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The Theoretical Minimum | Lecture 7

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Stecher University are we
started to explore entanglement I had actually started to think about doing Bell's theorem missed a tonight but John I decided not to do it yet it's a complicated story of Bells series and very lovely but more important than the pyramids of what's so I wanted to get as much as possible tonight what it says and 1 entanglement Thurman but it has to tell us about the nature of systems sometimes people say the nature of reality I don't like the the word reality I think it it's misleading Okayama looking on the internet and the messages you present back and forth the very interesting in and many of them have some very interesting points in them visit terminology which I would discourage we start must know what it means go with greater precision and I've seen this all workplace people say entanglement Sharon to quantum systems share the same status share state I'd never understood that means the share state our quantum systems have states that composite quantum systems that have states are you could say that a quantum system of quantum systems when they make given quantum system Charest if you like but it just means the state of the composite system on these important idea above entanglement is not so much that system sharer state that they always do it's important thing that they are not described by a proper state a product state being a state in which the wave function of the system will talk about Waco have we talk about wave functions very much I don't think we have let's begin by talking about wave function and then I want more to the concept of the density matrix this will make it much easier for us to think about entangled Brazil notion of a wave function the caught a wave function may or may not have anything to do with way these concept rule historically out of quantum mechanics of particles and the relation probability description of them in terms away years but he would be the logical concept of a wave function is 1 that you could apply to any quantum systems and in general doesn't have to do with wave motion any particular kind of basic idea of a wave functions of following you have a quantum state let's call the quantum state an back capital sigh use the capital pCi no account a generic quantum state of a system I think previously had been using a seasoned bees with by using side as a generic quantum state for of pretty much any system and also I have a system the bassist system is typically labeled by a collection of I get values of something off it might be labeled by 1 or more variables if there are several commuting operators in the system you could specify simultaneously the Eigen values more than 1 thing that in many cases you want to do that and bases vectors in those cases all labeled by a collection of our observables our collection are ideal values of perhaps several quantities of give you an example arms if you're talking about the motion of a particle the X and and Z coordinates of the particle are commuting variables the things that don't commute other components of momentum with the components of position I know we haven't done this yet there are never hand B 3 components a position or commute with each other that means they can be specified simultaneously a useful basis for studying the quantum mechanics of a particle is a basis which is labeled by the 3 coordinates of the particle away for her broke down during I still have a basis on the bases label but collection a bargain values works label them as peace a BDC however many commuting variables aura that need to be specified emotive specifies system the meat in a Prague part of the basis sector with the state of the system itself is cold of the wave function in fact we can write sigh and these objects former basis because this is a summer storm and little of that some of the bases factors which is a sum collection of eigenvalue using we thought of we inner a product on the basis vectors with the state vector kinds of basis sectors we've written this equation is kind of equation them before what it says is that the expansion coefficients that you use to expand the vector in bases vectors adjusting in the product of the vector would base effect which this subject is called the wave function right here and as a function of the eighties business sees so it's given a Mascorro little sigh of a C himself what was called away function and that defined in such a way but the state vector side is equal to the sum of 80 all these indices which takes a specify the basis crimes sigh of a B and C and so forth L a B and C of according Serbia quarter an example if we just talking about a single spin and a complete set of commuting variables necessary to specify vector is for example just Sigma z z component of stand and I would just be the probability amplitudes for the spin to be up for the spin to be down be a functional 1 variable led variable would take on the values of the narrow if we had to stands then we would label a basis of actors with up for ups and downs for once been ups and downs for the other spin and with take tool labels the label state vector not just 1 but 2 labels would be the labels of the 2 spins and that's all that's all we're talking about here might take multiple labels labels the label of a bases vector and if so just display in here and then by definition good definition the wave function of the system is something of a two-way use you could think of it as the set of coefficients a set of coefficients that used to expand an arbitrary vector In terms of the bases that there's all you could think of them as In the products a product of the state vector would be basis vectors themselves that is called away function and Schroeder is wave function as a
special case of it which we will not get to it at the moment but it is a special case but questions OK so if you know the way function of the system you know state sector and you could compute all sorts of things Sukur computers power expectation values in all sorts of things this look 1 is for transformed effort case a position moment of various hour of the the less for well not quite transform is going back and forth between different basis at here I've chosen the particular basis at supposing I want to use a different basis instead of Sigma Z next that I will have to transform these wave functions somehow who look different the Knicks basis and that that let's see analog before transform OK now I wonder to introduce a concept called the density matrix if we have just a single system this it is the way we think about the state of the system what local expectation values such talk about expectations supposing we have some observable let's call it arm but scored killed if some observable end it has matrix elements were led matrix elements need they need this is a bases vector I won't bother writing out multiple labels but just but 1 label that summarize all labels and another label over here a climb he called the matrix elements of the observable make its elements of the observer observable acts on a basis factor in new projected on the aware of the bases factor and that's called L I suppose a prime just notation looking us a puzzle I want to calculate the expectation value of In a state whose wave function this side would see how we do that we take as I said I'm just going to restrict myself to a single a single a year but the same thing would hold of still multiple days and we do we take state sector on the left video of the kid back there became labor which arrived at vector he sorry Bravo and Brava this is kept the bra vector is the sum of all that a and B again I want to use different indices here or cloned a prime and be prime the being some Nova so it doesn't matter what I call them distorted computers movies and then we're gonna have our brought vectors ABC 8 times precise yet sigh star so exactly the same kind of expression for the brought accept complex conjugated coefficients now let's calculator but signed side that's the expectation value or the average value of L in the quantum state site but we do it just by plugging left left abroad that here is the sum of the single index a prime a prime sigh of a prime where constructs a brawl that Cadillac yeah I will have to write another some over 80 of sorry a firms a summing on aII turns this into these kids sigh capital side something over primed turns us into a brawl vector the corresponding brought back and in between we stand which held that the school somebody please correct me or wrong so I start we take the size and the size starters out of the expression they just numbers and the whole thing is a sort of a prime in a size star of a prime society times in a product or a sandwich a prime L. A. that's exactly the matrix element of a between In a prime solve that which 1 is it a primate center and this reduces every expectation value to a calculation in terms of wave functions assigned always sunny and always size start times a matrix element of the observable that were interested I'm so this is a basic formula for the expectation value of anything that we might be interested in basically bulk of quantum mechanics is contained in that the expectation value of everything in the world that that's just about everything we ever may want to be interested in and so let's but equation over here and keep it over here now supposing we have a composite system the composed of 2 subsystems let's call when our and Bob subsystem Alice hazard control of 1 of the Bob has control of the other and maybe even there are times when her in there they should be don't have control of them but there which is called to subsystems Bob then fact let's use 18 now to represent a complete set of observables for Alice's subsystem Everything that Alice meets the and Arthur specify a basis for a subsystem and be a set of variables for Bob subsystem we talk about and I think we called Alice's variable Powell and Bob's variable stigma for remember correctly but there is a a complete set of things complete set of variables that us and this is a complete set of variables for Bob Alice's states for subsystem would look like this Bob's would look like that we maybe we want use different patients so we don't warned that confusing to vector spaces but there is a calm compound vector space the composite vector space is a vector space describing the composite systems and that has a basis which consists of an AT and the when they runs over all its values and be runs at all its values we have a complete basis Of the combined system OK so now let's talk about wave functions of the composite system again we have a state vector the state that the is a stage victory even the calm positive in the tensor product space space made by concerning product being season bees together it has as its basis vectors a bases vector from AT and they suspect from B and now I want that defined in the same way we note the notion of a composite wave function so what is it it's just being in a product of siding with the the bases vectors was a function of it's now a function of 2 sets of variables auto variables sigh of a and B different be here this year we are clear it that's a function on the variables Of the 2 subsystems depends on 2 of them focus that's the way function however notion of a wave function of a compost system and let's talk about
