away from collapse was observed by but In observing this entire system here you want say that be electron wave function collapses you will say save combination away puncture pet theory and strode or whatever is collapsed by way of making expiring Horn was was a cabaret care ensuring we get the idea and the only question is whether it's consist whether it's consistent Harlem whether at the end of the day day he wears Rhodia comes through ignorance as what did you see me think and we just as I thought that I thought I saw you think the a cat was made by ensuring reserve yes that's what I saw the cat was then our that's consistent game Kwan Canucks hooked up and such a way that you don't run and inconsistencies in let's rug worked with world course with talk about entanglement well we wanted a break free from for lawyer measures were diversifying number starts from a problem course while prior entropy fraud discuss entropy really discussed is how you define a measure Of the degree of entanglement between 2 sisters if I have to electrons and both this beans are up and are entangled yuan nothing about 1 now by looking good looking at as nor is no sense of road 1 and finding of some piece of information you didn't know about the other soft so let's not Pangle straight to his balls are generally on entangle states a core products states and products they each of the subsystems by itself has a very well-defined state that that were you marvel disregard altogether but they can get entangled Ricky get and the amount of entanglement is a quantitative measure of many quantitative measures of entanglement but the 1 we talk about the usual 1 is called entanglement entropy so our our explain what entanglement venture for years it is being if you are find out where some description of a calm posits system involve entangled in the test as the calculate entanglement entropy a 1st of all let's talk about classical entropy what classical arm imagine we have some system and the system has some collection state classical request of logic is Boolean algebra it's just classical set theory a space of all possible states of some system and let's take the states to be discreet so some finite number of states what's called normal state capital a mail entropy is not just a property of system it's a property of system together With your state of knowledge about the system another words it's something it has to do with both you and your state of knowledge on the 1 hand and the system itself the less you know about a system the Moritz entropy more you know about the less be according to the definition of entropy that we're God talk about life so let's see how Alcoa quantify back far out of his capital and and let's suppose that we know is that the system is an
1 nite after little and states 1 of little in the state capital and car where remembers role from more always know is is system is in 1 hour little and states with equal probability with equal probability for all those who and state we would say were ignorant and degree of ignorance is we landed and we could be in any of the lands states the bigger the lane near the less we know all about the system if we know that little if little in which is 1 of the world's that would be a situation where we know exactly what the state of the system there's been 1 of the state's 1 not only state of the system than we we would save the 1 my case see the entropy is 0 0 with no ignorance we know everything there is to know about them and servers nor nor I know 1 begin voters begin it means we know nothing about which state the system that corresponds to the maximum entropy I stole money give you it 1 definition of the entropy which is good for this situation here it is simply by definition logarithm called S and the logarithm of the number of states are that at your ignorance allows shorts little and for you nor certain why we states that the entropy is simply logarithm of will and farms would probably wondering why you think lottery rain or if you are a proof at issue you multiply numbers of states whoever entropy that first were where saying it another way of saying it would give you your example suppose my system is again our system of Spain's far and I know something about whole bunch of em ups and downs and ups and downs and so forth end so I know nothing whatever about the system I know nothing whatever about how many possible states are while the answer is cool the number of now of a completely different tone the number family spends are caught him and was totally and the was the total number of states I was the logarithm of food and if I nothing at all about the system and is here which is between 3 and any 1 of 2 civilians state the logarithm of the number of states would just be proportional and or what no longer and measures is the number of bits of information a bit being a spared the number of bits now a bit basically means of two-level system the number of bits father is number but the number of bits get exponentially aided the determine the number of states and is just a useful definition to go back the logarithm which is basically a measure of the number or or a little bits of information among the number of yes nor a number of sound decisions that those contained in your degree of ignorance OK really case that's a definition as is equal water the now why happens if you have a more general situation instead of having are always in this case he the probability of strikebound from From probability of probability score only states body to the probability for party the probability for the Iife configuration is 1 although little M ah 18 a subset M this is a subset and argues of the label both the number of elements and also the subset of self peace analyzes while and if why if in piece of 0 the yeah Hari is not equal bureaucrat fabric that was not equal not in a bus the means to say we know all the system is in but we don't know anything else I don't have a more interesting probability distribution more complex problem distribution center having more of probabilities we are all equal if not 0 we can imagine a more general situations where we just have a probability PORT Korea that might be big in some regions of the phase Of these configurations basin might be small other regions and my approximately be such that there are some subset year which all have equal probability 0 on outside they may not just some PR I would like some kind of a