Transcript for:
Quantum Operations - Lecture Notes

okay so we continue our discussion on Quantum operations so we'll kind of uh uh this will kind of be uh what we'll do for most of I mean for today and maybe for half of tomorrow's lecture uh so let's quickly recap what we did last week right we were looking at the most General kind of quantum operation on Quantum State and we reach this conclusion that they must be completely positive trace preserving Maps um this was a kind of both this was sort of both uh well why cptp Maps primarily from physical considerations right and then that kind of became a mathematical characterization of um this most General Quantum operations right the most General way in which a Quantum State can transform okay so now there were two things uh which we did uh last Tuesday's lecture one is this mathematical statement that if well it's an if and only if e is a cptp map then there is a representation of this form there exist these operators EI let's say I goes from one to some capital N right such that e of row can be represented as Su over I EI row EI dagger so essentially this set of operators completely characterizes the map and I gave this a name I said this is the Choy kuss Sudan theorem uh representation theorem for cptp maps and physically we actually derived this kind of an operator some representation so I just keep repeating these words for some time because it takes a while to see SE in right this is some this is all absolutely new for I think almost all of you um yeah so this is what is called an operator some representation physically we actually showed that this happens provided you look at this kind of a system bath picture you have a system environment unitary so this is one way to draw this there is a system which is interacting with a bath or environment and system itself is no longer an isolated Quantum system but this joined um system plus bath is an isolated system so the system plus bath together join Point L they evolve according to a unitary transformation and that is what we call this usse right um so physically system environment picture I will often use this environment and bath interchangeably so the idea is that I start with some initial state of the system and initially when I prepare the system state it is not interacting with the environment yet so we pick some fixed pure State for the environment so the idea is that initially they start out in a product state right where the system State could be Pure or mixed but we assume that the environment starts out in a fixed pure our state and then we do not have control over the environment degrees of freedom um so the question is what is the final state of the system alone right and we sort of showed that if you trace out the environment so if you trace out the environment then that's essentially uh you know so this is often drawn like this it's sort of like discarding the system right so you're discarding the information in E so this is a fair thing to do because to begin with you never had any access to that information anyway because the environment is typically assumed to be some very large system with many many degrees of freedom that the experimenter does not have any control over right so in if it's a super conducting Cubit which is etched on a chip then this chip is placed in vacuum and then there is of course thermal dissipation that happens with the surroundings the surroundings are a much much larger macroscopic system than this little Cubit which is etched on the chip right so this large system is not something that you have control over neither do you really have control over this interaction right the only thing that you have control over is the system state right and without loss of generality you can assume that the environment state is pure because all always you can append another system and make the environment State pure so if you recall your basic uqi course we did something called purification where you can take any density operator on a smaller system you can append another system to it and always make it pure right so it doesn't hurt to assume that the environment is in a fixed pure State the system state is all that you have control over the question is what is the final state of the system without any knowledge of what the environment of what happened to the environment okay and then we showed that by doing this partial Trace so this was the final State this is uh I put a subscript s to say that this is System state right um and we said that the way to get this is to do a partial Trace over what state well you started out in some State row s okay and then that's stor product e not e not that's the initial state of system and bath and then you have this joint unitary usse which is acting on this state and of course it acts on a density Matrix as U row U dagger right and then you are doing a partial trace of this state right this is a joint state of system and environment initially product but after the action of this unitary no longer product right because the unitary the whole point is that this is a unitary that is going to cause some interaction between system and environment right it is an operator that is going to somehow entangle these two systems so obviously this final state of join state of system and environment is going to be entangled it's going to be an entangled mixed state in general an entangle density operator and then you do a partial Trace over the environment this is how mathematically you depict this idea that you're discarding the system in probability here it's like taking a marginal right it's like saying that I have this joint distribution but I don't know or I don't care about what's happening to one of the two systems so I'm uh basically tracing out that part so I'm taking the marginal State and this we showed