observables of a composite system the observables of a composite system 1st of all among those variables are the ones which would describe separate subsystems by themselves so let's suppose Ill In a AM till observables 1 of them having to do with Alysa subsystem powers and the other 1 having to do with Bob subsist I could call Elsa Beatty and of BU but I won't call a mail in Ann Arbor how we represent observables acting on the composite basis states began so it is simple we operators associated with Alice subsystem on Alice's piece of the state vector and leave Bob's piece of the state vector unchanged as so we did what we did spans we said Bob stigma matrices acted on his half of the wave functions on his hair from the bases vectors and were completely passive or of Alice's piece of the state vector was completely passages from bypassed by operators so what that's said what that says is that Alice's observables are matrices their matrices rewritten S L our heavy prying be prime just like Hill here as a private variable prime variable but now the single index is replaced by 2 industries and where this correspond to corresponds to be sandwich a prime be primed L. ADB wherever a where are the stands for but NO only belongs that Alice's space of states only belongs Salas system so the question is what do we say about this matrix he was said to indicate that it has no effect armed mobs variables into simple we make it proportional to the identity matrix in Bob's variables the identity operator and identity operator and in a space is the operator whose matrix elements are the comet Delta symbol as identity operator the ACGIH ditty operated on a subsystem does nothing that of the identity apparatus simply gives a saying back-to-back other words if you apply the identity Britain does nothing but now we're doing is applying the identity operator on Bob half of the system but the observable that's supposed to be associated with Alice's have of the subsystem is represented by the same matrix that she would have used had she not even known about Bob as you not even known about Bob she would have used the Matrix 80 a prime an 82 describe her observable house she described it in the bigger Hilbert space she says it does not nothing think the variables and so it's just the identity matrix of barbs over half of the space that's palaces variables we could say the same thing about bombs in lieu of a prime and be primed In a B just equal chronic Delta of Alice's variables kind the matrix of Bob would've used to describe the subsystem heading not knowing anything about us or let's proved some things about them about the 1st of all it's calculate expectation values of Alice's variables in a product state a prostate is 1 in which the 2 systems are not entangled with each other and we get what would you expect to get your hand and on entangled state describing Alice isn't Bob subsystems and you consider the expectation value of Alice's a veritable get exactly what you will die and Bob Bonilla altogether sorts check that out was mean forcible by a proper state I explained that the products they was now explain what products state it is in the language of wave function is very simple it means that the wave function is a product of two-way functions 1 Fallis 1 for Bob that sort of product product state is 1 whose wave functions sigh of 18 be houses Bob variables is just a product so let's how showing note that much Call From is courtside Jerry and Safari Bob of as fact arises such a product of 2 functions as a function of 2 variables a product of 2 functions each of 1 variable it's pretty clear that if a fact arises like this Alaskan think of this as the wave function for her subsystems and Bob thinks of this as the wave function for his subsystem OK let's calculate now Alice's expectation value of killed if this happens to be the wave function really do we 1st XI eyewear maker sandwich which oversized on 1 side the and the other side and ill in between exercise star that size star Alice of 80 times our fi star Bob of prime this is where we ride yeah I love by sea complex conjugate of the wave function as a function of a primed be prime what next next week putting a sandwich L a prime be prime a and B matrix 2 0 1 entry he but simple just tell of a prime and aII trans delta B prime and then we close it up with the other half of the sandwich which is this Pierre sigh fly of being an only son as always we some over everything inside a prime ADB prime that's the rover calculating expectation that you will get 1st of all the dealt the prime that tells us the set be primed Eagle Tobey Village do that 1st offered direct this erased Chuck set a price equal to be with you but found notes few want said a prime equal to be and remove the Konica Delta that is equal to be prime another silly thing that depends on B five-star and fight me what this summer would be prime was done Mississippi could be primed analysis some all the B of five-star fire was active more under the assumption of course the Bob's wave functions normalized under the assumption that Bob here for the wave functions normalized His toggle probability for his all his measurements he could do was 1 some of would-be of 5 star of beef times fire B is just war was whole thing we do still the whole thing reduces to exactly what's up there we can remove the
fires quick amount from down here that's some was just 1 and the rest of it is just exactly what's up on top of a black for another word in a prostate the expectation value any of palaces operators are exactly the same as if Bob was there and it doesn't matter what Bob's wave function is as long as it's a prop for about 1 barbs the expectation values or happen if start sucker Bob operators will then instead of the doctor of bees but go back and be in be prime William B. Bar B & B prime prime over here and set of Ellen a prize l rumor has it that B & B prime Delta AT a price and now we would set a price equal to a some that would again give us 1 family we get the same who give basically the same thing that I rode up here but Aug said it would be Bob's wave function 5 star Bob of prime a of be primed B Price flight Bob of beef which is exactly what Bob we're done I'm not sure about so proud of states are really just that this state in which 1 party just doesn't know about the other is no correlation is no noaa information in 1 of them about the other chick what is a little bit more about correlation let's suppose now what we were interested in the trial of an Alice variable and above variables were interested in the expectation value L kind and let's see what we get expectation in the plot strictly products dates for moment with go back up the here and now instead of writing and beef is b b problems should be prime set of built of a prize of we're taking the product of limit were ripe prettier L if I'm interested in the operate which represents the product of 2 variables 1 Fallis for Bob instead of putting the unit operator over he of fat Fallis I will simply put in Alice's operator not a house elf yeah that is just factorize was literally a dependence here's a dependence here's a dependents here's a dependence that fact arises that's completely separate from the B dependents I could take everything will be dependent on 1 side everything with an 80 dependence the other side and only a and B summations independently we see that that which starts right out but son of a prime ADB primed B of sigh Alice star of a prime L. prime sigh hours all times by Bob star of be pride in our Bibi prime end flight of B so they can be done quite independently from the summer we hear same thing I haven't changed it was something over everything is received all we get is just the product we get the product of expectation value of failed times the expectation value of and not surprising if we have a product state and Alice does her measurement and Bob does his measurement and we can come together and compare the measurements of mess hall was a product of the measurements the expectation value will just be the product the expectation value of the product operator will be the product of expectation values with right this is like write it look for them that it's as product states for product with prompt states the expectation value of L and times em belongs to a palace in belongs so Bob and they operate in different timbres of the system the expectation value of times and it is just the expectation value L kind expectation value event now Is is something that we expect always be true for every state that the expectation value and that even in classical probability theory if we have a probability distribution for 2 variables in general is it true that the average of the product is the product of the average which means what terms of the probability distribution it means that after the news of the probability distribution of talked about probably rack well what thought that's not true what probability distribution does not factor arrives in this quantity here the difference between average of El times and and the product of averages of element is cold the correlation between and as call the correlation and it is a measure of how much these variables are correlated with Gould will see some examples this is called the correlation and whenever a coalition wherever there was anything in the system on Alice variable and a Bob variable who's correlations not Zero that's an indication of entanglement you could find any pair of variables 1 of Alice's and 1 Bob's such that the average of the product is not the product of the average's then you're experiencing entanglement I think we do example less burning tho we took what I call the single state which woes on down minus down up over the square root of 2 visits i.e. and this is not a product state and in this state we took the expectation value of Bob's variable Singh Missouri we took the expectation value of Alice's variable which I think was Tozzi demand for these war with the average value of a Bob's variable but say the 1st entrance here Thursday treat there's a lot as Bob Bob His Alice What you guessed mistake the averages Sigma Xavier's not the state of the combined state not
