we are going to look at the generalized control gate for example here so any quantum gate
u u means unitary can be converted into a control gate how do you control that what
do you need is a control qubit associated in addition to the the other gate that you
have then you can produce the hm generalized control gate and that's the reason why c not
is a very important gate because taking any kind of control unit you can actually produce
another gate so c not is the controlled x gate which is a zor gate you also have
we also have c c not ok which were we use another control to sort of have condition
on the c not gate and as you know this is our starting gate so you can add on controls
to any particular gate that you have so for example i have the zor gate or the not gate
the starting one i had a control i get a c not i put in another control then it becomes
a c c not gate ok all these are possible so we can also generalized saying that hm whenever
we take one control bit or in this case qubit we will produce a generalized control gate
so that's the idea now this is actually useful because in quantum
computing having or adding a control bit or a control qubit on the existing gates can
become very use it and that is the very important parameter as we will see now the next very
important thing is the measurement part hm again before also we have discussed this that
generally speaking we do not want to make any measurement in quantum conditions because
once you make the measurement typically that is like the end of the line hm you cannot
any more talk in terms of what else was possible because that's a definitive answer so whenever
you make the measurement you have essentially decided to terminate or come to the result
that you are interested but at some point of time anyway you have make measurement and
it is best to be able to make quantum measurements so that you are able to get the solution that
you are looking for now in terms of the circuitry we have been
drawing the measurement in a quantum circuit is drawn in this fashion that i have the
state that we are talking about this is our measurement process and once you have done
the measurement you end up producing the classical bit representing the outcome of the measurement
please note that now i have specifically mention that that's the classical bit now that's the
very important part of this entire exercise making it clear that the movement you have
come here you have decided to look at your results and whenever you look at a result
in a quantum system you are essentially coming to the classical world looking is always in
the classical world that's how you understand so for example hm for any two states any two
possibilities as long as you don't make the measurement as we have seen even in this case
for example rotation the the spinning top can be anywhere until i measure right its
schrodingers cat all the possibilities everybody who has been talking about are all
based on this principle that until i make the measurement the outcome is not clear hm
just reminded shordingers cat is that experiment taught experiment which essentially talks
about the case where if inside a box a cat and poison is put together what is the possibility
that once you open the box you will find the cat alive or dead depends on the condition
whether the cat eats the poison or does not ok
so that's always the possibility because a cat might not be hungry might be sleeping
would not interested in that food whatever be the case it is you can find it alive but
the other possibility is equally true that it might be very hungry immediately gets the
food eats it and dies and when you open you see its dead but as long as you don't make
the measurement which is opening the box and seeing the condition of the cat you have no
no similarly for all quantum systems as long as you don't make the measurement which is
finally the classical bit you are not getting the result and until then you only talk about
probability so whenever you have two states and they can
be represented either in times of alpha or beta which are my corresponding amplitudes
then the probability will be the square of them which is associated with getting either
one of the answers so that is the idea so this is generally an important question
to ask how much information is there in a quantum state at hm because all these logic
seems like if you don't measure the information [qua/quatent] quantent of a quantum states
can be infinite is that a correct statement we should explore that so that's why this
particular situation has to what do we mean when we say how much information is there
in a quantum state so in order to make or extract information
out of a quantum system you have to perform a physical measurement so that's a whole important
point then this is what we have been trying to tell by making measurement of a quantum
system you automatically change its state that we have already said so once you make
the measurement you have change its state you can no longer get the original condition
you obtain in general a random result which may be different from the original state now
that's also a very important point so just making a measurement does not make sure that
you are actually seeing what was there before ok once again going back to the shordingers
cat it might be that in spite of a having the poison the can miracles had no effects
that's also possibility right the poison was meant for let say cockroach it was not big
enough for potent enough for the cat nothing happen to it right it was still fine
so the original state cannot be talked about it will be the other way round no matter what
happens every time you open the hm box you find the cat is dead may be it just died out
of suffocation that are nothing to do with the poison you don't give the poison even
then it is dead that could also be the case so the original case cannot be reflected by
making this measurement so that's the reason why it is important to notice the difference
between the fact that the measurement has not or may not always give raise to the original
statement or the information about the original statement so we have to actually known as
a quantum measurement to make sure that we can get repeatable answers and that's the
more important thing because given all these uncertainty if he is now want to claim that
we have no knowledge about what is going to do when we make a measurement then we cannot