actually has a form like this right where these operators EI were basically obtained from this joint system bath unitary by taking its uh by acting it on just the environment state so on both sides you act on environment States and then this I think this is what it was right if you look at this then yeah so this was the EI and EI dagger would correspondingly be the um transpose conjugate of this right so we explicitly obtained an operator sum of this form now I don't know whether I pointed this fact out I've been putting some over I going from 1 to n here right so now can you tell me what is this capital N actually what does this represent from this system bath picture system environment picture what is this capital N number of degrees of freedom of the environment exactly good so this capital N is nothing but the dimension of the environment so there is some Hilbert space remember we associate with this some he so it's basically the uh dimension of this hbert space right um physically you can think of this as the number of degrees of freedom of the environment um and why is this because remember the way you get this operator sum is from this partial trees and the partial trees you're going to trace over the environment basis right so basically this EI this set EI that I wrote down here k e right so these SK Eis the they form a basis for the environment or this hbert space H right so this set forms a basis for the environment and so obviously the number of operators you get here in this operator some decomposition is related to the number of elements you have in this basis right if it is an infinite dimensional environment which can happen for example the environment in a if I have a cubit sitting in something like a cavity right all of you know a little bit of U you know Optical cavities and stuff then the cavity is your environment right in which case what is the dimension of this environment well the cavity basically um is a is an electromagnetic field right it it is modeled as an electromagnetic field so that has infinite degrees of freedom right but but the question then becomes how many degrees of freedom can actually interact with your Cubit system and if it's a single Cubit system and your cavity is basically photons the electromagnetic field then you can imagine that there will only be two modes that will two in in in terms of two in terms of the photon number space I'm talking about now so you can either have the no Photon State or the single Photon State and this is all that can interact with the single Cubit because this is a two dimensional system right so the interaction with the higher um phon uh the higher elements of the photon number basis will be minimal if at all so then you can truncate the environment appropriately right and that truncation will give you these operators the number of cross operators so when although I say dimension of the environment Hilbert space should not be taken literally comes from the physics of the problem right how many modes how many degrees of freedom of the environment can your system really interact with Beyond a point there's no point uh putting in all those degrees of freedom you can truncate the environment appropriately and that will give you the number of operators of course note that these operators I gave a name to them so these are called um crous operators right so I said that these are called cruss operators and this is one way to understand how the kuss operator originates in a system bath interaction and that's why I said these are called open Quantum systems right system environment picture or so-called open Quantum systems okay so I left uh the Tuesday's lecture with an example circuit and I said let's try to work out this kind of a picture right where there's a join system bath unitary uh acting on some state so let's just do this example I don't know how many of you work this example out already so I said let this join system bath unitary p a c not okay this is obviously a toy picture um so this is my system State and I said this is now they're both cubits right system and environment are cubits and there's a c not with the system access the control and the environment access the Target and you start with the zero state in the environment and some mixed state some density operator on the system and then you have to trace this out right and like I said you typically denote this as saying that this information is being junked right and the question was now what what is this um what is this uh map right can you get the operator some representation can you get the crow operators of this map so anyone solve this Crow operators are projections on zero and one very good yes exactly so the answer indeed is that this map is nothing but the sum of so you you you will find out that the final state is actually the sum of two states so I'm going to write it like this it's p 0 row p 0 plus P1 row P1 where p 0 is nothing but a projection on zero and P1 is a projection operator onto K one so I'll quickly work through the steps of this um has everyone worked this out or do you want me to go through the steps quick please go through this step okay let me go through the a few steps and I'll leave a few blanks for you to fill up okay so what's the first thing you need to do so first take this join system bath unitary and write it in an outer product notation right because cubits so what is the outer product representation for the C not so it does nothing to 0 0 it does nothing to no sorry it does nothing to 01 right but it takes 1 one to one Z and one Z to 1 one right so this is basically this Matrix no right so the last two things here are flipped right and all other elements are zero so there are only four nonzero elements I'm writing it in this outer product notation so this is the first order of business write