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Quantum Entanglements, Part 1  Lecture 8
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talking about less time was the density matrix description of probability is known as a matrix is another way of speaking about probability from probability postulated and it's a situation which is more general than knowing the state vector of a To roughly speaking when a system has been prepared but you don't know how it be prepared and sold it may very well have been prepared in a very definite state of some sort I give you an example West kind where electron might be prepared along a given along the direction by placing electron strong magnetic field but somebody else did the preparing and didn't tell you what direction the magnetic field was perhaps they gave you a probability distribution for different directions of magnetic field Thank you for a coffee tonight I really need of apoptosis coffee so it's a situation where a system does has been prepared but you simply don't know how was prepared and how described has assessed some description to what may have some degree of knowledge about how was prepared humane and not incidentally if you have none no knowledge at all in the density matrix has it all equal I give values its proportional to the unit matrix each I get value was 1 In way In his total number of state and that just means complete random information you simply don't know all the possibilities are equally weighted but let's just go back and discuss a little bit more of world the review the review will be equations of the basic concepts of Golden density matrix the density matrix is at a lot of knowing the probability distribution a classical states classical states have some probability distribution in any context you would call a probability density density in phase space density in coordinate space a density of probability that's where the word density matrix came from and it's basically a quantum analog Of the emotional probability density of probability distribution on condition that the sum of probabilities adds up to 1 here is denoted by saying the trace Of the density matrix is equal to 1 to remember the phrase is the of the Eigen values the eigenvalues themselves of the density matrix can be thought of as the probabilities a different states 1st of all the faces of the density matrix is equal to 1 of the density matrix itself is a her mission operator that's basically just a condition that the probability is a real probabilities is a real Sony Classical Ecuador's we'll probabilities only somebody with a very bad disorder would would decide our complex probabilities would make any sense the analog of that require mechanics Is the Eigen values of the density matrix recalled 1 of us are valued values of density matrix are real let's label them land that body where I is equal to on equals 1 0 . 5 0 in ways in the number of dimensions of a vector space on the end all positive spirit and there or positive right each Eigen value goes with some Eigen state Her parents happy some bases some particular bases army in the space of states so off normal basis of states are corresponds the eigenvectors we remember that be eigenvectors of mission operator are all orthogonal to each other so they form the basis of some basis in the space of states which has just been mutually orthogonal vectors each 1 of them being in eigenvectors and each 1 having value and survive you could simply thinkable Landis a used as the probabilities that the system was prepared in eye eigenvectors Harvey a density matrix if all the eigenvectors of all eigenvalues are equal effectively means you know know nothing about you don't know what state system is the density matrix then it is proportional you make all same argued values each Eigen value being 1 of the best 1 women at the limit of complete ignorance the limit of maximal knowledge is a pure state that means you know what state the system has been created in the neck tasty and say the system is described by some pure state award value which 1 but solely was so we know which 1 of pure stateside end when the system is described by a pure state distiller density matrix you could still introduced a density matrix and the density matrix is simply the projection onto that fewer which state the projection operator project on that pure estate written this way both was only 1 state and you know it is probability 1 being in that state the density matrix is just of them is that this is a lot of the death of all know what are known or the concept he density matrix being permission operator has a set of mutual your final eigenvectors only makes sense to discuss the probability for which state in that based on basis and the only situation you would talk about now we can we can answer the question we come back to it on where you will know how the system was prepared somebody may have pay put magnetic field this way or they have put 45 degrees we can forget about we figure out later are compact but still it will be the case that the density matrix always has mutual fund and values so I I will pick case now yard or next time if if there is a big state means a density matrix which simply has more than 1 9 0 I get New York and are Yao me that's what OK in Germany which called good fish toward I learned when I was a young student because German was still 1 the important languages of physics now watches called mix state at Haiti by Germans trimmed Germans now but others curious whether Germany in Germany the call mixed they expect there will be issues the word villas they could buy the fathers of or McCarrick said the German word that are OK now what other yes let's just point out 1 thing take this matrix what it this operated it obviously has a single eigenvectors with I get value 1 what it the idea that there was major providing everyone it's just not the same backed might take this and operate on it but assumed an upside is is is normal assumed the normalized if I plight is the size and type size equal to 1 disk is that side so that
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size and eigenvectors value 1 Paul any other state or orthogonal the party let's some other state which is off signed a qualified little orthogonal renewed is 0 so if the density matrix is a simple projection operator onto 1 particular state that has won a nonzero Eigen value the density matrix is representing the purest 8 and appear state is just try know whether or not the state is pure Omega makes these roles horror average values they usually called even in this case expectation values Our I don't wanna confused determine our confused the patients or just indicated an average value but the average value of the observable and which all indicated by putting a bar top is just a trace of mind your faces the traces of the density matrix primed em and incidentally it's not hard to prove that a trace product to matrices whether or not the matrices commute whether or not the doesn't matter which border you put new Brooke beyond are the operators so you could also write trace palms role and will make a difference that's the average Our and definition of the trace for me remind you of definition of the trace of a matrix you take they what's to make a quick you take any basis whatever the traced has the property that it doesn't matter which basis you calculated its in variant of the changes of basis most of the year and vector space right but any case we take some bases it doesn't have to be the basis on which rose diagonal it can be the basis which rose bag but our it is not that no space nevertheless the traces defined as all right Rolland whatever the matrix in question is planned by some of ice OK just II matrix elements from the operator question EU summit for war that's the definition of a trace Emirates work it out just the very simple case in which