have a competition so in terms of competition we have to define it in such a way that the
quantum measurement is a specific kind of measurement which will give rise to a repeatable
answers that's what we are after so when we try to make a measurement say
away the given state which is super position up to states let say you will never be able
to so this is one of the very important reason why this is true you will be never be able
to make the independent measures of only the alpha or the beta you will be making a measure
of their squares right i mean since alpha and beta can be complex numbers knowing theirs
squares does not again necessarily give the values of their original alphas and betas
ok that's the other way of looking at the so hm it is also important therefore to know
the basic idea of the basis set that's why when you make a measurement in the frame that
you are sitting in typically or roughly known as the laboratory frame the basis set is chosen
by you in terms of the laboratory frame it might have solve the entire problem in some
other frame and so when you make a final measurement and that is coming to you as a result of a
particular basis that is due to the frames of the measurement also so that's another
very important point to may understand so in this context when you have many many systems
together then we talk about an easy another approach which is known as the density operator
ok so how do we define density operator the density
operator is essentially a projection of one frame into the other so if there is a state
says v cat state and i have the w cat state is the other one ok they are at some angles
with respect to each other the projection of v cat on to the w cat is sort of like the
density given for the particular condition that we are looking at so this particular
projection the length of the projection is a scalar product using the fact that all the
probabilities all the w is are kind of normalized ok so this is the case where we
are taking advantage of the fact that we are in a frame of reference where that frame of
reference is only composed of normalized states so anything else which is happening in any
other frame of reference when we look at them we only see their projections hm those of
you who have been dealing with hm engineering drawings and many other forms of doing things
whenever you take a solid picture and you draw its two d projection on a piece of
paper you are doing the exact same thing ok hm you are representing a three d picture
on a two d plain so here also we are not really able to get the actual state in its form
the particular frame that we are using we are projecting it on that frame and its looking
at so that's why you need an operation which will do the job for you and this is typically
known as the density operator which gives you the length of the projection in terms
of the scalar product ok clear so actually an operator right so this is all
we are doing in terms of measurement because at the end of it when wherever we are whatever
we are we have to get the measurement so if we measure with the respect to lets say zero
one basis and psi by wave function is essentially just zero vector then the answer will be zero
with the probability of hundred percent similarly if it is going to be only state one then the
answer will be with probability hundred percent the value of state one anytime i measure
wave function i either get one because its only one or i will get zero because its only
zero ok but in all other cases the result will be probabilistic it can be exactly in
the middle which means that i have a equal mixture
so in the large number game the typical case is always that either one of them is
equally likely so like the coin toss problem ok without for a bias coin unbiased coin
the probability of getting a head or a tail is almost always equal so we give the probability
as half right so no matter what you do you will always get the probability of half and
this is exactly like saying that that it will get a alpha square and beta square will always
be equal to point five so that's the simple thing
the point however to note here which is interesting which you do not really think about when do
classical measurement even say coin toss thinkings in terms of measurement after the point of
making the measurement the value permanently changes to the result obtained ok its like
saying that hm if psi at some point of time change from zero to one but at the point of
time that you make the measurement it was found to be one then after measurement this
state will always remain in one because it has been made into a classical state now right
the measurement essentially takes it there so once you have made the measurement original
it could be have been an zero or one with some probability but but just by chance that
you measured it at a time when you just converted into the zero case if that's the way it is
then the measurement will always show forever its going to be one ok that's the point which
is very important to note that its not exactly like the classical case there is a certain
difference there ok if you measure with respect to a different
basis things can become very complicated so hm for example here is the case when you are
measuring this is something which which i think we have there are practice which has
been done on a one of the practice problems will do in class it will become clear also
this is something which we will doing here hm same measurement of a function psi which
is having the hm mixtures of alpha times zero and beta times one now we are going to measure
it with respect to the plus minus basis which is some other basis the this will give you
one of the results plus and the other one minus with some other particular probabilities
ok now this is different from our earlier measurement
because there we were measuring with respect to zero and one now we are suddenly decided
to measure with the respective alpha sorry plus and minus and so the measurements found
from the this result will not be the same that we are gotten for the other basis so
if you make a basis transform and you get answers for a quantum system they need not
be necessarily the same ok if by chance they are same then you can be lucky but there is
no guaranty that they will be the same most likely they will not so and this is always
true whenever you make a measurement it changes anyways this is all way have been telling