down your unitary uh in this outer product notation right then what do you need to do you're starting with row s tensor 0 0 correct this is your initial state so what is the final state after the C not well it's just this whole map and of course the map is self adjoint right the adjoint of this is the same no because you have this two one Z 1 one and 1 1 1 0 okay so this has to act on row tensor 0 0 and then there is c not again right and what you need which is the final state of the system alone is simply a trace over e but what is a trace over e if it is a uh projection on both sides to get zero of the environment and get one of the environment right so it's 0 c not row s tensor 0 0 c not 0 + 1 I'm putting this e out here finally just to say that the zero and the one here are just of the environment so you should end up with something which is a system state right so you have to end up with something which is a system State here now what you can do is to immediately note that when you have the C not this is system environment system environment Etc so you should already see um that only certain uh outer products of this c not expression Will Survive do you see that remember that the first kit is the system kit the second one is the environment kit and then this is acting on both sides of this now your environment is starting out in zero I told you that right so there is only certain terms of the C not expression which are are going to survive which are the terms which are going to survive and on the other hand you have sandwiched here 0 0 1 1 so take your expression like this 0 0 0 0 now I'm saying that you have row S 0 e0 e and here you have 0 0 0 0 right so this will clearly survive but if I have 01 um 01 this is not going to survive because this is your taking the inner product now between z e and 1 E here right this is s e remember so you already see that this term is not going to survive so similarly you can see that uh this term will survive but this term will not survive so you can write down on both sides which are the terms that are going to survive okay is this step clear so this is something you have to work out okay I will just write down the final uh expression here to say which are the terms which will survive so what will survive is so let me write this still as Trace over e let's come to that but first on on the first hand itself what will survive is only terms like this so I will have um so in my working out what I've done is I have written down the C not explicitly like this okay this might help you so maybe this is a clue or a trick so note that this 000000 0 I'm just rewriting this as system tensor environment right each of these terms so there is again 0 0 on system tensor 1 one on the environment then there is 0 1 on the system and so oops there is one one of system tensor 01 and one one tensor 1 Z right so I hope once you write like this the first part you can do clearly and then you have to sandwich on both sides with the zero e and the zero e so when you do the first step so let me write down so once you do this then the state at the end of step two which is the final joint state will look like this um so this is what I claim that it will look like and you please work this out to see this for yourselves so what am I doing I'm now taking this um c not is on one side and this is now acting on row s tensor 0 0 right so obviously on this side it is only going to pick this state and this state on the other side it's going to pick this state and the other state right and then you can separate out the parts which are system and environment right because everywhere there are tensor products so you remember that things only act on the corresponding systems and then you will see that the first term actually looks like this the second term has a z00 system row system one one system tensor a 01 of the environment and then you have the one one terms you have one one system row S one one system and now you have the one one state of uh projection on the environment and finally you will have the 0 0 term here and then you will have the one Z so you understand you have these four projections on the environment I mean not projections the four Matrix elements of the environment state right which is 0 0 0 1 1 1 1 0 these are the four Matrix elements in the 01 basis right and each of them is accompanied by a different system State System state sandwiched between two different um operators on either side right each of these is an operator 0 is a projection 0 0 is a projection uh sorry this should yeah these are all like projection 0 0 1 1 uh 1 1 0 0 Etc right this is the final join State the final step is to take the partial trace of this right so after tracing e now I hope it becomes clear which are the terms which will survive right after you take the trace clearly this term will go away and this term will go away and so what remains will only be the final state of system which is what we want this will just be 0 0 row S 0 0 plus 1 one row s 1 one and this is what I've written as p 0 row s p 0 plus P1 row s P1 okay okay I hope that was clear I didn't want to uh write down every step but the key thing is to maybe maybe this is the key step to identify write down the C not like this separate out term by term and do this term by term then okay um is this clear any questions so NES I guess you want me to work it out is this okay yeah yeah it's okay okay good fine so this is just an example to show how you can start with some system bath unitary or or you know a joint unitary between two systems even you don't have to necessarily think of this as environment because now it's the same Dimension as a system but fine so this is one uh way to see how you actually get this cruss representation right now what is this final state it actually look like it's a sum of