0 why is 0 because it has much probability of being not as being now has probability while has probability 1 he to be up and 1 one-half to be now the average of Bob's herbal 0 I think we were up last time where about Alice valuable it's also got as much as down it's also expectation values what about sites or 1st that both of these are 0 in this case With where about the problem what about the product of indeed what its average value would work it out last unanswered and remember the incident minus 1 but it does look at this and say in both sectors he the bizarre opposite so Casey the product of the spins is minus 1 it's pretty clear that this whole thing is an idea that the root of the problem because either vector here the product is minus 1 after the city was a key factors in the zeal Geiger nor will happen so just change sign so yes this has the product of the signal Aziz Eagle toward minus 1 excuse me so they are entitled effect in this case there were well cold maximally delivery entangled our if you could find and of variables which are correlated in this form the quantum state is entangled do you remember now here it's completely obvious that the average of Sigma 0 for just a reasons I said Our former 1st Tunisia mentality op appears as much as down for Bob and appears as much as our Fallis so it's clear the averages 0 what about the averages Sigma XT remember it is 0 but that's not obvious isn't a the book we did last time go back and you discover that not only it is the expectation value of sigma zeal 0 2 Next is also recalled 0 the average on the other hand you'll also find that the prime of Bob Sigma X Alice's how X is again equal minus 1 song the same easily Kaposi's of correlated stigma axes and italics is a correlated about why saying exactly work that out when Ortiz operators do you could apply to the states and calculate with their expectations are so this state is a highly a highly entangled or a highly correlated state being correlated the being entangle the basically the same idea each other were will be L is breath fresh Canada Oh said found out that they are at the point there where chairman mention all operators associated with 1 system commute with all operators associated with the other system so it doesn't matter in which order you right that's independent of whether the system is entangled the mayor which or rare direct that killed this what it showed it last where else not than this is equal to 0 what is it that there is a not don't what the state is these other the states easily observables right now don't know ill the names among states young parasites in here is a statement about the idea of state vectors look bright hope that 1 point it was just not think what I say things correlated that this spins of correlated I mean what I mean is that you could learn something about 1 variable by looking at the other if you know this is a state if you happen to know that this is the state then if you know Bob that you measure Bob's variable and you find out that it's up enact component of the state which is the only ones left 50 measure Bob Alice's down Anaprox you learned nothing about 1 system by measuring anything about the other highly state you learned a great deal about 1 system by measuring the other 1 that's what tangled with me and the other hand it may also follow may also be true that in the highly entangled state beforehand if you happen to know the state you'll may know very little about each individual piece for example in a singlet states here areas was is the probability that Bob's variable was up 1 half was a probability that Bob variable is then 1 half so in this state no link to knowing everything about this state you still know nothing about whether Bob measures are put down a few dozen measuring curious you know nothing about whether Bob measures up or down you know nothing at all about whether Alice measures up or down but you know everything about the product of them that if Bob those his measurement our stores for measurement and you compare the afterward every single time if this is the state that you put the system into the product will be my so this is an an example of this Bond fleeing quantum mechanics that you could know everything about a composite system and know nothing about the pieces as telling you you know everything or is when I say you know everything I mean you know as much as quantum mechanics allows you to know this is a state of the system is not gonna do any better than that that's all there is to know about it would you know about it Is that Sigma times Powell whether it be Sigma extra power next is true for the ex-wife Ozie components separately that the product of Alice's variable and Bob's variable His minus 1 but a lotta not but not as much as you would know classically classically if you know everything about a system he would know everything about the constituents as well as things that you could measure about but you could combine together and quite mechanics you can't know you could you you could say you can't know everything or to the extent that everything is everything you can know in this case you know everything about the combined system and you know nothing about constituents that's the really are odd thing about entangled it extremely odd property of of quantum mechanics allows such nonsense but To did he said he was not yard it did that does mean that where you were right about shot review is that both letter area
and measures mural Europe is rile village that's been very clear that they like the operation the operation mathematical operations of multiplying the state vector by operator it is not got anything to do with the actual physical act the measures and all of that but I'm not sure the syntax is not the same as the state I'm not sure what impact means are assuming the work right after an but the yes yes to me act 1st with them and then but yes yeah dip them but doesn't mean that you've measured veteran means is just on mathematical then write bad debts the same is true in classical mechanics I can multiply the momentum of something by the position of something else that doesn't mean measured vendors because of multiplied the 2 things together physical of measurement and not operators which act on on the state's it's the detectors or apparatuses which act on the system so in words when you go measurement detector or an apparatus acts on the system and use a result be In the space of states AT operator and so on bet now not a detector acting on a system quite different concept or we on a less we all know that there is a case that gave birth may be stressed he as it job but before we know that that much of a little bit but America we will have expressed the probabilities and get eastward a correlation to states that a jerk world now Oct . 5 minutes that a series of some 8 8 is somewhat so it is then that they gave scrappy out of it so that it it is ready to sign orca knowledge that we show what you're getting at but maybe this will help calm if we have a quantum state of the single spend let's opposed somebody told us what the state of the quantum state of the art than for any quantum state there was always and observable In particular some component of stigma which is that which is predictable every part of that's obvious for the state's up and down on the use of the state's left and right for the state up images seem was it's death if you measure if you know that the state is right and you measures Sigma X definite Mr. similar general given quantum state ever of Spain there exists a direction you control over the exists a direction for which the spins definite curiously when you combine the system together into an entangled state like this there is no direction in which the definite in fact there is no directions in next step for 1 of the each 1 of us things Alice's all Harris average values you roll every direction no matter what direction you pick XYC or anything in between the average of the spin of Bob spin in this state 0 so here's a quantum state in which Bob conceding nothing wanted know the plans they fully and as I said has equal probability for up and down in every direction on and thus that's what your address or not both players address those cult plus OK so we put that would put a big number here a small number here we we love the others he said there is still a yes we can calculate or probabilities calculated on talking what is what would be there all but but Burke import obviously important question another way to estimate how would you prepare a state system with right arm but I'll tell you something about the same state the singlet state has the property that it's and eigenvectors of a certain operated cited victims many operators but in particular it's 0 eigenvectors of Signetics power X you could prove it it's exactly eigenvectors next politics it's of eigenvectors of signal White How wiII Anderson eigenvectors and received houses what's the Eigen value for each 1 of these minus 1 and adding value the same for all of them that means that eigenvectors annually in any combination of these what I did value minus warned any 1 of them acts on the with state because that might assist what state any combination of them will be back my state so a particular if I take this subject here leasing status also an eigenvectors that's now this is a funny thing Allison Bob can't easily measured this whole quantity wine because Alice can measure How extant how y simultaneously by separate measurements this would be hard to measure KP measured by separate measures are wide because no separate separate measurement to measure power and power while at the same time more Tauziat but this object is a perfectly good her mission operator and they must exist the ways of measuring it but what the point right now a way of preparing a state which is eigenvectors now what states the single State single state is eigenvectors of the subject with Eigen value my history as a bargain value of this 1 of minus 1 this will minus 1 this 1 with minus 1 the whole thing has I get value minus 3 point is the eigenvectors Of this product over some of products include the same state as a particular 1 it could also write this is just signal that check next how it's like wants to us see that With ordinary classical vectors you arrive Sigma how it take electronic each 1 has been and the very very far apart they don't interact with each other is no energy associated with the interaction between them but as you bring together they all sorts of energies 1 energy for example just call on energy between product of the charges there is also an energy associated with this spin and 1 of the terms of the energy associated with the is just a byproduct of the spin the spins are aligned you deal with if this is aligned you get 1 value there and Pioline GDP of value and so about how it is a piece of the Hamiltonian Of the 2 spins system in the Cillit state minimum this 3 other