measure of the number of state there is included in this probability distribution would like it to be such that it cannot the state only a little bit of its improbable encountered a great deal it was probable I'm going give you have a definition of the entropy in Austria how it works which is true of a general probability distribution for general probable cause of I'll show you are used Mahmoud erupts just fine the entropy for a general probability distribution is minus the sum of all the states of the probability of EI state times a Zelaya of probability of the arts it but CL works on where our works all were sold in a white might Waterloo a lot borrowed a problem Road always 1 of the larger of the or something of Western make so this is really a positive quantity is really a positive quantity rather be so buyers or let's go let's an example calculators that go back to the 1st example here in the 1st example of probability was 0 if you not and 1 already in Waterloo and a few Oregon so let's calculate s uses some or all of prices so let's concentrate on a point inside and hear what's a probability it's 1 of life what's a lot of probability logarithm of 1 over my assigned and that's the contribution any 1 of the states in here any what other states outside the states and the on-site give 0 because their probability is 0 0 0 times logarithm of zeros 0 0 0 a quorum was the offer of PU 0 but he uses the p log p 0 so so you get every configuration in here you get 1 0 away and walked 1 over and there was a lot of RAM as minus a library here so we get rid vests and make it just slaughter but this is just a contribution from 1 of the points in years on many points or their India and so we have the multiplied by which is just a corporal organ Sonic case where we have a probability distribution which is you're all want reproduces the original definition hero just like where all Falwell about the situation where there's only 1 point which has nonzero probability all the others being 0 mother was when you know with certainty exactly where the system is interfaced based In that case only 1 of the piece of ice is not equal to 0 all the other P survive Oracle 0 OK so let's go back there about 1 point which is not equal to 0 of label JT now might what PUF copies of 0 piece of 0 art piece of abuse
spread out over about states if it makes sense to say spread out over and states that was just Library of America otherwise in more general definition which captures which captures the 2 were examples of a the really the capture it was some additional why piece a bar I want peace survive Rex definition of resisted the bigger the entropy years ago lest you know about the system our that's the maximum entropy that this system but a lot of capital while in the capital where would be a situation where you know nothing about anything so each 1 of the state's has probability 1 began you know nothing here where such just Laura of cash this is a yacht of Yoda which of those states is at best a urinal which respect yes you if you know what the system is but you don't know in which of the state's bathroom maximizes is simply the difference between maximum entropy an actual entropy is called information Our arms if you happen at the entropy were exactly 0 0 you have a lot of information you have a lot states in you know which 1 it seemed so Oh the difference between maximum entropy entropy disease amount of information available to you or find some sense information and entropy opposite sides of the same Corning are entropy Is the information which is unavailable to you because I believe stop it too lazy or wherever reasons for the object of small city years worries broker values know what those entropy have to do with entanglements that calm I will come it is a measure of arm of the degree of entanglement before we get on that we need a definition 1 definition 1 mathematical definition mathematical I will give the mathematical definition prepared a break for some getting fired take break and them will come back and talk about the density matrix I don't think will actually get entanglement in the Beacon nite but I wanna set up the the basic ingredients are going it basic mathematical operations bomb on matrices matrices OVER operators you could think of rivers 80 operation on our on our farm operators you have some linear operator watches caught him them spans from maker also stand for abstract operator and we're not suppose we have 8 basis basis formally to assume I on most of our let's take a basis Our vectors in space of states workspace Estrada's of course or and take the br II matrix the diagonal matrix elements of air if Emma is a matrix Baroness just stands for the or diagonal matrix on 1 1 element to 3 3 elements and add them altogether but called the phrase corporate phrase Bob end now you might think red mist depends awfully much on which choice of bases vectors there are many choices of bases that Romania are based it could choose three-dimensional space you could figure axes for Unicode promises is something exercise to use office easy for you Camargo your favorite Warialda book the traces of a matrix it is independent of a choice I've From a suspected it's invariant a quantity which does not depend on choice of a suspect there's an so all choices of bases that give the same value for the triceps import will be around plays an important role in what follows but you you look up for all produced over not OK that's the definition of a trace reviews acts are over but so pulsing the Matrix In some basis happens be bad doesn't know what basis we computed here and we can look for um basis in which the matrix was bad if Amazon her mission operate there will always be a basis which is Rivera's not her mission there may not be but if there is a bases where is that all the bad elements on nothing but the Eigen values of a matrix of more matrix does expressed in bad form if it can be expressed them back form the entries are the Eigen values of matrix that may not the 1st because may have an interest of any of the end of the entries of 0 it won't as long as injuries not the zeros part that our Nowak who want to work means is a function of our mission operators were me off but it had flag nor its characterized by a eigenvalue in then the traces nothing but some of the ideas about that's have describing the trace which doesn't depend on the choice of a base service the Eigen values of the parameters bases just if you happen if you happened to use the right basis and the matrix will consist of a back diagonal were bag entries will be the Eigen values of the concept of an eigenvalue view is transcends any particular basis so of you know all of the Eigen values you the mob and that the trace of some of the Eigen values are particular of our mission directs particular permission matrix which always has a complete set of eigenvectors of ideas about used some of the Eigen values Is he traced resistor plays a reporter rock let's take a break from successor but