formally all the time that any time you make a measurement you will get their probability
and the movement you change the basis of measurement the results will be one of the basis sets
with different probabilities now when you have many qubits as of now we
had been talking about this one or the other state so they were essentially single qubits
one state or the other when you go for many qubits then you are now talking about different
probabilities of their each combinations right so hm alpha zero so the since they their amplitudes
are difference now they becomes different states right hm and when when you are looking
at the wave function they are basically as we have discussed before they are hm tensor
products the probabilities are the final function is a larger basis set and if you look out
for what they look like then you will find that they are composites of say the zero zero
state zero one state one zero state and one one state they are all different ok
so we started off with zero and on one all right ok but we ended up with having all these
different combinations because they are tensor products so the probabilities will then have
given rise to different products so the probability of the first measurements will reduce psi
to one of these smaller states are given by these probabilities right you can actually
do these very simply and you can prove it yourself that this is how it works once you
have made the measurements once if you make the second measurement then you will reduce
this to one of the four states which is these four states right
now once you do that measurement then you get another set of probabilities which will
be of this kind because every time you are making one set of measurement we are basically
converting it to go to that particular basis that you are measure right so each of them
will have different probabilities as has been shown here you can go ahead and do these maths
so by multiplying the branches of the overall tree the way they are breaking up every time
you make the measurement we can obtain the probability of each result so for a state
which is given by this combination which means that it started off with just two qubits coming
together with all these different probabilities two consecutive measurements will give rise
to a result which is zero zero with a probabilities of this result which is zero one with the
probabilities of this result which is one zero with a probability of this and the result
of one one with the probability of beta delta square mode of beta delta square
so now there is a specific thing that i wanted to mention here that in this particular case
when we talked about we were able to measure each of them individually that was important
right we measured zero zero one zero zero one one one with certain probabilities although
we started off with states which were zero and one ok ok just remember that
now that was possible which mean that we were going and looking at how their probabilities
where when we looked at them individually in different basis now they can be also states
in the mid in this many qubit situations which cannot be broken down into a tensor product
of a kind where i can associate individual probability to each of these kinds of states
that you that we kind of discuss so let us consider for example is a condition
were it looks like this now when this happens see what i am trying to say is that these
zero zero and one one are the states where you were able to break them into these individual
cases on these tensor cases but in some cases it is such that you cannot break them and
then they are known as the entangled states i cannot really go back to my original states
no matter what they are that is because of the rule of the tensor product every time
you multiply you matrix ends up multiplying every element of the other matrix that's the
basic idea behind the tensor product and a vector product right
so the point what i am trying to do is that it is when you do simple matrix multiplications
you can always breaks them up easily but when you are doing matrix multiplications when
they are involving tensors what we are doing is every element of the matrix is getting
multiplied by the other all the elements of the matrix right and so that is different
as compare to so these cases where we are not able to get back to the original states
where they came from they are known as entangled states now this is a mathematical issue which
makes quantum systems behave very different from classical system because this doesnt
happen in classical systems ok right so in this period the most important condition
arises which was given by bell john bell hm is the one who looked at the e p r problem
which basically hm asked the question about actually we will do this in detail right now
let me just do the mathematical part of way hm for a two qubit system the four possible
entangled states are named as bell states because these states are the once which are
possible to be transmitted and measured in a very specific manner ok and then this is
very important because this one helps in quantum teleportation
what is quantum teleportation quantum teleportation is the case where you are transmitting qubits
across a quantum path way and unless and until you know how to look back at it or you
have the exact code to understand the particular set that has been transported you will not
be able to know what has come to you so that's why it is extremely important in
quantum teleportation and cryptography where it is a absolutely important to able to transfer
qubits securely so this is the place where quantum teleportation case where bell states
are very important ok if you go with more than two states
so this was with two quibit situation if you go with say three quibit situation or generalized
m qubit situation then the similar states which are the entangled states were given
by zeilinger and greenberger so these are the three so here we had bell states which
were associated with e p r conditions and here we have many many states which were eventually
generalized by greenberger horne and zeilinger where they had used m states to create the
same kind of entanglement conditions which can be transmitted across securely so entangled
quantum states which involve at least three subsytems so these has to be at least three
or more ok anything up to two is the bells condition satisfies the bells states three