row is a projection operator In that case and is equal trace our to trace mnemonic sigh sorry and now we take a complete basis whether or not has said it does not matter whether the basis is the basis on which role was dead wrong party are had to know we have to do a little bit of gymnastics post offices some old the right somewhat wired going it's almost always the case that when you have index like repeated twice almost in every equation I will ever write down its implicit or are they got out explicit ever forget remind me but it's almost always the case that it will be some Nova so that I is really just summation index are college a few like now lets check this out with the snakes and on weekend we write this by just shifting this factor over to side when you do it efforts rewrite fight so I am sorry are now they have to remember the rule and backtoback like there's some that's the unit matrix that's evaporator the sort of I might be matrix cannot always be canceled out with just a unit matrix it makes no difference images remove it and the result sigh and size was not a surprise the average of em is the expectation value of if you have a pure state now supposing you don't have a few estate but suppose we don't ever fewer questioned them but this is sorry I'm up here yes thank you good how yet OK let's says let's take more general case when we don't have a purer state and see what we get but let's evaluated colleges yup OK so again we have to take a trace of Rolland but let's still the calculation in the bases it which rose diagonal so to calculate the trace with put on role in that I saw no I am not going around estimations starring suppresses emissions for now the song although I know we can insert a complete set of states between role in and this is a trick we constantly keep doing in order to reduce the calculation of something involving matrix elements of both Rome and we stick In between there a complete set states away right at the former I maybe I'd Agassi said purpose or someone IEEE IEEE IEEE role and I'm going to insert in here a sum over shape analysis some of them both my NJ before now that and working in the bases In which rose diagonal then I know what it would be matrix element are world Javier this is just the these on a roll Jerry was that's just the Matrix has it I index OJT index the roll or column North Sea which was which is what it color whichever a young girl you column yacht OK but since role was bad this subject you're only has a matrix element of volleys people to Jerry and then the matrix element is just Lambda Lambda Jed the item W if so what this summer while jail becomes a song on only the over 1 of them and what they to to be Jagr because I has to equal Gennady costs was bad and then we have Jerry M. J for J. here because I has equal Jerry In the what is this this is just the expectation value of em in the G8 state this is as the probability of the G8 state this really exactly what you might expect it's the average of M. J state waned with the probability which is the probability of the Jewish state so you could think of the Eigen values role as the probabilities to be image a steady but only in the bases which rose backed down the game is a yes because I has to be equal to Jerry otherwise you don't get anything so for trivial reasons as a sound fiscal summing over JETS over Jerry crack yes he
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really broke my my own roses repeated 3 times right a there a little rules about city matrices them to remind you of 1 more definition is if you give a probability distribution probabilities let's say a piece a body says classical probabilities be supplies the Ike probability for I possibility and entropy the fighting S S for entropy everyone rituals were married guy we because it would they fifa energy off as entropy is equal by definition is a definition but we can say what it means for roughly its again the summer of what all of the year possible configurations of all the possible possibilities of Peace abide plans logarithm of piece of ice that is definition of getting it death only 1 of the pieces none 0 71 logarithm of 1 is 0 on the other hand all the other P is 0 0 0 so in that case if you have complete knowledge there entropy is 0 0 entropy means complete knowledge you know precisely the probability distribution is peaked at 1 particular point your everywhere else the other extreme is when you have complete ignorance and in that case on each piece art 1 of a number of things the same India he'd be supplies 1 Ave it and then you get past it is equal to 0 before you do some 108 in log 1 over and it's my lot 1 over and so that's actually saying his locker and but then you have In terms of Osama so you have multiplied despite In the answer is that the entropy is logarithm of the Member States they is roughly speaking the logarithm of the number of states which are important in the probability distribution that's as close as I can get to give you a with definition words but you see the definition of words is precisely what it is but nevertheless it is true that if you have a probability distribution which has steadily large or some states and in premiums wall everywhere else that the answer of the entropy will essentially be logarithm of the number of states for which the probability distributions significant so it's a measure RBE degree of ignorance the more ignorant more states that the probability distribution spread out all over me less you know about the system more you know about the system and a few of the states and small reinterpret case visually ignorant or a measure of world I know you don't wanna come to entanglement entanglements happens when we have more than 1 system so let's suppose we have a sip a system set up a system composed of 2 parts and let's suppose we have some basis and the basis is labeled body 80 state already on observable 84 1 of the subsystems and be other now for the case of does was a and B could be up and a beagle would both be up and down we care about them up and down are other possibilities we can have a spin with 2 states and Adam with 4 states or 3 states in that case a would go from wanted to up and down the EU would run over 1 2 3 0 1 performer and so altogether the total number of states would be father of the number of configurations Of the subsystem maybe plans a number of subsystems be so if we have a spin which could be in 1 of 2 directions and putting it together with Adam which could be in 3 possible states of the state will be CX but would just be 3 times to let some let's say that a delay runs from 1 the capital and and will be guns from 1 till it'll have to be the same thing could be took quite different systems could then the most general state only the combined system that we can write is Osama but Her workers some already a and B of some sort of coefficients called wave function Our AT & function of a and B that India discreet Jr list for us now a B are discrete variables the more general context AT&T could might be continuous but promised studies here was studying cuebid and discrete systems AT&T integers there it integer valued variables but we can still right sigh of a and B sticking in any pair of integers that gives us a complex number multiplied by the state that they may be summed over a was general stately right down everyone normalize it which I do then we want these songs over 80 and be of sigh star side they add up to 1 case right down some of size star I'm sorry should be equal to 1 that's on normalization condition take let know supposed that would doing an experiment which does not involve be it could be because the systems are separated from each other and Bob is down at that end of the room and he has been rocked by looking at a system all Alice's up this end analysis the experiment experiment is going bald only 18 degrees of freedom they are the observable which starts measureless call em it is NI observable for subsystem 80 that means mathematically that when em and my stated that a it may be an actor on it 8 coordinate but I will just leave them be coordinator long we've done this before we've done this was big news effectively is only a matrix in a system here that acts on a degree of freedom and leaders B alone it doesn't touch me whatever it does tho state vector B is completely passive and doesn't change matrix may act on a quarter sold a degree of variables are observable that's associated with me a subsystem end let's calculate the expectation value while in the state for the combined system so that we have to do we have to take 1st of all we have they are vector associated with this would probably right has brought some only a and B sigh star well it still need 2 sets of variables will climb variables and price variables with Sierra or did its young sigh star of 8 crying and be crying 8 prime prime some great prime be prime misses don't know where