states called the triplet state in which its maximum so energy is the lowest in the singlet state as spins are trying to get in the entire lined with each other but in every direction not just a Z direction every direction and they eventually compromise on the single state so you have to energy levels associated with Spain's electron these 2 energy left uh in hydrogen in positron Ian better and positron you have realigned the entire spins and they have different energy levels saying Triplette if you bring told electrons together than any state can be written in terms of England with states even the products they can be represented as a sum of singlet triplet states what happened when you bring electrons together after a period if they're in the triplet state with a higher energy why the higher energy state after a while they a photon angle to Lauren G. state Soviet take too spinners which may be in a product they you bring them together and wait a long time a photon nail may not be needed if the system is in the singlet state no emission limits in the triplet state its gives off a Fontana mourns works there so this would be a way out preparing the system in which they just take the 2 electrons await until they Radio 4 times week long enough the do adult radiator photon if you wait long enough you know that it's in the singlet state so that's that's a experiment that's an experiment with another way of knowing of his wounds this 1 is a very very far-fetched extremely far-fetched the energy in the singlet with state is less than the energy and the other 3 states Of course there for possible states altogether for possible basis states England 3 trip the energy of the singly lowest so you just take your calm passes system and you put it on a scale and new measure man masses energy when you measured its mass measures whether its energy is the energy of the sync with state of the trip with state and that constitute on measurement are constitutes a preparation devices are measurement of basically is variable but if like the difference between single and triplet so there are ways of preparing systems in the sync with singlet state always over effect if you bring systems together and you wait a while it's very difficult to have anything but the singlet states them once you need that Singh would state if you grab the electrons without turning this In you grab them somehow away that doesn't put port on spins and Paloma apart you never have to to separated Elektra for example what state sorts by terribly hard thing to do would be to energy yes in this case yes and this in in this case yes but that but that's well you but you could say is the lower energy is certainly entangled the lowest energy state the 1 that seeks wanted to get through by any it does radiating a photon any mechanism by which these systems and Laurette's energy in and many have fallen on our doing good interacting with the crystal lattice or anything lower energy will leave it in the singlet state so and I think directing Justice ideas that these Lauren G. state is entangled that's almost always the case law that there be states of lowest energy are highly entangled between systems OK so it can next question is there a measure 0 a quantitative measure of the degree of entanglement 0 way of taking the state and determining whether it's in tangled and measure how much it's entangled that brings us or concept called the density matrix which are probably go getting later talks or 1 over go over so quickly but I think we can do it let's define an object called Alice's density matrix Alice's density matrix is a quantity which even a N N entangled state is as complete a description of Alice's system for the purposes of Alice's measurements as Alaskan ever get its no wave function drawers on tangled the wave function would be the best description she could have but for entangled system you cannot extract out a wave function for you the peace of the entangled system the best you could do is something called the year of the density may actually had up here we go back to it let's talk about both Alice's experiments that she could do with out measuring anything about Bob's electronic or Bob system but imaginative separated their systems Bob has his house has Alaskan do all the experiment she likes and measure any 1 of these ill observables stands outstanding out art however she described all the possible expectation values of Everything that she can ever measures so so we have formula before the rewrite it expectation value of L is sighs star How offer since he was a Alpha Prime they are primed already a prime prime this is a combined system
going back to to the combined system on L a trial date signed a B Bowie said yeah we said equal be prime free likely we be prime here and play a clerk at the Delta of B & B prize this Alice's variables and so as far as Bob goes it's just a unit operator does nothing to Bob's piece of the wave function and stick Beagle to be prime OK look at this quantity right over here with stake L only outside stake L. a prime in Haiti out them isolated off after some over some of everything we some over B and his a quantity right here that depends on a prime AT and multiply something else that depends on a in a priority but doesn't depend on B with some Overby wanted some be it doesn't depend on B anymore so this object over here as a function only of Alice's coordinates alienate prime it of course to calculated you may have to know more about the way functional once you've calculated it that's all that Alice ever needs to know 0 to calculate anything about her half of the system that's given name it's some of sigh star of a prime be sigh of a B doesn't depend on B degrees of freedom Bob's degrees of freedom it's called the density matrix of function things which depends on a prime a matrix its electric service to entries any matrix we thought of as a function Our role column this is a density matrix of density matrix doesn't matter where it came from doesn't matter what Bob state was if we know wrote role than we know that the expectation was right formula the expectation value of is equal to the sum of the role of love a prime in a hell of a crime I think I have won backwards so I don't if she knew this density matrix could forget about Bob completely and say I know everything there is to know about my subsystem and here it is what is the role of AT and a crime of we start with the purest state supposing we started with Pugh state that nearly exhausted is a squatter camps for you started by but this is it OK I can make it me easier this is as easy as it could make it yes it down with it yellowthroat and penetrated but I'll do my best OK this is another server density matrix What if society is the state what can we say about density so let's go back and see exactly what if Sonny is what society is a product the signs our state means the wave function factor so let's go back UAR use the density matrix the wave function factor arises in 2 of b up not that not in the density matrix In this form yes it's a bubbles on double somebody only involves Rauscher sewage from around and see what we learn about the creature called the density matrix if site His a problem that there must have a special forms of so let's say society is a priority of Alice's wave function times bobsleigh function I forget which forms which that was sigh our source are but sigh was Bob sighs out circuit the matter would be changed some of the video of the complex conjugate here is Alice's wave function times is not the way function of Alice's is the wave function about suspicion currently the court ours is wave function Alice's wave function Bob's way function the complex conjugate and that we multiply that by they None complex conjugate saying it put the aurify of B must identified B would be here some B prime and but Prime deciding AD each 1 factorize factorize isn't function of a prime and be in the front function of be and now only some of baby was a summer be off I star 5 1 right OK so now we can take this up out and he is always find we find that fraud trial state only for products state that role is simply made up Out Alice's half of the wave function without any regard the Bob she doesn't need to know anything about Bob this is not a surprising that she didn't need to know anything about Bob after all is a product function and the density matrix is just 80 80 product Of the complex conjugate of the wave function with away from itself don't different 80 and AT and the price I want Tilley say about