two terms one term where the system density operator has been projected on by with the zero projection onto zero and the other term where the system has been projected onto one does this remind you of something else a different operation which can lead to this kind of a final state what other operation can lead you to a final State like this no I'm saying um think of just the system state is there some operation which can take you to this final State suppose this were a pure State suppose row s is nothing but SI s right this is a pure State now I'll drop the other thing I'm asking is there an operation for which the final state is of this form what no answer the Zed gate The Zed gate let's see what the Z gate does yeah close but not quite right so what does a z gear do it simply does Z on S no and if you want want to think of it this way then it does 0 0 SI S 0 0 right and then there is a minus 1 1 s s one one right because Z is 1 0 0 minus one right actually I'm sorry this will have more terms I yeah sorry this is not what I wanted to write so yeah okay so let's work out what is yeah maybe let's back up a little bit H so what is Zed acting on this so if I want to write it so if it's a pure State you will simply say it is 0 0 S minus 1 1 s right so now on a mixed state what's going to happen I hope you realize that you will have all four terms now so you will have yeah in fact this is a good example to do because we will soon see the difference between this and that um yeah so I don't know who gave me this answer is that sui or yes yes I understand yeah so you understand the difference now so you have all four terms so this is not not that not a z gate so what is it+ one okay yeah so we'll have four terms but here I have only two of these four terms no I have only these two terms this and this these two cross terms are not there so what has happened so you can apply a z gate now corresponding to the Z gate there is an observable the spin along the Zed what can what else can you do like the Zed what else can you do come on guys what operation will take your state to this where it's just a projection on 0 plus a projection on one there has to be only one answer no we've only seen unitaries and one other kind of transformation so the unitary is covered the Z gate is a unitary it is not the Z gate so what else can it be what else can you do with the Z basis measurement exactly that has to be the only other possibility so so this is actually a measurement in the Z basis why is that what ises a measurement do so this is what so this is a unitary so this so note again that this is unitary action by The Zed gate right that is what this is but a measurement in the Z basis what does measurement do it collapses your state onto either 0 0 or 1 one with certain probabilities all that is fine but how did we write down the post measurement state so I think you all have to go back and brush up your measurement Theory a little bit how do you write the post measurement State remember we said that SIII goes to a projection onto zero an action by a projection onto zero right in fact this is how someone described a density operator on first day of class and action with projection onto one right so if I now write this out in a different way this gives me 0 S Mod 2qu 0 0 plus 1 s uh mod squ 1 1 so then this uh is a projection on zero with this probability and this represents a projection on one this with this probability so this is a mixture so these two mod squares are now my probabilities right so this is p 0 probability of outcome zero this is P1 so this is exactly like a density operator with these two uh which is a mixture of these two states 0 0 with probability p 0 and 1 one with probability P1 yes is this clear or do people have questions is everyone asleep I don't know but the point is yeah was there a question no like it is clear okay good um so this C action of the C not and then tracing out the second system of the C not the target system of the C not essentially is like measuring the system in the Z basis what is 01 what is a measurement uh uh in 01 it's a measurement of the Z basis that's why the two possible outcomes are 0 0 the two which is basically like projecting um acting with these two projectors on your state right so that is what has happened so this is the final postmeasurement State remember we went over this when we did measurement Theory where we said we will not talk about collapse all the time we want to have a valid State transformation for measurement and the way to do that is to talk in terms of these projections right now observe an interesting difference between what the unitary on the state has achieved and what this c not and tracing out the second system has achieved so this we said is like a measurement okay so let us now contrast the two so it's very important that you understand the difference between these two operations and that is at the heart of understanding noise and Quantum channels and all that which are the two operations so one is this c not right system environment started in some pure State then got junked and then you're looking at the final State and we said that this it turns out looks exactly like a measurement right so this is the same as this process that I measure in the Z basis I started out with the system State rows and I measure in the Z basis and this is the output State I get so contrast this with a unitary Evolution so remember this is now a cptp map right this is the cptp map as far as the system State alone is concerned right because you have identified two cross operators the P not and the P1 right these are both The cruss Operators so if I didn't note that let me note that here already when we completed this exercise so and I wrote down the answer to this yes so now here note that there are two cross operators let me