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multiplied by area and now the cat vectors which looks exactly the same mix of little Zacarias a of a complex conjugate sigh of a and B B for portray orbits sector sorry and sticker outside here that we separately after some already trying to be primed to define the the band a and B or suffer some a and B defined kept back there and that's the expectation value but were interested in but no let's look at the Matrix M. M. does nothing to beat but it's a matrix as far as Eagles what does that mean that b Prime St equal beat be primed and be the Mary operate a and does nothing to beat leaves it alone and that leaves alone and be prime and be must be the same OK because it is this is that I use I and J is that it would vote no OK well we're not Gilliam on density matrices revealing state vectors so let's go back and expectation value firstaid supposing we have a wave functions on planned a single system and now we want to calculate the expectation and this is of course some already 2 watch then the expectation value and now we have to write the brother that some of a prime of sigh are a prime finally there was a prime OK so In calculating the expectation value the vectors on either side Our expressed as some of who have a double some to do in this case Call the left part of the double some pride in the right part prime so as a safe thing except we have to label but now the definition of the idea that it is an operator acts on a subsystem which is passive subsystem is simply the statement that the matrix is by Admiral In the beaches that be on this side and be primed to be the otherwise the Matrix managers and that case was bubbles this somewhat a B and B prime becomes a somber only although 1 of them so let's show you what the right idea now B & B prime have to be the same in is really just a matrix in me a space it to make directs with matrix elements ANA Prime hardware Warner Colora this work practices a matrix of borrowing 2 variables any in April I but be beat has the equal be primed and low dependence on B whatever no dependence on B & B has to equal be prime focus of asses Electric's here and all have those remember the instructions the rest of the expression he the BEST equal be prime so what star a prime B side of a mail voters on the left we have sigh a prime site but as far as me goes b ee ropes be hasn't changed bees just passive this is now schools step further assist some it's also sung a promenade but I wanna ignored for the moment the sum over a prime in AT and realize that this is a somewhat be also or some a primary but was so that later there are no group the 2 things that depend on led a group of together there was a racist other here and put it back right back yet sighs a private unimagined holding a ANA prime fixed in just something all the beat that gives you some expression he said that expression longer involves be because he's been some Nova Funds B is summed over it doesn't depend on these let's give this name what's likely sigh of ADB incised a prime beef its approach and only of in a prime is not a dragon is not are a prime that I don't like justified commander right letter wrote rally new ASICs pressure it was late in components written components written in terms of matrix elements is the place Of the product times role why is it to trace its a trace because I'm summoned over a prime but the product because I'm something this is the formula if I if I had arbitrary things he a double prime than this would come a double trying a prime matrix element of em times world is awaiting multiplying matrices terms of components this expression Louis traced of him conjure let's check that let's do it by walls in the calculation that really is Tracy now we're talking about a system will only talking about a system now now we've eliminated the beast system with some lowered its no longer even the expression this might have been something that made sense for just over a system without any reference to be it'll all colors so let's forget be passive it's gone we finished with it and this is some expression only involving 8 degrees of freedom I understood the trace Our the Matrix am I remember it really only depends on any b is completely passive be primacy equal be otherwise it just doesn't depend on sold what Van and as the identity on beat Yao Yao ride on acting on beat does nothing right OK so let's it's just calculate trees and we derided as a prime and role 0 a prying some a prime although this sound of AT&T prime a back of some time this is the trace the definition of a trace amount insert a complete set of states here it becomes somewhat already a and a prime are a prime and a E E role a prime that's just this year a a primary this and world O. a apron message I sulk local ways we found that by ignoring D ignoring it and asking what the rules when the system is a combined system with a unique state up Europe vector of purity it's a pure estate
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for the combined system nevertheless when you focus only on me a subsystem described by density matrix that density matrix and you get the density matrix you get a density matrix by taking the whole wave functions letting a N a prime run freely about some of them but something over that eliminates be out of the expression and it just becomes a matrix between 80 and a prime soul when you have a complaint composites Schroeder camped in the garden the shortest cat and the gun may themselves be described by a single state vector to which they were entangled but we just look at a cat along with the cat was described by density matrix prescribed by a density matrix and in general that density matrix will be a density matrix of a mixed stayed in general is a special circumstance under which you will be a pure estate ill described a system that moment what is the effect is worse for you right Mary questions up till now there is growth of more visible than would a trace of no sorry you chase a apart with a proper trees a bicycle what but the How great trace of a product up Prague Villarreal our side a some of ADB of sigh that is most visible general to state of the combined system OK so I missed it again you have a combined system it's in the purest possible state sigh when you look at a subsystem the subsystem will be In a mixed statistical members no analog of this for classical system if I have what would correspond to a pure estate from truck classical system draw classical system with the idea of a purist estate would be a probability distribution which has only 1 point man 0 another words a definite piece of information about which they carry that's the analog of pure estate Lewis complete Norwich if you have complete knowledge about the combined system you have complete knowledge about either of the subsystems in classical