a matrix like this let's call this matrix what's called a is actually role of court role of ANA prime score switch your but that's just a definition let's take this role it's a matrix and act with on a column ideal act if you know the components of a matrix patio lacked our column vector sorts oppose this a matrix now role 1 1 just 2nd rower warned to solve off road to warn make make up picture a yes I do support Turks what Ochoa have wrong it's made up of Alice's wave function and I maintain that this density matrix thought as a matrix has eigenvectors I maintain moreover there has won in only 1 eigenvectors with non-zero Eigen value and that that I get value was plus 1 so we say right good OK let's truly up tell you exactly what that I did victories its society hits sigh itself is itself so let's check that out let's multiplied by signing of a prime factor and some already prime this is matrix This is the process of multiplying column vector by a matrix that's way you do it you write down matrix you're right down the column vector somewhat over the year the element the between an OK with see figures you we get all this is the action of Rome strike OK so as do it role he is wrong now it's multiplied by sigh of a prime and now some of a primary get sigh star of a prime-time sigh of a prime summed over a prime 1 right some of it all turns out just be site of a society where they tell you about eigenvectors Amargon values of the density matrix it tells you that the original wave function is in eigenvectors density matrix of eigenvalue plus 1 and April now all vectors perpendicular direct and minus 1 the dimensionality of the spaces and and this is 1 eigenvectors added In which has argued are you want let's take indeed sector now which is perpendicular which is orthogonal society the site AD Dallas's wave function let's call it site killed till the narrow means it orthogonal to the original Alice's vector now losers in a product 0 right so we find out that a role in on-site twin is 0 that means that Eddie that the perpendicular factor is an eigenvectors like value 0 the original state of our suspicion is an eigenvectors with plus 1 conclusion the density matrix for a pure state has 1 and only 1 9 0 I get value equal toward it where does not approximated it is not product stated that you find a color of collection I get values and factor in general there will be all of the eigenvalues in general with none 0 of the density matrix you can try this out for these young for to spin system work out the density matrix of Alice's spin density matrix of Bob spin and you will find out that they're quite different they don't have a single eigenvectors which is non-zero they have a pair of eigenvectors which monster or so what we what we learned In that is information in the density matrix of particularly in Neil Eigen values of the density matrix which tell you where there is a great deal of entanglement on not the more eigenvectors which I'm on 0 more entangled it is only 1 eigenvectors which is 9 0 it's not entangle eh Alice half of the wave function is just that I give back if their entangled then 1 finds and the more they entangled the more eigenvalues you'll find maximum entanglement is where all the eigenvectors are equal all the eigenvectors are equal although I did value where all the Eigen values are equal bats maximal entanglement take the singlet state work dealt the density matrix of our suspicion what you'll find it 1st tool Eigen values each of which is a half equal and equal tore hair so that's it steps awaited test whether system does or doesn't have entanglement calculated density matrix and study its idea values maximal entanglement is when they're all in equal Mary called the toward Yukon prove that blames the dimensionality of the space for it OK that's a that's entanglement quantified more
quantitative me more quantitative now let's not do that our our our right a more quantitative condition notes Burma to quit are not firm enough for calculation for Mormon but very quickly go through the connection between measurement entanglement was a deep connection between measurement and entanglement basically measurement is the process Of the strangling of starting with the product state appellate state of 1 what a profit state of the system and the apparatus the apparatus is a definite state to begin with and the system is a definite state what state the apparatus in well the apparatus might b something which detect something or other animals not detecting you might put in some neutral state some neutral state which is starting state on detected the stayed neutral state of the apparatus and that you bring me system that you're interested in right up to the apparatus it does something to the apparatus something happens the system itself and what's left over His entangled state between the 2 of them so I'll give you an example I'm going to think of the apparatus and a very simple system with only 2 states but a different now my system as spin and my apparatus also happens the have only 2 states you could think of it as another spin if you like the fact that could be ours is Bob's but it could also think of it as a more complex system but for simplicity only has 2 states it is the apparatus and has only 2 states because of what zeroing 1 now 0 is a state before the apparatus His switched on awards which was switched on not detected anything it always starts the state 0 here is not the state 0 0 give it a shot and make it put into the state 0 otherwise it's not her otherwise you have starter apparatus the home state otherwise Kabila measurement so 0 0 1 but starts at 0 and we have our space is a spin Morgan to a measurement on support to the spin is in upstate and the apparatus of course be dealers in the 0 state still represent that by combined kept their combined factory is in the hands of a product of this being Hilbert space and the detector Hilbert space and 0 yes the space comes in to begin with then only then you start out space down 0 White because you start out with a detective in the neutral states now are not going to tell you in detail what the interaction is between the detector and and a step that could involve magnetic fields and other kinds of things causing an influence between the detected but I am going to tell you that if you bring us into the detector and then take a spin out of the detector I'll tell you what state is leftover particularly are making this up of course I'm making up a rule which would be 0 good detector which would be a good detect namely if the system starts the combined system starts in the state you 0 van the detective feels a spade and jumps the state 1 because they you want nothing Abbas suspended just activates the detector on the other hand if the spill is that it doesn't activate the detector that's my role so this goes that that don't worry not found 0 down 0 that's a very good detector of up to now but now let's start with a superposition of up-and-down stake by now thought up time's up plus often plans down somebody now brought us a spin in in a superposition general quantum state but of course the detector starts out 0 so that's a start state what happened after the detector acts well if the state is out of his you 0 it goes to you 1 so that means a Costa alphasub you kind you want plus out for some bearing down 0 0 His all of a sudden has become entangled state it's entangle state if it up the spinners up the detector was out of the detector is 1 of the spinners down the detector is 0 that is and entangled state applies in final status and Pangle state because by looking at the detector you could tell what has been is going in fact that sort of measurement is a measurement is in interaction between system and an apparatus that leaves the apparatus in a state that depends on being whatever it is measuring about the system so from want point of view you could just say Look this is exactly what I know measurement always was adjusted release just a process that leaves our record on the apparatus What initial state laws of the state but from another point of view it's the physics of our combined
system the combined system being spin and the apparatus which leaves them in Ohio entangled state so tangled by looking at these detector you control what spent so entanglement and measurement a really the same thing reflects 2 measure a spin you really only need another spin a survey detecting it could just be a spin in 1 and 0 could be up and down so we had this kind of concoction here where exactly this happened it would take only 1 Spain to be a detector or apparatus which could measure but in general this b nature of measurement that it and triangles systems together tables a particular apparatus was systems circling Muri clear now solids that's 1 way of thinking about as rebound this language it what you think of the system and the apparatus together you never speak about collapse of away from you speak about the development of entanglement if you don't want to include the apparatus In your system the only thing to say is that it after the experiment is done the system as either up or down and you get better and better which 1 do you pick you pick the 1 that you recorded authority recorded is the 1 recorded by the apparatus so thinking about justice system by itself you have that this rule of the wave function collapses thinking about the system plus apparatus the same physics is described by the development of entanglement between the system apparatus that's a lot of what you see on your what OK now you'll are another detector that this is the hierarchy of detection on hierarchy of measures is a spin the apparatus but when I finished we still have a look at the apparatus we haven't got anywhere as until we look at the emperor right alright well what's gonna happen again will have to say we look at the apparatus we want the collapsing the wave function 1 of these 2 pieces of the entangled state or we can say include the eyeball into the system in that case we should have included under the system here the eyeball I bowled I don't know how it started started I it also has to states 2 states corresponds to the 2 2 possible answers that that the brain up there was also a brain appear do states of the brain 1 state of the brain says I saw the apparatus with 0 of the state of the brains says I saw the apparatus with 1 if we want to include the observer as well as the apparatus we have to include another factor here has to depend on the state of the brain let's call it be that's could be it stops being knocked which is the brain before it is looked at anything and then it goes there be 1 over here will be not only here it's now become a system which is entangled with me I would be fine and bring it is a way America now we now have this entangled state we describe all by the quantum mechanics of entanglement between an apparatus them later on between apparatus eyeball it's all been described by quantum mechanics but he may not be satisfied you may worry about this other fellow comes in new Rule Texola or a asks you what'd you see somebody else comes in ask you OK Charlie what did you measure and in the process collapses the wave function of media and Spain's apparatus with his name given the name not much could this is jolly he becomes an afterward we live in a wiki the when novices a strutting showed I don't know which 1 of these has ears this in the Adams of the the cap deals look at the yellow buddy gets hit said something happened something happens to me so easily detect captors the detector prime Minister owning the looks into the box and his Wignall looks a troubling there is a hierarchy here the hierarchy weight you decide that break the hierarchy where you decide to break the hierarchy Steve Germans at what stage you will say the wave function collapsed life arms I suppose at some stage you have to stay away from collapsed over Norwest person looked at it but you could always include that last person the system and that no collapse happened but then a what makes sense Out of the whole thing again you have to imagine some other person so you always have to mention a hierarchy of observation is such a hierarchy of observation the world is built like a year like an onion rings inside Mexico now next thing our next and each thing gets to observe inside of it the profit where are you but what at that time well and armed but I don't