call them e not which is p notop and E1 which is P1 and because these are rank one projectors their adjoints are the same so these are your cruss operators for this map right where after the action of C not you've traced out the environment so contrast this with unitary Evolution via the Z gate right okay so I have the zgate I have row s so what happens I get Zed row Z dagger Z dagger is the same as Z what is the contrast now we are talking cubits right so we can write down the final State as two cross two matrices right so let's write down the final state in both the cases these are two cross two matrices right the initial state would also have been some 2 cross two Matrix it has evolved to this final State now what is the structure of these finals matrices right look at this case where you have this measurement in in the Z basis or you can think of this as the cross operators coming out of this uh joint Evolution are all four entries of this 2 cross2 Matrix populated no no which are the entries which are populated only on diagonal you only have the diagonal entries indeed good the off diagonal entries are actually zero there is no one z01 term here you do have the diagonal terms and the diagonal terms essentially are of the form zero expectation of this density Matrix with zero if it is a pure State this will just become that probability right mod um size 0 squ even now it's a probability it is Trace row S 0 0 which is the probability of outcome zero and in this case it is 1 row s 1 right so you have a diagonal m Matrix and the off diagonals have completely vanished now note that this is the case no matter what your initial state is and your initial State typically would have all four entries populated so what has happened is these off diagonal terms have completely vanished right and if I start out if I represent this uh row as suppose this started out from a pure State then you will realize that these off diagonal terms contain some kind of superposition information why so I'm saying this very Loosely because they are the cross terms right they are the 0 1 1 0 terms right and if you write this as outer product so these are the cross terms so in some sense they contain information about the superposition or they represent the coherence in the state so what has happened out here is what we physicists call decoherence which is that you have completely lost the information about these two of diagonal terms initial density Matrix might have had it might not have had it maybe you were already diagonal in the 01 basis to begin with well in which case a measurement in the Z basis doesn't do much if you're already diagonal in the 01 basis right it's like saying I'm already in an nigen state of Zed what happens if I measure Zed well I get a fixed outcome there is no coin tossing it's a fixed outcome if I'm already in an igen state of Zed and I measure Zed it's a trivial transformation but if I'm in a general Matrix which has off diagonal entries this measurement of Zed or this cptp map coming from doing the C not and tracing out the environment remember the operation joint operation was unitary but this act of tracing out the environment and looking at the reduced state has led to a loss of information wherein these off diagonal terms are just fixed at zero now so this is basically this this entire transformation has taken you from what was potentially a coherent superp position or a state which had this off diagonal information which represents coherence has now decohered and youve completely lost the information in the off diagonal terms on the other hand contrast this with the unitary Evolution and see we worked this out that's why I I wanted to work this out although the answer was wrong wrong you notice that you have these two off diagonal terms right you have this um have these two off diagonal terms these are the off diagonal terms right so if you started out with something which had octagonal terms then you would continue to have so in this case the final state will have all four entries right with different probabilities the pro with I mean the entries will be different based on what your initial state was so you have zero size SI one one size SI zero so you see so long as SII has these entries then they will feature in in this final density Matrix as well right so I hope you understand so you'll have all four entries okay and there is no loss of coherence okay so is this contrast clear between evolving a density Matrix with a Quantum gate or a unitary transformation and transforming a density Matrix either via measurement or in this fashion where you have a join system bath unitary and then then you uh Trace out the one of the systems and you look at what remains then you notice that you have lost some information in the process and this is why this is why Quantum operations like this are set to model decoherence or loss of coherence and already you can see this aspect of loss of information coming in right and that is why this is this represents noise right questions now physically what information does the off diagonal elements have yeah so I can give you an example simple example suppose s was a plus State okay yeah so I have 0 + 1 by < tk2 what is siai please tell me the Matrix quickly what is this Matrix 1 by two on yes now imagine you have another state five just get minus so what is this density Matrix 55 diagonals are negative exactly now this under goes this evolution by this map which has these two cross operators p 0 and P1 right what do you get and this under goes Evolution via the same map what do you get you'll get same result you will get the same result and you will get half half here correct so now do you understand this idea of information loss yes ma'am yeah exactly so you notice that there is crucial information sitting here this is often