some on fire but what Aurora traces where our 3rd column rescue worker back for Minatare classical situation can't who who the classical or probability distributions from is remind you that if you have a P body and you have some observable classical observables our classical observable just means some functions of the points in the configuration space some functions which takes on a different values with different states so we can call body Joe as just a function are the configurations just 6 on different values when you observer to find out something about which for direct plus a B average of f give a probability distribution piece of body is of course the son of What are you ah piece of RAF survive you just wait different values of with the appropriate probability the probabilities or the 1 and this is this is the classical classical average or expectation value of are the court a war of course average that that's the definition of my because of recalled expectation value courtside necessarily expected on Beach Rice was the trace is the analog of the storm over configuration replaces in quantum mechanics summoned all possibilities becomes a R are arm or configurations becomes your works as we go along now there is a very similar formula bears in quantum mechanics which of course involved a faith and draw involves a trace of the observable kind of something and that something's called density matrix where are you now or in our very simple situation 1st surpassing somebody prepares state of assist on any 1 of the state's over off normal bases are behind your back somebody creates a system In some state I but doesn't tell you which state where created so it's definitely an electron or spin nor a composite system or whatever happens to be being created by somebody else In a particular state 5 all they have called you is a probability distribution for what state they create a system because dual Woolacombe legend is so possibly creating electron by putting it into magnetic field and the magnetic field could be or could be down and make value His electron it's either up Obama my which give you probability of 75 % the likelihood up 25 per cent likelihood that found this is not a situation where you would write off plus down or even 3 quarters up plus 1 quarter and right of the urine are you will add the state's together or definite raised you just say you don't know what stated in parents the system is described statistically with a greater degree of statistics then if it had been in a state of it then you know what state it was even know what state it was in our certain measurements would still be probabilistic North state it was in certain experiments might be definite others might be be probabilistic but he is and a degree of problems probabilities headed probability aspect affairs not only if the system was a definite state might do not know the answer to every question but it might not even know what state the system on the most circumstances use an object called the density matrix instead of state vectors density matrices are kind of looser concept for concept when you don't know what state the system is in it's a probability distribution of different states quite salt let's opposed but we have that somebody has Penta system definitely in 1 of the bases Bates party with a probability let's call it rose by using wrong announced that P. are just because of the standard convention for the situation the core court row so speed however roll band is the probability for the arts now we can make a matrix out of world could make makeshift several years away on the committee of Roh I'm just going to this is the matrix in the RIA basis so desired calls 1 articles 3 articles our likewise the Rose it's going to be a diagonal matrix and as entries just gonna be Rome role 1 rose on the back row 1 row to row 3 all other elements 0 Roy woes a real numbers positive numbers because the probabilities and all
less 1 and 1 more thing and this is called celebrity this is called the density matrix Caldwell who called event make Seattle rotting classical physics Ava probability density Arthur of work or piece a body but now just been organized and gourmet trips were bag or matrix elements the real and obviously they are the Eigen values Of this matrix because whatever make it it has been written bag form the Bago cited 1 thing you could say the Of the density matrix is 1 wires act because probability should add up to 1 or for some probabilities of the 1 traces of the density matrix adds up the 1 4 now suppose we have some observable politically observable F and it's also are mission operator no observable F astonished an operator has always been a finally average value of F if fully head on ordinary state of yours if we knew which state we were with probability warned then the average of F would just be the expectation value of dash in that state if wearing a Iike state the average of F would just be the expectation value abreast of sovereign strike while I say more generally when you have this uncertainty about which they during the right answer is the average of I want indicated the same way I 1 indicated by a Bronner because that there was matter particular state of Israel as the average of F it is equal to the price of a product of S and raw let's see if we can make any sense ever formula formula is quite similar corresponding classical formula starvation or what are arm of High Times piece of art more former are face being replaced bar some being replaced by classical function being replaced by an operator and the probability distribution being replaced by ribbons be makers Corbin City make it because it's like probability density robbery letter what's so work out what Australia's work over the years the trace we can pick any basis we like but let's pick the basis on which role was back in office was be assuming the