physics throwing classical physics Idaho and complete knowledge about a combined system is equal I'm a very good will but it implies complete knowledge are either of the subsystems that's never been quantum mechanics and quantum mechanics you may have complete knowledge about the combined system and yet be subsystem by itself must be described by a density matrix now you state might be that that density matrix is special and better combine corresponds to a purer state not in general are sure you some examples of examples are already familiar with our work out the density matrix for summer familiar example but let just Yacht Club and said it had nothing to do with a bit of a movie that they will be here said tho that he does not believe that the voluntary and would you consider going agree to work a combined velocity and position is classified as part of a plan you have a combined velocity and end position distribution renewed make it it's it's similar it's similar but the difference is but if you had complete not in that case the velocity in the position which is correspond to subsystems and I'm really 2 subsystems but you could think of them as 2 different systems to different coordinate OK and that like completes the knowledge about the system would will the velocity and the position and there's no problem with knowing the velocity of the position and so will not only with the combined system being state but you just look at position as you would have better knowledge a position and knowledge over it was a thing called velocity your really that's a difference that the Kaoopulu states of the combined system which can be states of subsystems OK let's me just tell you what is the condition that is an interesting image what is a condition on sigh of indeed that these individuals subsystems actually following states is there a situation but old or a wave functions of be in which the 2 individual subsystems armor and a purist the answer is yes indeed time that the wave function at the arises Antwerp product of a a product of a function of a the price of a function of beat let's just analyze that exhibit that situation a little bit and see what it says a special case not all functions of 2 variables are products of functions of 1 variable times a function of mother variable but only when the wave function is a product that each individual subsystem is itself pure estate of Hitler's look at them more carefully let's Apollo's Messiah Beatty I happen to be for Prada of a functional and another function and be an clerk all the other function Kiam really Willis is saying is that the combined system is a purist 8 5 Lee a subsystem and which they caII would be subsystem will assume the 5 carline normalized so that Kaiser want some Nova beat is equal to 1 staff I some way aids equal toward those conditions are equivalent to the year the normalization of ties itself both sold 80 80 s rock edits plantains Oregon for the cats yes that's right I noted but but it up a gap as thus is a proxy right back products OK so far I haven't factorize like that the let's calculate the density matrix itself the density matrix for the a subsystem me a subsystem Lee after multiplying million beat times I star you know something I think they actually have something backward outside perfect rift ivory and be done star of AT&T that's fine of a car be times fiery star of a fourlane for him a N a prime but be goes will be transcribed Christ all over being rested density matrix rome 0 a the same formula except substituted for signing the problem I have is easy are the only thing the some Nova is kind he a isn't even in the Yukon polar finds outside the summation summation was about a altogether and it just becomes a defiant if I star of 80 times someone be our be plans Chi star the letters that some be Kichai that's just 1 as normalization of ties be so old as 1 of the density matrix from product is very simple it's just the product of wave functions 5 of 80 times fivestar let's calculate now in this case the
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expectation value of but it's just equal wears if of a prime and sorry I missed the prime here somewhere right right In from a 5 of a and let me put Vice star of a prime over here well this is nothing but a written a components this is nothing but foggy and flight In other words the expectation value of em and not just In but any of that involves only Eddie is exactly the the same as if the state were just fine we a degrees of freedom of our completely entirely described by the purest state 5 so when the wave function as a product of 2 factors each subsystem itself In a pure state and the purest state is just the fire they will cry beat that's a special case when the combined system is the purest stake in each subsystem purist but that's hardly general Murrell General not every function of 2 variables is a problem Our functions of 1 variable pay incidentally in this case the density matrix while here is the density matrix the density matrix has won in only 1 9 0 eigenvectors by itself all the other Eigen values are equal to 0 and entropy is equal to 0 on Aragan values 1 and all we have is 0 so that's the case that's 1 way or the diagnosing House fewer state vectors or how well the the wave function resembles a problem is to calculate the entropy if the entropy is small it means you near the situation of the purest of of of a product if the entropy is a large sum would be the maximum logarithm of it that dude deeply entangled and deeply entangled means you far from a proper often a product from should just wasn't factorize let's work out 1 particular example where we would have 1 particular example of Let's do it leased 1 example a familiar things that we already studied calculated density matrix and calculate the year of our lead the entropy we should do tool we should make 1 I to different examples the 1st example is not going to be a purist take the 1st example is gonna be offering the singlet state be highly entangled Singh would state of 2 electrons OK so what is it by definition up not sorry down minors em up delighted by square root of 2 but let's like that in terms Our 80 wave function of 2 variables sigh has 2 entries the 1st entry represents No. 1 the sedimentary represents No. 2 and he or the values its side up is equal to 0 I don't see up here no OK about sigh of them is but the coefficient of this 1 sticker boy was quoted till arrests I doubt trapped miners 1 over square to and sigh down narrative 0 OK let's calculate the density matrix FOOD system a system now earnings the No. 1 so we have to do we have to a right down Thursday as density matrix right here right it's not over yet this is the density matrix Starting some I we begin by writing down next size star it but without which the studio rolled up our member Rosa matrix matrix just in the space of 1 game so let's calculates up up and fixed element of matrix element we get by writing signed up side by us this doesn't make sense yet it will in a moment that somewhat over the beat coordinator so let's up up down Norris is a lot of calculating beat up I make Excel from back here with calculating AT a price matrix government sold 1 place we put aside with aII the other place we put aside star a prime should be started star In this case both a N a primal up so we have a sigh of a size star of a side of any size star aid but we have some more would be so sold is up up to down there that's the destruction of sliced our society would be a is being evaluated as matrix elements of the density matrix and the being some Nova pumped up close down down Piercy up up matrix element would what it is sigh off after 0 so this is 0 but sigh up then is what Querida too so this is just 1 half right about wrote down down the road band that we do exactly the same thing side down sigh star damaging last saying things side down started aside I started down but they you fill up up down there story right now OK waterside my 1 was squares to its squared forget get 1 have turned them down 0 . 