called face information in this particular case it happens to be face information and the face information is completely lost under the action of this EO uh let me call this Sigma or something so I started with two different initial States this was row this was Sigma okay and both of them underwent the same Evolution uh the same c not followed by tracing out the environment or you can think of this simply as a measurement in the Z basis but you see this information about the phase is now completely lost on the other hand if they had undergone unitary evolution of any form right this face information would have been preserved in some way going forward but if something like a cptp map of this kind acts on them then you notice that you have completely lost this pH information just a very simple example right you can imagine constructing more non-trivial examples of this kind okay so in fact what we have just um seen uh can be thought of as some kind of a phase damping okay sometimes this is also called phase damping the idea is that you have lost information about the in fact what you have really lost is this relative phase information whether it was a plus or a minus right so this is a classic way to understand why cptp Maps represent noise on Quantum systems and why unitary Evolution does not represent noise okay that is the contrast all right so this was all some you know to give you some physical idea of what's happening uh what this operator some uh how to get this operator some representation for a concrete example so in our assignment we will do slightly more um complicated uh uh channels like operations like this okay now um I have still to complete one part which is to prove the statement right so I simply stated this that they're always exists uh such a set of operators which characterizes the channel and as a pH true physicist I just showed you uh how this can happen from a physical point of view but mathematically I have not really proved The Cho kuss sashan theorem I just want to go over the steps of this proof because I think this also builds some intuition in playing around with this um idea of cptp kind of evolution okay so what does this theorem says it says that if this map e is a completely positive trace preserving map then there exist this set of operators which we call cross operators such that e of row is Su over I EI row e dag right and uh know where these operators must satisfy yes what is this condition please remember this is an important condition that this must be equal to Identity right these are operators purely on one Hilbert space which is the system hillt space in this in defining cptp maps and The scuss Operators there is no reference to an environment or bath it's just to understand physically how does such an evolution even come about that is why this entire discussion on you put attach a bath and it's as if tracing out the bath then you will get this kind of a system Evolution this is that is when you will get this kind of a map for Quantum States so all of this discussion was to physically establish this fact so we have physically established this but now I just want to show you mathematically this holds right recall that we proved the converse last time we already proved the converse which is that if the map has this form with the CR operator satisfying this then such a map is always completely positive and Trace preserving okay and we already argued this last time okay so let's try to prove the this forward Direction now so the proof relies on an interesting idea that the map well it basically takes off from the complete positivity part right so since e is CP I can extend the system right so we can consider an addition [Music] system uh so let me call this additional system like an you can think of this as an Ancilla sometimes this is called a reference system okay it's called a reference which we will call head subr okay so all of this action is happening in one Hilbert space right so let me say the cptp map on one Hilbert space HS okay so now I'm going to to consider an additional system HR and I can consider this kind of a map right which is identity in R map e acting on S and some join State between the um reference and the system I can always do this because the map is CP this is a valid map right this is a valid positive map so let let us say that this reference system is identical to the system itself right so they are both what are called isomorphic spaces right I think the correct symbol is to put a TIA here but essentially for for our purpose all it means is that they are of the same Dimension right so they are of the same Dimension it's as if I have another copy of my system hir space okay so now I'm going to consider a particular state of this joint system which I'm going to call as uh I think I call this Alpha RS so consider this kind of a joint state which is simply so they are both identical so now let this be a basis so obviously I is going to be a basis for system and then I will be a basis for the reference right for the system and the reference okay so then I can construct this joint state which is IR tensor is now these are all basis vectors of the joint system and I'm going to sum over all of them like this the of dimens D or something right what kind of state is this so try and visualize this just for a for suppose the system was a cubit and the reference is also a cubit right then if you try and visualize this what kind of state is this so think for example if it's a cubit then what happens uh so let me say 0 to - 1 okay so 0 0 + 1 1 plus exactly plus 3 3 Etc d minus1 d minus1 that's what the state is right so this is in fact a maximally entangled state of this joint bir space uh r p is right up to normalization I not normalized it I can put a one over what should I put to normalize this exactly I should