role was back from bases are just becomes a trace tells us the definition the trace cover compared the basis take nothing was faced with taking F times role and close up again and some are that's a tracer reference row OK valid Electric which we talk about last time we have really used much of music now if we have a basis of states unit operator can be represented as some of the states are the ads made up Our over basis vectors so let's and that is what this means is you can't always stick an expression like Paris between anything else and not change from particularly stick between ethanol acts around you right now we have 2 summers ago 1 for the definition of the trace in 1 for resolving the identity of called resolving the identity by using a complete basis state that would give us F. J. Jerry role on solemn for not only ORE but also a some raw for the some Robert J. to resolve a identity God now raw water In the basis of my musings assuming his bag that means that I have the equal JT otherwise we don't get expression right in that case is just becomes EU I G matrix on really sorry diagonal matrix element of role I has to equal will get nothing cycles J. this just become ZrOsub RIT behind entry in the matrix and this becomes F R R why was says this is the expectation value of Athens state party This is the probability that the state was in configuration or ii and this is just as some of the expectation values of F waited with a probability distribution of different salt this makes sense costly average about we see now or any i.e. quantum mechanical version of this formula roller here in wedge as I said solemnly is replaced by traders end probability distribution replaced by role is a little paramedical prolific whose 2nd proved to have a product of collaborators and the trace of a B is the same as the trace of being a what about even if baby daughter from you the trace of the product was my which or and
on factor matter which way you water at the world price of times role as the average OK that gives the concept of a density matrix let's just look at Sun Myung analogies with key 1st of all the somewhat of piece of are a 1 the L is the trace of world equal 1 trace of rope to Google Sewell purchase orders airport of trace area burst being essentially the same as very good now I'm going to give you the concept of quantum mechanical concept of entropy the quantum a cackle entropy of a given density matrix 1st of all the definition is such that if you know what state really only 1 of these roses 1 the others 0 and the interview 0 any quantum mechanical state quite real quantum mechanical state of usual crying the entropy 0 Arthur theater is not 0 not because quantum mechanics has a built-in probabilistic aspect but because you simply didn't know what state the system was made you want for our that's the source of entropy quantum mechanics lack of knowledge about state not lack of knowledge even if you know it's OK the answer for the entropy the definition of be given a density matrix role 1st of all he is I use it to calculate apertures but 2nd of all the definition of the density matrix well Red Baron classical order of classical form new or summer untidy miners of piece of I walked peace quite well the answer is racist minus the traits of the density matrix role Panza logarithm of the world but that was a lover of the February well there's a general and wrapped but let me give you the answer in a simple case where the matrix can be brought by bag form if the matrix and some basis is diagonal the Malaga of a matrix is just the matrix made out of the locker rooms of the ideal Water of World War and Barbara of cool and so forth and so on so what is this this is Of course just disarm although survives long rows survives exactly classically but I've written in the war and now Ave that 1st calm as more appropriate for quantum mechanics as a trace overall Logrono so as being in a vector space not just as a classical probability distribution This is the definition of theater it has a property but only 1 and only 1 of the Rose are in which case it has to be 1 because trace of role has be 1 if 1 of only 1 of them is nonzero than entropy 0 1 Carnes Lord of War musical 0 if the probability distributions uniform all states having equal probability that would be the case with this would be proportional unit matrix that the entropy here is logarithm of the number of states logarithm of the dimensionality of the space serve you know nothing about system from our was prepared earning him like back then we attribute is logarithmic logarithm of probable number of states which means the logarithm of the space of states lover nature of the state bats take your work practically directly from the classical concert there should be a places and its indications of lack of knowledge of the system knowledge that you could have but that you don't know the words knowledge that there was therefore we got but which you may be too lazy to get what's all this have to do with entanglements Well I think would never have quit now next time I'll show you what this has to do with entanglements but basically I just outlined for you when you have a system which is composed of many parts or what to pipes to be specific you may be interested in only 1 of the parts and not look at your papa manner system which consists of toll and triangle parts they may or may not be entangled particularly interesting when they are entangled AMB there is some state which is an entangled state-of-the but you're only going to do experiments on a D and ignored be altogether and the question is how he described 8 without reference to be there must be some description that you could use are Eddie which contains all possible experimental information that you could do on a if you're never interested and also looking at beat it all what could be answer is he use density matrix to describe a if AT&T Pangle then the density matrix will have entropy if any not entangled it will not have a trip so want diagnostic looking an system or at composite system defined our things are handled is to determine what's called the entanglement entropy of the toast it's a measure of how badly Pangle they will complement with about your work out some examples or that real reject calls to show you how entanglement should be codes fact systems on
eye Key Cheung Chau the preceding programmers
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