think there is a short line don't think a short 1 we can't describe big systems uh as having little uncertainty but I'm not sure that anything really different peppers where there are electron looks set another electron account books electron I think the physics of the mathematical description deserve the same description or at least the 1st half we want to say something special happened when the cat look that's not will describe require mechanics we have to introduce something about camps it is contained within quite mechanics and personally I don't believe there is anything about cats which is not contained corner mechanics so happy with this description I'm happy with this description of what makes me happy is that it doesn't matter where you draw the line you could draw the line in 1 place the next place all the next place the results will be consistent if they can get so the spin up throwing the will see the cat having seen the spin-off and William Miller will have ceased Rhode Inge having seen the cap having seen spin-off of the nature of the state so that they find satisfying but you don't ever have to say where you draw the line and until you draw the line you put line as far as you like until you draw the line everything is described by the rules over conventional corn attacks because of of bad luck but OK now yeah on book I think it's some stage you have to decide for for yourself how you think about this I'm not going to try to push money interpretation of quantum mechanics when you particularly since I don't know what to say about it except this seems to disturb a lot of people the small business of systems which shuffle which you could know it as much as can be not only know nothing about the parts of the system seems disturbed people disturbing I got 1 final was asked whether he thought it was a problem but Kwan mechanics Zinta was the problem was so big that he couldn't tell of the was a problem he normally use it he normally predict with it could predict anything that he wanted to predict had a better and that he did not expect that Quam actually speaking about myself per year no demand for him he did not expect the quantum mechanics would ever go wrong that the rules would always be satisfied but is there a problem because of this sense of unreality about where actual object with objectively something happen because the objective death of the cat happened when the capital of the Alamo was it when Schroeder looks can always a way when you look Schroeder go with the object phenomenon happen and maybe would just it is playing with words that maybe will playing with words trying to use words in context we don't apply you have to decide that yourself my goal was to give you the tools the be able to ask the question and uh I doubt seriously that even a hint of the question even for yourself you probably come away feeling uncomfortable about it but you know why you're a comfortable what it is with some precision Mickey you want comfortable about it with hot respect my goal right now is he no where do you with 5 that is what he where she wrote that the correlation of 1 they is that the arrests last I guess that would know has become entangled with the rest of the system but Due to the fact that this article a week measurement outlook none of that can happen they can be partial there can be partial entanglements issuer I'm giving you a kind of idealized situation right now I'm not too serious question people people who watch it well you right now no OK there could be several reasons why I was the 1st of all it could be that the interaction of the apparatus was simply too weak but it wasn't enough the interaction was tour I was just wasn't enough coupling between the 2 systems and then you only get partial information could be your is dead because you have cataracts but in no case be I did a measurement it became entangled with with the system completely it's just the connection between these 2 love this stuff in your eyes and you brain wasn't the good enough to send the signal Bakhtiar brains of the correlation have our blood it was a big hit was a bit messy is but uh sometimes to detector efficiency problem and sometimes it's problem while detector efficiency could have to deal with the strength of interaction with the system or it could just be that it's too hard to get a signal out so I'm not sure what Hector efficiency really means but I could think of several things 1 is a signal was varied imprinted itself on the system and you personally have a hard time getting out the other is that the interaction was not strong enough to really establish a full entanglement with systems both could happen but it really worked for a right would it added that the figure will also observe very this to the that's that
what that I will be a little bit of growth is this I guess the weakest link is where you right let's talk about which took about stock about locality and entanglement which is really where people get hung up and the villain arch villain here will really triangle things badly was Einstein Einstein inactivity as the question of course as a wonderful questions about entangled systems really are understanding of entanglement in much of what we think about it curiosity about the cross back the wines thanks question 1933 who was very much puzzled by this idea that you could have a system about which you know everything in it you knew nothing about the components and he wrote a paper similar complaining about it I don't know if he was the first one to notice the phenomenon of entanglement I had a feeling not use the 1st one get really exercised by saying something is very peculiar here that about I find were quite OK he then used the turned what was the 10 again bookie action at a distance and that got everybody confused after because I signed know exactly what I actually distance may end it didnt mean what he meant when he was using it in this context action at a distance literally means forces between systems at a distance the ability tho affect the outcome of an experiment that 1 place by doing something at another place action at a distance of the Cologne force between electron now of course in reading the real world it's not actually distance but In his world but did know about a lecture you know about their the gravity and no as world the gravitational interaction was action at a distance and the meaning of that from our point of view is that if something came along and gave the we have the sort a sudden shot we would know it instantly for the simple reason that that the orbit of the earth would instantly react to the change in position of the sun the force on Europe would instantly react and so every measurement of everything that you might want to measure would be to some extent there another modified by the suddenness of the sudden change in the orbit of the sun that's what action at a distance may be ability if you like to send a signal by showing the son you're sending a signal to new sending it instantaneously if his action of distant Einstein must have known I am certain that he knew that entanglement did not entail action at a distance thing that stands on what 1 can do what we can work out is supposing you start to do something you could try to play with Steve Waugh surpassing is start with a system which is a two-party system has a wave functions sigh of a and B and from that system you could construct did city making Allison Bob you could construct a density matrix rolled out that density matrix containers every possible for statistical prediction that Alaskan ever make for her subsystem let's imagine that Allison Bob took their subsystems house when the so distant planet Bob stayed in power and are very far from 1 another so far that it would take light for years to get from 1 place to another Bob as an experiment that means is and this could be an entangled status is not none entangled state Bob does which permanent and that has and effect on his half of the system he had but it has no effect what ever on Alice's density matrix perhaps I could prove that prove that tonight is a snub Park prove that 80 physically allowable change in this wave function which only involves bobbed hair the process of interacting with some of the system or whatever it is Bob uses electron knock whatever every Augusta you will not have any effect on Alice's density matrix that silicone mechanics is really put together and the implication is that anything that Bob does suddenly at a fixed instant of Torrington will not have any effect on prediction or any of the success of any prediction will have no effect on the statistical probabilities of Alice measuring anything about 8 full not armed so in that sense it was impossible for Bob to send the message that ours suddenly not he can't do an experiment on this subsystem and yet to Estonia experiment taken send the telegram Tallis far away in the standard way these send messages and tell horror with again wasn't entangled state it could be up and