0 services again a half 8 what about sigh of all of them the matrix element of the density matrix the them which Excel those same thing society up so I started down up there plus riding again sorry up size star now but affiliated up up there OK what's that but sigh up proper 0 and down barriers equal 0 0 and of course other was also 0 wrote down rappers also 0 so in this case the density matrix is just equal to 0 is equal or half and half if when matrix has equal injuries entries can be thought of as probabilities of you like but it's traces 1 if you add up the Eigen values of the 1 and that's the Emirates Stadium the entropy not if you you want to Bob Hill I want to wanna have of onehalf was long 1 they have long warned him appropriately is a apply 1 logger warned here is my a lot of tool but we are trying starring definition reinterpret brought to us maximum Libby in Topeka I have their hands stressed there but it's an easy here on our remote tribal work that out the maximum entropy ever can be is where all the elements of the density matrix are equal maximum amount of ignorance and so in this case here we have a pure state with a combined system and complete ignorant for either of the year that fighter of the components are so if either of the spin I concentrated on the left spin he has been No. 1 and what we found is that
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the density matrix described complete ignorance In this case what is the expectation that you Of the components of the space particular spin No. 1 week a calculator them by using the full wave function here always can calculate them using the density matrix the density matrix for spin No. 1 world is equal to onehalf times unit matrix disclose onehalf which is 1 have EU matrix the expectation value of aII component of spin sped along any axis sped along any extras is equal to which placed on world kind is hard to see about certain about a trace of row Back in and that's the average the average are simmer about a trace of wrote the observable back In a trace of wrote that he observable plans trees role which wastes are the product of ruled punch observable that's the average observable well Raul is proportional to the Unity matrix you could take that out here matrix doesn't do anything it just urges gives you have a factor have missed you your have plans the of Sigma got in where's traces seemed not to Nevada and anybody else 2 of the trace every 1 of the matrices 0 or just check that traces of we is 1 minus 1 His traces of the others are
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off and we also have no trace so whether we learn we've learned that in the singlet state something we already know because literary work it out in the singlet state betrays every component of Spain in any direction your which means in directions whatever the spin is equally likely to be all along that direction or opposite direction Will you know this and this was an example of a highly entangled state we're also known as no pews state of a single spin is the average of the spill equal to 0 in every direction so this is another way of seeing that there is cannot be a purist but the easy way to see that it's not appear state visit to Wagon values here are both nonzero so it has to be a mixed state so similar state as a maximally mixed state we take another 1 . 2 x let's let's try the following above found heard figure won't Whitewater square Touvier list were worn half way worn a yacht yet showed your hair yacht notes normalize right size thus I would leave a quarter a quarter for Sequoyah accord was a quarter would give 1 services and normalize state with all interest equal or have led to to recalculate will we we get here if we had this situation this is not a single state similarly have we have popped up because they have so this is a quarter right direct and this is also a quarter who oppose a quarter was given him no difference tour have this will restore have MSDOS this 1 all the matrix elements are equal to half with a halfdozen Force have they have built or legal Tway after time dear factors were was also recall him no this is the density matrix it's traced his skill equal to that that's good some probably is recalled 1 4 1 0 eigenvalues on the matrix of equal not actually not initially ordered 2 0 I get values are was calculated to incidentally do you know the says that they properly Eigen values of a matrix is equal foods determinant as 1 way today us 1 way to check the eigenvalues he got that but that fact is a fact be product of the Eigen values of a matrix it is equal to its determination and the sum of the Eigen values is equal to its trace so we know now is somewhat to widen values of this matrix are equal to 1 that's good summary eigenvectors equaled 1 of 1 of the determinant of right a test times I have my have found so the product of the Eigen values it up the 2 0 that means 1 of the values 0 0 and 1 of the eigenvalues values is 1 With set yet this is a box that this is a product state can anybody guesswork tho it corresponds to this a year and a half high stormed electrons warmly individual electrons ruin mile yeah that measures they worry for stretches in what direction were acts of sex but calculate the expectation value of army but follow the expectation value of segment of seen disease with stars in 3 acts of that we take place Our role times symmetry see get sick 3 and doesn't matter which order 3 is 1 minus 1 0 0 and roll is 1 half of 1 half 1 have 1 here OK let's see what the product is 1 Transylvania was 0 times this as you would for half on 1 times ahead was air for Europe problem half canal minus a staff now I have to take place in face 0 so the expectation value of 3 is again 0 the same ritual signature wool let's calculate the of the product will plants signal 1 now but could signal 1 to get what was the matrix 1 0 1 1 0 0 0 OK Top terms a 1st column we get a half earlier road listeria halfyear also now we get I think everybody is a half right the traces equal toward so the expectation value of single 1 is 1 of the expectation value of the other signal is is you know this is a spin this is a pair of things that happened to both be lying along the xaxis it's a product its entropy is yearold because it has 1 0 Eigen value and 1 value equal to 1 so some gave you this density made Syria and said Tell me is it appears that he always have to do is calculated citing values and calculate its entropy RTS Help worries you could characterize abides entropy in this case the contributed 0 if you change these numbers a little bit for example make this is the 1 half naked . make just 1 point for some very bad argued change things a little bit and instead of finding a 0 0 entropy density matrix it will have a small amount of the mowers abuse small degree of entanglement it further centipede entanglement go together yeah there are sometimes called be entropy that 1 gets this way because systems are entangled scold dangle entanglement in Tripoli Cordaro break West or in the case cases
1:07:15
doing we assumed that the combined system is the state a combined composite system has a definite wave function that means the combined system should be thought of as having a pure estate as far as the combined system goes that means that its entropy is 0 there's no certainty in the wave function on state vector of a combined 0 interpret the entropy that we're talking about is the entropy of the subsystem subsystem now which will looking at which a subsystem by virtue of the fact that it's and Tanglewood with B E a entropy if you didn't look it be a hazard interpret notice 1 thing and this is a general property our from this kind of entanglement entropy of this kind of entropy it doesn't end the combined system has 0 beat the 2 subsystems each had as a matter of fact we have entered B of logarithm of tool in a single state so entropy in general does not a under the special circumstances of everybody general entropy is not an editor of quantity in particular entanglement in trip is not a good thing because typical situation it will be that the combined system is a purist 8 0 entropy the 2 subsystems each will have some positive perfect so OF distinguish between the entropy of the subsystem and the entropy of composite systems can resist Garcia once more is not and if stay they they all want a law that I think probably area say