put one/ square root D and then it becomes a normalized in tangle State okay so now this is the state I want to start with and I want to look at the action of this map which is identity on the reference and the map on the system on this joint entangled State okay this will give me a valid final state of reference and system and because this is CP I'm guaranteed that this final state is positive right and what is more this state this final state it completely characterizes the map e the map e on system right because I've have started with a fixed state of the reference and the system and it's identity on the reference and the map on the system I get a specific state so what does it mean to say that it completely characterizes the map it means that there is a one toone mapping or a one toone correspondence between the map e and this join state in an larger system and this correspondence often goes by the name Choy josi correspondence or Choy josi isomorphism it's an isomorphism between maps and States maps on a smaller system and States on a larger system okay and it kind of presents a very nice unifying picture so the map what you think of as a map on the system HS on the Hilbert space HS can be associated with a particular density Matrix on a larger system which is HR tensor HS okay so this is a very powerful and important idea in Quantum Computing and Quantum information um it is also at the heart of this cross operator representation and all of that because once you have Associated a state like this then it's just a matter of reducing it back to your system right and then you can use this you can use this Choice State this is often called the choice State okay it's called U it's called the choice State associated with the map e so you can use this state to now get to the crow representation of this map okay so yeah this is a very deep idea mathematically also so if you do any course on functional analysis and so on this is a very important aspect of this thing that will be discussed this this unified picture between maps in one dimension and then when you extend the space to a larger Dimension that essentially is like a state okay and there is a onetoone correspondence between these so let's make this one to one correspondence a bit more concrete so now Define a state s uh on your system so please note that I keep putting these subsystems now S and R okay and I'll keep track of this so this is my system um I'm defining a typical superp position state of all my basis States okay now I want to define a state C okay in the reference which is nothing but a superposition of the same remember we said they isomorphic so I have these basis states in the reference but I'm going to take the complex conjugate of the coefficients here okay so I Define a different state of the reference if you want I can put a different label here but this is related to the state s and that's why I call it stier of the reference where I'm simply taking a complex conjugate of all these coefficients okay and now you can note this beautiful fact that if I take this joint Choice state right which has been defined here yeah this so I take this state and I act on both sides with this slda of the reference what should I get I'm basically acting on I'm I'm taking an expectation with respect to a particular state only of the reference right so then it's like once again truncating this just to the system so I should get some state of the system what state of the system do I get so let's work this out please put pen to paper and work this out with me otherwise this is all going to be totally uh overhead transmission so what do I have Sigma RS so let me let me write this out so remember what was Sigma RS it's identity in R censor the map acting on some over let me call this I I r i s okay uh yeah actually you guys should have stopped me right here uh I wrote down a pure State here but what I intended to write was actually a mix State okay so this should have actually been this Alpha RS Alpha RS right right so this should actually have been of this form um I I this should be okay so let me start with this so this would be i j this will be I uh i r i s okay and then there is the uh this is the K and the bra right so there'll be a double sum here there should be a j s and a Jr okay so I'll write this out properly again so remember that this is this acts on this density operator the projection onto this maximally Mi State and remember so let's sandwich on both sides with site in the now let's write this out explicitly yeaha e it should also be on the right side which one yeah here this I TAA e Alpha Al alha and after that I should we should also have this Ira e d tensor e you mean well this is a map okay when the map acts using CR operators and so on then you have a conjugation but this is a map okay so if you're confused about this the way I do this right uro U dagger so when I say that I act by a unitary and I get uro U dagger this is represented as a map U of do you understand so when I WR right this is the action of a unitary on a density operator and it acts by conjugation if I write it as a map I will simply write it as U of row it is like saying some function of X it's just that this function is now represented by the action of a single unitary operator like this on the other hand if it's a cptp map then it's represented like this if it's a measurement let's say on the Z operator on the Z basis then it is represented like this you understand now the idea so it's just e instead of E I now have I tensor e this is the map acting on this state I've not yet expanded the action of the map okay so this state here is like argument passing to this correct exactly that's why I said it's like f of x it's like this yeah this is the argument for this exactly yeah okay good I hope that clarified the notation