down mine is down about when Alan Bond does a measure on his spin it has no effect on our assistance he convinced Allison messages say look I slowed down and say got as because I saw up your correlated I thought they were correlated you just confirmed that I measured up both of fewer few months ago and fact now getting a telegram tells me that you have them and that's consistent reveres find nobody sent any message speed of light the mere fact that when Bob measures his variable he instantly knows something about Alice's is no it's no source of anxiety forever localities this would happen nickels and dimes and for perfectly classical systems be peculiarities is different than that the peculiarity has to do with this idea that you could know everything about a system and yet nothing about part on what what is it that if this localities standard physicist definition of localities not risk and are you can send signals Fastow what is it that Einstein Maine what did Bell Maine and what the only other people who talk about these things mean brand they said entanglement entails a degree of normal calories tried explained to you by imagining a game but the earth we are here plotters Alice with their information alone should be enough to want to cheat us if our sends a real sends a real message in Alice hasn't looked yet object so it's a bond Alice is not a lot she doesn't know anything yet as far as he's concerned the density matrix is consistent with that and basically tells or nothing at the density matrix which hasn't got very much information about 1st and then she gets a telegram from Bob saying My spin was up as far as concerned that collapses her density matrix Intel's 1st spins down so thereafter she will use a density matrix which says the been down yet wants a messages that kind of go between them yes she will work you will know something which you know before and she will therefore update with density matrix but that will only take place after a after used in sufficient time to signal properties young but that will be saved you are but it's expected value of yes wants you we expect to get here but that's all also were area not a no expected a expectation values a reviewer called on average values if you know the average values Of all the variables in Oneco noise what's OK supposing we have 1 variable that were interested in what's called again if we know the average value of failed the average value of El squared the average value of L Q dot dot dot we know the whole probability distribution of is called knowing the moment all of the moments of a probability distribution if we know all the expectation values of real ale squared actually for single spin you only need spin and spin squared off then you know the full probability distribution once you know the full probability distribution that so you cannot sell so knowing all knowing a full set of expectation values like this is knowing everything that Alaskan or on actually the 4 that the checked yes young yet Kim Jong that's the strike Gary they are guaranteed the opposite guests but cannot find Alice Doesn't up data information about until she gets before Ellis gets information from she knows it's half a half but that's what try to better Alice and Obama are now going to each have minis so Charlie as Villa give each of them many entangled pairs of house has been No. 1 and Bob has his spin number warned and they are can a relationship of being entangled also Alice has been No. 2 which is not entangled was the No. 1 but it happens to be entangled with Bob spin No. 2 Charlie gave now Hall series of so they can do statistical experiments to collect statistics on when they've collected statistics they will discover that every time at our barbers up houses down every time barbers houses up but now they could do a lot more than that they can now see what's what's a good example for redundant column
young right guard Bob does all of his experiments some instant of time and find out every 1 of these find out exactly what every 1 of them is doing those that convey any information about which she knows that just about half of them are up and half of them Obama Bob goes in makes his measurement he does is measurement on every single 1 of them those Alice know anything afterward that she didn't know before or not those anything surprising happen when she measures these things does the statistics of these things over here in any way change because bond measure these things now will be no statistical difference Popkin those measurement measurement the wastepaper basket is measured every single 1 of these it has no influence and wet Alice measures but if Bob sends a message that hours before she doesn't measurement and says spin 1 came out plus came minus blah blah blah there is simply Alice knows everything about the fact he could say Sigma Z equals plus 1 but he decided he and not the measures signal 87 measure Signetics only here minus 1 hearing measures signal Eagles plus 1 having made these measurements without informing Allison anyway he does nothing whatever to the statistics of her measurements but he now says there she can now decide if you decide to measure and things you want but if she decides the measure signal because she know that Bob plus 1 here and Signetics for this 1 signal for this 1 will find it every case the adopts saying that if the telegram I hope of course yes but not the important point it takes time to send the message Yoshi the I missed the game a lot while you're serves problem is he's but you guessed was not a big deal out of nothing else that gives us it really well I am I will try to give you a precise definition now I there is a theorem lurking here I don't know that the here has ever been discussed in the computer science community circuit that could computer science theorem whether anybody is actually discussed it in the language of the Italian there I don't know and I don't really quite know how formulated as of my never tried love but I think I can capture the essence of what the work steering would say if only Iran could be made precise but we're 10 minutes to try to do this for goal ruled the earth will have to go will that is again I or play I want to find the right a computer program my computer my computer screen obtained screens and all the usual things a computers have over here on the screen is a little black box and black boxes called speed you can't see inside but something in it and in here is apparatus a An apparatus you could see to it that works doing so for me it's gotta 0 can measures your old 1 now is again on the memory storage things the memory stores information about the spin misses a completely classical computers the game will be played I'm not a fool you into thinking is really a quantum mechanical system in here how you do that we could do it I am going to write a computer program which completely simulates classical computer program completely stimulates what happens 1 apparatus look have a it OK his awaits gotta go on we're going to start just the just the convenience will start with spin in a known state so the apparatus are computer program it's a portent of the computer program let's say it starts out with this spin Opel these apparatus can either read 0 0 1 there's a button over here in this as a but the button over here that touched and that but his them means you measure measure and so when you touch that but not here the apparatus measures the spin a computer recourse things they recorded the state vectors of the spin hazard do that by providing till complex numbers stored in the computer until complex numbers out an alpha down subject to the constraints that Alpha it was star offers eagled the 1 so it stores and information which is equivalent to the state vector of the spin and the other thing it's stores is the direction that appointee apparatus you can move the apparatus around by playing with a computer number around Whittaker Marie oriented any direction that you like you start with a spin art now this didn't make be interacting with a magnetic field or whatever is happening to it the computer solves the Schroedinger equation solving the Schroedinger equation for the spin that means it updates the state of the start with a particular state was nowhere near as an effort time he updates the spin the updates the state so the computer knows what how often is after a certain time has elapsed then you punch the measure but and the computer take the state turned violent will pair of probabilities now the star Alpha style offer down the computer now knows the probability for up-and-down there also has a random number generator they number generators are a dime a dozen and they work very well With a random number generator event takes the probabilities Alpha Kappa Alpha Darren and spits out a number plus or minus 1 with probability star out an off so just simulates the quantum mechanics by having a random number generator that is that's our plus 1 0 minus 1 but with a probability that's given by office star off but then the computer then whatever the answer is whatever the answer that the apparatus spit out the computer then updates once again the state of the spin and says it is in the state of the apparatus measures of the apparatus it's out plus 1 along some access you you get turned and the apparatus but because when you get apparatus you have to do a slightly complicated calculations of things final with the statistics are but once done it whatever the answer it the computer goes back and updates the state of the spin make it consistent with what the apparatus measures you could do you could try writer program the program that could be written probably not even terribly complicated and if you are the only got to look at the measuring but decide to measure and look at the outcome you would get Everything exactly equivalent to the extent that a random number generator for me of good faith for random number generator you completely simulate the quantum the mechanics of the system bus in this that quantum mechanics can be a special case of classical physics so what's going on what the Cana can be a special case of classical mechanics what's all this confusion about corn mechanics being the logically different right now I want to play the same theme with a pair of Spain if I but they it