1:09:16
it really is always entanglement entropy but sometimes it's a little bit obscure Watson Watson tangle what arms but if you count the magnets and people who are prepared the system himself all of our the only reason they would be not knowing which way a spin was prepared would be that the preparation was entangled with the thoughts of the preparer or something but everyone I keep but keep observer rather Luger saying that that is different probabilities that's 1 origins of our arms around the density matrix but the other origins entanglement suspect probably entanglement covers all but then you would have a think about entanglement with what her and complicated but a will leave subject of entanglement for wild creature will come back to it but is 1 thing we haven't talked about all how how States change with time In classical physics if my belief classical logic and if we dealing with discrete system it would dealing with a discrete system with a set of wouldbe configuration space being system of points could be just 2 points that details the only kind of evolution that makes into discrete evolution where time workers and steps and at each step this system moves from 1 state vector took not state vector from 1 configuration to another configuration and you could simply represent the evolution permit thought this that record followed the state you go he had in my car here are here here here here here and perhaps back here that would define the evolution of the system just a permutation which tells you where every state that it does not go connected like this you got for states and the role could be this 1 goes this 1 when goes this 1 as we go through this one's when this 1 also Kay has played well because how we're gonna go from 1 discrete state to another discrete state continuously Hi how you know from heads sales but you could turn Nova slowly but none of that would mean that the way intermediate configurations that would really be a tow configurations to say that there are discrete configurations that I think you have to say that you have that evolution worker discrete steps In quantum mechanics In some sense you could have a well and we certainly have systems which are discrete a finite number of also normal state but Our D evolution of the system can be continuous why is that you you could have well why is is it you could continuously go round up down up means a magnetic field pointing along the upper Texas his lucky electronic place bound electrons has been locked into place by that magnate and of course you have an infinite a continuous number of possibilities of between although those only 2 miserable possibilities for the ops so in that sense the system is discreet nevertheless you could continuously interpolate between them by and about allows the possibility that the state vectors can vary continuously with time even for a system which Our is discreet in the stands repeat dimension of the vector space is finite let's have struck about how state vectors change with time now assumption and the assumption of the basic assumption is that the logical relationships between states doesn't change with time for example if you start with 2 states which have a senior state then allow the system to all costs 2 states will evolve in the same state that trio you start there you start start with a particular state annually Bobbitt evolves at the state level but if there 2 states of the system which orthogonal whichever speed up and spin that well some a tool of final states from the same system to allow them to evolve would evolve into toward final states that distinction is the fact that measurably different will be preserved with time you could imagine in insolently bets that's also truly the classical description that you don't have the situation where 2 different states could vaulted saying state that never happens in classical physics you would lose mineral you and maybe they're here Free stunning he'd go here and maybe from evil back here well then you would lose memory these started here here because will you start here here you wonder Peter that doesn't happen classical physics we of memory where there has been a life patient choice not who is not microscope equable not the microscopic level like this is a principle of physics in classical physics it's called conservation baseball you that these things volume stays the same facts or expands but so 0 far I think what the mechanics of the mother many of the territory but what it says is peak In a product between 2 states status saying what kind of 2 things orthogonal they will remain orthogonal each state will change in its own way but if state No. 1 it is the tools are allowed to evolve with time they will remain orthogonal suffer example if you start with a spin off and you put does some it feel that could be doing complicated things after a while the state will change with time if you start with stay down after a while it will change and become some of the state budget to final states will remain off orthogonal which other principle is that they are on the same right right yes absolutely deal with members of the plant is yet what happened what happened as if it put into magnetic field along the xaxis and you start with viewers spin op or down they will begin the process that will rotate so that my levitate after half the period of something to 1 being along this axis of being along this axis but those orthogonal richer OK so orthogonality is maintained with time even stronger in a product is Romania is is over it a product between collectors is time independent of works but actually I should say that's the 2nd principle of time evolution the 1st principle of time evolution is governed by a linear operator let's start with the 1st principle 1st Stalinist stateside it could be any state of the system and let's Maginot at times 0 after a certain amount of time it evolved Toulouse state which is a state of Parliament seat the principle of quantum mechanics Our saying no is 1st of all that the evolution theory is Comey summarized very easily by saying that
1:18:48