for everybody uh all right so now this becomes Su over i j okay there should be Alpha is what I I when I take the bra it becomes JJ okay with corresponding subsystems and all that and then that's why there's a double summation here okay I hope this part is clear uh and then I can put a whole bracket here if you wish all right now this Su over IG I'm collecting all the things that belong to a single subsystem right so the first IG belongs to R so this is RS r s we have done these kind of exercises in the basic course so I hope you are familiar with this so I'm just this is exactly what I did with the c not above also right I'm taking all the parts which belong to subsystem R and all the parts subst put it as a product and of course the summation over IG outside right now I will write the expansion for the uh clda R and note that this is only identity on the r subsystem right so now if I write this so let me write that with a difference different index k out here right so this becomes CK well so stier was with the star so when I take Clea bra obviously the Star goes away right so this will just become CK K on the reference and then I have this now right I have this summation over IG I on the reference tenser the map e acting on each IG so I hope you understand that this is linear right so I can pull this inside the summation so the summation comes out there is identity on the first part and then there is the map e acting on every one of these outer products of the second part right and once again I have I can expand the c r so I will write this as Su over L CL L star k l acting on the reference okay so once again make sure that you take the bras and the ks acting on a given system and you collapse them together which means these two will collapse together these two will collapse together so this gives me the double sum here the K and the L right here it becomes K and I must be the same and here you see that J and L must be the same so instead of a sum over four indices this reduces to a sum over only two indices because I have Delta Ki so this becomes CI I have Delta JL so this becomes CJ star and everything on the r system has now coll right this inner product is done this inner product is done so the r subsystem only has a scaler and that scalar finally is this c i CJ star and then I have e acting on i j so this I can now take these again linearity okay I'm now going to pull the map e out by linearity so this is is nothing but map e acting on Su over i j CI I and then there is RA J CJ star okay what is this what do I have now in the argument of e what is the map e acting on which state is it acting on what is the state CII this is the state s that and so it's nothing but k s draai e acting on S so what have we shown in this somewhat laborious exercise is that this remember this was we started with this joint state of reference and system and we projected it onto a particular state in the reference or we sandwiched we took the expectation with respect to a particular state in the reference that is related to or that is exactly equal to the action of the map on a particular state of the system so let me write this fact here that if I take the expectation of this joint state with respect to the reference that is nothing but map acting on system this is why there is a one to one correspondence so this is what characterizes the one to one correspondence between the map e and the state RS is a broad idea clear the details can be worked out and the details have to be worked out patiently but the idea is that there exists this way to represent the map using a single state on a larger system right and this single state on the larger system completely captures the action of the map because if I want to know the action of a map on a particular State what do I do I take the corresponding Tia state in the reference and I simply have to take the expect ation value with respect to the Tilda state of this joint State and what is this joint State well it is the action of identity tensor the map on a particular maximally entangled state of the reference and the system right so this is the first part of the proof right where you associate so where did we start with we started with there exists a cptp map on some system we need to show there exist a bunch of operators whose action like this captures the action of the map so the first step is to establish that there exists this state on a larger system which captures the action of the m so the second part or the next step will be to Define operator EI okay and these operators we have to somehow get these crous operators right and these operators will actually be related to uh pure State decomposition of this joint state so remember that this is a density Matrix on this larger system so this can be decomposed in terms of some set of pure States right because a density Matrix is nothing but a convex sum of some kind of pure States so it turn out that there is a way to relate this to the brow operators so this we will do tomorrow okay so let me stop here and questions doubts okay no questions either it was all very clear or it was all too abstract okay so let me stop here and please go over the lecture again or go over the notes if things are not clear if there's anything in the background material which I'm assuming which you need me to explain you have to let me know okay uh yeah so let's proceed tomorrow and Define the CR operator so in a sense we are the exercise that we did for the C not was a particular example and then we did this system bath again that's a particular example now we are doing that in full generality we are saying that for any cptp map we have a way to construct these grous operators VI are this isomorphism or this m or this extension to a larger space okay all right we stop here I'll see you all tomorrow thank you