I they talk that immediately however OK but it's all game but it's all this is a classical computer and using the classical number generator classical random number 2 Eric Seidel day in and but the difference between Kwan mechanics in classical mechanics is random you could generate all sorts of random number generators would easily fool you into thinking they were really random because any real random number generator the numbers come from somewhere else so they're not strictly random but that you could sure make a random number generator would fool anybody solo yacht you could you could build you could build a computer program to simulate the quantum mechanics basically of any system that you like but saying the same method but now supposing you wanted to simulate a system consisting of 2 electrons 2 electrons the boxes and you wanted to do it with tool computers and in the process of taking electrons apart from each other so as to ask this question about the you want to pick the computers you want coldly information about Alice's in 1 computer and barbs Spain In the other computer physically separated them and ask Can these computers simulate quantum mechanics or play with the answers as long as these on spins are in a product state as long as they stopped and product state and they don't interact with each other and therefore sustained a product state thing you could do it you could put all of the information about Bob's Spain in here all the information about Alice's spin and he likewise you could have Bob's apparatus over here you can have Alice's apparatus over here little black boxes and is just basically tool identical versions of what I said before you could take the computers physically separated do experiments on each 1 and they will completely agree with the results of quantum mechanics because their products state 1 doesn't care about the other but they say 1 if I wanted to put these 2 spins and tangled state well 1st of all to put them in the tangled state you cannot do it without bringing the computers together bringing the computers together so that they talk to each other you could reap Caloocan recorded in the combined computer system now brought together many companies numbers as cake to specify the state of to the system but 6 sulk you recourse 6 numbers in the computer he 6 numbers us Our recorded the computer end some numbers to reflect the orientations of various various apparatus respect you might conceive of and then he tried but the computer into 2 keeping Alice's information in here and Bob's information in here separated and try to stimulate the quantum mechanics of the year of the entangled system Bell's theorem is with tells you that you cannot do this on the less there are instantaneous actually distant phenomena are taking place between the compile wires between the computer server speak hidden wires that have to be part of the system connecting the computers together you can stimulate Riyadh the quantum mechanics for the classical computer system you can't simulate the quantum mechanics of and entangled system on a less it's really just 1 computer but really Moon 1 computer means that signals could be transmitted back and forth basically instantly that's what Bell's theorem really says now is this is this a form of None locality will you might think that if if you really have to have these white use them to simulate car mechanics them send instantaneous mission to these wires of course you can but not if you will only allow yourself to to do the things which quantum mechanics permits for example there are things acquire mechanics doesn't permit because no I committed the X and Y component of the spin simultaneously all kinds of things that does not permit due to do because they correspond to tonight commuting operators if you restrict yourself to the measurements which quantum mechanics allows you to do either 1 of these or any combination of them then it's a very magical special fact that you cannot send information festival Spiegel cannot send information instantaneously by this computer set up he was of the visual art all around it does America if you want to stimulate a quantum system with the classical system of computers and you want to study and Pangle systems the computers have to be able to communicate with each other instantaneously that that's the content of Bell's theorem there does that mean that there's any action that spooky action at a distance or does it just means that you got yourself into trouble by trying to simulate quantum mechanics of classical mechanics my view is that you've got yourself into trouble by trying to simulate something which just doesn't want to be described by a classical computer system and deep an interesting fact about quantum mechanics Einstein's view was that this is basically evidence that however that the description incomplete and that there are other variables of the system and bells conclusion was those other variables have to be hidden wires that connect distant places so what you could take a photo of however you see it yourself yes if you had a quantum computers they we had our quantum computer you could door separated right now we are better off here not figure but out the right but you can feel that it is not the same problems but look all of quantum computers is an electron this computer electric is just a quantum system it's just a true quantum system quantum system and each 1 of these which has been entangled that saw quantum computers is quantum mechanical systems which can reflect these these logical structures of quantum mechanics in the way you make it back actually making a pair of entangled electrons and something real husband triangle between them and key be stimulated by state computer program on West the computer of poses property of Ojeda car crash full don't you had Cher's right at the spot where if is Deuteronomy areas where eager to Ashley not yet on jails we got the other boat which can't do what you can do it synchronize them in advance and then not let them interact with each other that won't do it with a random numbers have to be a random number generators have sought with this 1 measures responds to it that 1 that the ran the city and whatever measures because they were always generate up and now reversed was the name of love you know it's simply it's simply not enough somewhat says it is not enough to to synchronize them in advance and then take them apart they've got continue to interact with each other that's what that that's what that they must continue to interact with each other and feed information back for or ages if it were some breeds of problem short the ball
changed Alice's there at Todd everything I know about it what I know about it there is nothing but do which can influence the statistics that was measurements without sending her measures they are hot track were moment 1 2 more questions it added that you have to to say deathly plaid yes Ukraine classically simulate quantum mechanics but you basically have to do it with in a single computer where all parts of its speak to every other part you cannot physically take them apart the way you would want to take the quantum systems apart all of but part of it or else or they were better if Mel a tangle system Bob condone his measurements Alaskan armature but there are other things that can be measured I just the things hours would measure under a system or the things of Bob would measure for example neither Bob no Alice by themselves can measure stigma X Times Tower acts thanks both of them to do it it takes both detectives text both apparatus make a measurement of X Times Tower this is something that you cannot measure now you commensurate with 2 apparatus the question is are things that you could measure that you couldn't measures even with those 2 apparatus is yes there are next how experts the White Tower why forcing eat housing is something that cannot even be measured by using both apparatuses Dixon no kind of apparatus which is not a combination of Alice's and Bob's original apparatus which each measure of Bob things about us yet nobody could measure entangled no but can discover that he's got she has nobody can do anything with 1 of their but Navajo replication described the way where where describe we have many replicas of the same thing many replicas of the same things Bob can discover the fact the His state is entangled with something and the way he does it busy says Let me see for a direction space for which the spins
of definitely aligned in some direction his hours he knows all the systems were preparing the same way the role identical each parent identical to each other Perez identical to each other and wondering could each 1 of these B in a product states St. products paid for this 1 saying products that once in private if the products they bodkin search around he's got enough versions of the trials that 1 with experiments forever and researcher and for a direction where when he measures to spend his days he always discovers that it up now the 1st time he had you may be in the wrong direction refined our next time you may find out who tried again it will be down wrong direction and keep searching around if it's a product state he will eventually find a direction where the spin will continue to be up up up up after he's found Everton entangled state no matter what direction he measures you will always find a spins around that's that's a character of it In a separate separately that is not the same Ewing you've got have them ensemble or over identical systems and to do anything like just prepare each won the same way to experiment and doesn't care about us to just discuss measures along some axis of the axis of other axis and Kylie finds 1 weary keeps repetitively getting the same man and says that's the direction that that's been polarized Anthony keeps doing it finds consistent never find that was year products 30 British surged off porch for more please visit us at
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Metadaten

Formale Metadaten

Titel The Theoretical Minimum | Lecture 7
Serientitel Lecture Collection | The Theoretical Minimum: Quantum Mechanics
Teil 7
Anzahl der Teile 10
Autor Susskind, Leonard
Lizenz CC-Namensnennung 3.0 Deutschland:
Sie dürfen das Werk bzw. den Inhalt zu jedem legalen Zweck nutzen, verändern und in unveränderter oder veränderter Form vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen.
DOI 10.5446/15016
Herausgeber Stanford University
Erscheinungsjahr 2012
Sprache Englisch

Inhaltliche Metadaten

Fachgebiet Physik
Abstract (February 20, 2012) Leonard Susskind continues to discuss entanglement and what the concept can tell us about the nature of systems and the nature of reality.

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