society is always equal to sum linear operator call you which is a function of time cut side of Zero what is your of time chief 0 was you of 0 1 of the state's side at times 0 is just tickled status that time equal to 0 so want EU is equal to 1 . 2 years ago 0 but at some later time the state vector is changed and the assumption that this is forget 1 of these things which is justified by Pericles success could justify empirical success but you'll see as we go along how this connect star of things in physics and why her why might be there how we know this is true but let's just take it as an assumption that there exists a linear operator which is dependent on time spent just a fixed linear operator it also time whatever the state is a time 0 you hit it with a view of the find out what it's doing at a later time that's a 1st principle of time of evolution Pesek did principle of time evolution is that we have to sectors and brought kept allow them to involved with time both of of them buffer that the enough product between them is change with time and other words that equal 5 0 plans sigh of 0 for any value of the time this is a statement that distinctions between states on maintained would kind of orthogonal it's enough actually just another of pogroms states go orthogonal state from that it could move in a Stern arm remain invariant back in particular for example if flight sigh of the saying that this is just the normalization of is just a normalization which we normally set equal to 1 this is just a statement and a flight is equal to society just a statement that the total probability doesn't change with time the normalization of papers change return on the other hand the fires of bottled precise then what it says is the science fire measurably different an experiment you could do to distinguish them uniquely and that will be truly either private Straw a tyrant equals 0 it will continue to be true at a later time so the logical relations between states stay in the areas with time that's an assumption of course but it is true but let's see what this condition here and this is true that indeed pair vectors science now let's see what it says about you OK society is able to you will tee times side but let's look at the dog What about brought sure IUE yet Uganda the Commission conjured level when you were when you missing conjugate or complex conjugate ADDS expression you have to put in the year mission conjured by now let's see what happens the fivetime Cy this is equal equal on the 1 hand and 2 5 0 0 you Yo car a loose sigh of Zero but also week will adjust 5 0 Zero and this is resumed the beach full for every vector repair besides your 5 0 well that equation that I want for us this is more or less obvious if an equation like this is true for every sector every pair of actors in the vector space His only 1 conclusion and that conclusion is that you'd dagger you must be the unit operator that's that's something easy the pool very easy April elite few proved farms back if his upper later rules matrix elements between any pair of states is same as just in a product between it follows that are operating must b the unit matrix cell is what we learned we learned you'll dagger you it is equal to 1 matrices with this property or operators with this property the product of them find their emission conjure 1st before we talk about that if these were just numbers it would correspond to a number of times its complex conjugate remember times its complex conjugate is 1 would art Sabre if they dead as a face its appeal babies Eaton EIA data of some sort under complex plane says that the unit circle if you live in late so if few diary was equal to 1 if were a number of this property accord unitary 0 1 would say that the time evolution of quantum systems is governed by unitary operators which depend on time let's say a magic a very small time interval time Kabul magnitude Epsilon where epsilon is a very small number so small that we can ignore the square Epsilon Bush but with another words in our equations will retained things just a 1st order in epsilon let's see what it says From matrix To be on unitary but 1st of all if is equal to 0 0 footers of epsilon is equal to 0 that's a lot of the time of is equal to 0 then the EU was equal to 1 so right can you of Epsilon enchiladas warned of all time slots out being worn and then this small correction to reconnect correction is water Epsilon was sort of water Epsilon but prime some operator which is restored correction right now I am going to just to avoid changing definitions in midstream Winterbrook might signed here but that's purely a definition recall the operated here each also grew provide easier this I had not all special meaning on its arbitrary I could absorb it into the H I like Michael also absorbed the minus sign him and I could just written 1 minus Epsilon each of later purposes I know little 1 of the flying our so that is all I hear on my side by side is completely arbitrary the RIA news orders and advice definitions Wilsey said let's see what the condition on a beaches for you to be unitary but that you Daddy you always you that you dagger of his eagle to 1 plus ii Epsilon remember left the complex conjugate everything when we pick commission conjugated and then we have to write mission conjugated H. Parisian conduit of 1 is just 1 of the Commission's complex Condrieu by epsilon is SIAPs alarms assigned changes here we substitute for each we substitute multiply 2 of them and eventually lose the approximation that we can ignore epsilon square keep things or Epsilon acts of the product is 1 plus RIA epsilon age diners oneliners R E
1:28:22
epsilon H what is at half the equal well was unitary that has equal 1 so let's expand allow water now we have 1 times might decide epsilon each SIAPs sorry each and then we have plus I Epsilon aged dagger and then we have something of water epsilon square all of that has to equal 1 would always ignore epsilon square epsilon so small but we can ignore it make we don't know ignore Epsilon the reason being that the warns cancel on both sides the once cancel on both sides and so the leading terms is proportional perhaps more aware we get we get i Epsilon age beggar Midas H should be equal to 0 what are the words to leading order in epsilon age that for conclusion is that age that recalls Willard's H is permission debt features a name from him intermission Soto observable the observable corresponds to is the energy of the system the idea the values of the matrix H the energy levels of a system workers were main point that defined in this way with the idea H should be her mission we get like this another way let's try remember rate of foreign lorry works right sigh of epsilon might sigh of 0 what is that on sigh of epsilon is 1 miners RIA Epsilon 8 times sigh of 0 my newest of 0 Meza 1 canceled sigh of epsilon minus 5 0 0 is just equal time miners epsilon each side just a difference between them the extra little piece in New Waterford comes from here this is a difference but let me divided now lamps or 5 everything both sides of the equation ramps on well this looks like a derivative of Saudi militants 1 very small limited epsilon is very small is the difference of something it epsilon zeroed divided by epsilon this it is a derivative of society With respect epsilon always respected T he is epsilon is just a small interval of time change in Epsilon provided by a small group of all of time at a derivative of the Of these state vector with respect to time is equal come minors by each arms from this tells you how the wave function of system have a state vector commercial how it changes with time HP permission guarantees that you is unitary his it guarantees that a logical relations between states maintained good time and this is that this is the equation which governs the evolution of family system of crust you may get to pick what a teacher because I haven't told you what a shares may depend on the particular set up for example age may depend on the magnitude of the magnetic field it could depend on all kinds of things the only thing we learned from general principles is that the Hamiltonian should be our mission operator Our mission operator has eigenvalues and eigenvectors family Eigen values are called the energy levels is away a state vector changes with time In principle H can be a function of time for example if you have a electron magnetic field and you change the magnetic field but if you have a situation where nothing is changing magnetic field or whatever happens the governing system happens to be time independent under that circumstance it is simply a fixed definite matrix affixed operator that is called Hamiltonian crude if if requests from the blood shed a day ahead of a key cog in the wheel of a cross over your call E. growth for the only thing you could conclude is if you are 2 different state that there's been a product of the remains invariant return the Bucks that doesn't you Pangle Ripper some are no members that was not correct to her 2 things if you want correct you would not be able to an annual thing if they want entangled they would remain entangled and measurements could not be done so that was out of town of him as that the entropy unit of entanglement is West not long rural and Inter beat FC OK entropy is a pure this is to who has no dimensions dimensions measured in dates but it's our return units walk to worse from Frank I know the location here there are now let's let's us with the units the standard thermodynamic units standard thermodynamics where the thermodynamic equation is a thermodynamic equations need direct differential change in me energy in the heat of a system is the temperature at times d s but let's see we usually came they're not OK in their eyes not the units of temperature well OK degrees usually Brewer said another equation another equation would be for the fact that the kinetic energy of a particle are Connecticut energy of a particle in against 3 hands Katie which is Boltzmann's constant temperature so the temperature always has units energy temperature has units of energy by energy divided by Boltzmann's constant whatever the Boltzmann's constant are believed that Boltzmann's constant virology divided by temperature but but all but you know the only reason that it was stupid Boltzmann's constant and physics is because people measured temperatures with stupid they bought the hardware store this match or real units for temperature Our energy get rid of the Boltzmann constant that's just the conversion factor from units of energy to units of Celsius or whatever you want you make once you just do although the Boltzmann constant and say temperature is measured in units of Energy Energy Energy dimensionless entropy becomes dimensionless an entropy why is dimensionless it's a logarithm of an integer how could have dimensions of gross you you you
1:37:55
assigned the dimensions of bit bit is just the meaning of your unit dimensionless was unit so low entropy is dimensionless I was you want to put back the Boltzmann constant and bring her in room with girl harder I can't figure out what goes energy the latter but temperature need case but there gap yet but rather are purposes we want defining entropy and information theoretical 80 hour all information purists would call and trippy absolutely dimensions it's only when they want to make the thermodynamics being used for the unit the temperature of the Marmara that their Boltzmann's constant enters into the story the talks good at it Hill I don't think Boltzmann Avenue was on constant instantly a hot for it wasn't until Einstein in 1905 heard Boltzmann's constant was actually measured Einstein begins about measure on the Boltzmann committed suicide before because she was very discouraged because nobody knew what his constant real say heads is physics McLemore Boltzmann visible very great physicist um warily outlet there were jump was there was a total of agony values of age but probably next time and they are of course just call the energies energy level now others like a team play made it is called a Y because serve Bulgaria but colors it goes back dying starting warehouse everything goes back Einstein for the same thing OK you know let go the way the wave function varies with time is with an exponential phase the depends on energies and that's that's equals Newell right of it should be at a plant comes in here I really want to keep track of some of our wears a plants constant earlier but by judge quiet this summer's multiplying manner made there Tobias depending on where you use featurerich bar and each borrower is part of a broader reform plan of Breyer's me but yes over over our supporters are a debate the sold which are but the hot Overridge yacht energy levels of the recent Wall things in quantum mechanics and a new divided by age for perform services change with time OK on could let's let's all suppose that society isn't eigenvectors of H on sorry His equal Melanie call it only major comes out to really be a maker of very happy but scored only major on on major be Inc not energy E Energy East energy he is a No. size of eigenvectors Rachel would I get value you can't just assumed that news book getting tired of interface on if signed is initially and eigenvectors energy it will remain eigenvectors energy for all time let's see if we can solve this equation ceiling solve this equation assuming that society remains an eigenvectors of the energy for all time that remains of eigenvectors energy fulltime say women wait a that means change it doesn't change is not not Sison eigenvectors but remember that you could multiply and eigenvectors my number in particular complex numbers still remains an idea that so let's write society of TD is equal to 0 by a function of time French side of 0 therefore sigh of 0 eigenvectors the energy so will ever he plans IBM eigenvectors energy and therefore still left inside I back to the same value just not lets it wouldn't solve this by saying if we can find after tea which will allow to be solved but so what does it say we just factor outside of 0 and what it says is that the effort by TV is a quota minus all I now H on society whose he turned aside from workers in E provided by each bar priced at all their multiplying the vector side in the this do them on a vector spy at turntables 0 seek lazy just becomes an equation for the function efforts pF by ID please ii over after all I've done substitute society at 50 times sigh 0 and then used a chance I'll here Is NYTimes side last August solution Oliver equation like fears is just that a half goes like each of miners RED over age after 0 so what it says is if you start with an eigenvectors the energy your eigenvectors Hamiltonian than it evolves with time just by multiplying by a timedependent phase after OK buddy a colleague coefficient in each of the RIA something he would you call or coefficient of tea with it's easily Kotite passenger revealed that it would be the case constant you call the frequency quota for another frequency but media the angular frequency major so we have the wave function balls with a frequency with angular frequency or major which is equal to the energy already each bar all he yet energy use age part terms or maker that Severe familiar required it also could be written energy use ordinary frequency kinds of our observers plant books original constant differs by factor to pipe couldn't really tired recalls getting worse and worse sewer quit now but that just show you this will come back to it the media are phases and the fact that frequency is related to energy is equivalent to this equation which tells you how States change with time this equation also has a the name is showing their equation is these Schroeder equation it's not a particular Schroedinger equation that Schroeder wrote down for a particular free particle that was our particle a potential was a special case of theirs is a generalized Schroedinger equation for any system but you will have to pick some of them potentially important for you 1
1:47:27
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Formale Metadaten
Titel  Quantum Entanglements, Part 1  Lecture 8 
Serientitel  Course  Quantum Entanglements: Part 1 
Teil  8 
Anzahl der Teile  9 
Autor 
Susskind, Leonard

Lizenz 
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DOI  10.5446/15128 
Herausgeber  Stanford University 
Erscheinungsjahr  2008 
Sprache  Englisch 
Produktionsjahr  2006 
Inhaltliche Metadaten
Fachgebiet  Physik 
Abstract  Lecture 8 of Leonard Susskind's course concentrating on Quantum Entanglements (Part 1, Fall 2006). Recorded November 13, 2006 at Stanford University. This Stanford Continuing Studies course is the first of a threequarter sequence of classes exploring the "quantum entanglements" in modern theoretical physics. Leonard Susskind is the Felix Bloch Professor of Physics at Stanford University. 