or more are the once which i given by g h z g h z or the greenberger horneberg zeiliger
conditions many measures define g h z to be maximally entangled
now this term maximally entangled is something once again we will come back to it this is
theoretical principle but the idea of maximally entangled means no matter what happens you
will not be able to get to a scenario where they can be separated out so let me actually
tell you this the right a way here that entanglement has its own degree also there has some cases
where under certain basis set under certain conditions you might be able to hm separate
the states to some level then they not really maximally entangled but they are sort of entangle
to level as long as you are not doing the states transforms to a certain levels
in a given condition you can get them to be entangle but when they are maximally entangle
no matter what happens they are always going to remain entangled in the entangles state
you cannot actually treat them as individual units for two dimensional subsytems g h z
can be as simple as zero raise to the power the tensor product to m and its a super
position with the other state over phi two and the simplest three qubit state is exactly
like this actually the reason why it is at least three more subsystems is because if
you go to the two qubit state it essentially reduces down to bells state because its essentially
the same all the signs and everything put together the fact that this as to this tensor
of the matrix on top also correct for the science that you want at a in a middle so
that's how it works out so if you compare them similar terms exist this is always
with two states where as in this case you have at least three or more but they have
the same characters so and they cannot be sort of never taken
out together so maximally entangled all of them all the bell states as well as the or
maximal entangle if you are hm doing multiple sub states or sub systems you can have other
kind of situation where they are not maximally entangle conditions there we do not talk about
g h z but if you want to get bell kind of situations taken on to larger number of qubits
then this is the only kind of conditions that will be able to use and they have to be maximally
entangled now what is the amazing thing about hm entangles states what is it which makes
entangles states so important in quantum mechanics and quantum computing is a term associated
with this spooky or in common language ok hm [laughter] if we up to super position which
has a classical analogue waves you can always justify whatever you are seeing with some
sort of a physical analogue when you come to entanglement since you cannot ever fall
back to the individual states there is almost know particular condition where you can say
that i have a classical analogue is just really not there right
and therefore everything in this area looks very spooky but for the movement if you don't
worry about the spooky pair there are some amazing fun associated with entangled states
so the first thing is after measuring an entangled pair for the first time the outcome of the
second measurement is known hundred percent its like this and that's why this question
arises that lie that information can move faster than speed of light because according
to this theory what did i just say i said that i have an entangled pair which i have
just created right and now i let the other so i have two of them lets say the simplest
case two quantum systems i have made in a entangled pair so i get two one i keep with
myself and the other one i put it on say anything i mean lets say the the rocket which is
going to moon i put it on there it goes away to moon its there
now i measure the one that i have it with me the theory says the movement you measure
you that you have the information of other one because that's how it is mathematically
done mathematics says that if that's the way it is then you have just violate it what do
otherwise things is speed of light you got information faster than the speed of light
because the movement you measure the hm one of the pairs here you have the information
of other one which is now sit residing in say moon and you can do thought experiments
where you can say that the it has been send to the other galaxy lets say right and
yet the movement you measure it here you have the fully information of the other ok
now this is one of the fallacy which bell was able to prove there is something known
as bells in quality which proves that the e p r paradox is is not leading to this
kind of a situation which says that you can have faster than light transformer information
that's not possible the reason being the concept of this entire information quantum that we
just talked about is decided a priory at the time of the creation of this information whatever
you do after that has no many so its a little difficult to come to it that's why it came
all these names spooky and everything associated with it but the basic act of the idea which
is kind of fun to at least think about it is the fact that if you have state and it
is a bells state for example like this one then the movement you make any measurement
of this then you have the other measurement perfectly known to you ok so that's the that's
the idea then the outcome of the second measurement is known with hundred percent precision and
this one definitely is hundred percent there is no probability associated with this at
all this is deterministic right that's the basic idea
so we are going to review what we have done until know because there is kind of heavy
staff that we did though it might not look heavy because i avoided much of the math here
hm and hence whatever i tried to write somewhere it didnt come out properly but i think will
some of the math little bit more hm but what we try to do as we we showed you in these
last two lectures that today at the last one how qubits are represented how many qubits
can be combined together which essentially means that can be any but for practical purposes
there is some limits that also we should know what happens when you measure one or the more
one or more qubits where entangled pair come from what happens when you measure them now
these are also very important right so this is the review until now
now based on these that we just did we should be able to take one example which i have already
alluded to which is teleportation now teleportation you must have known science fiction then everything
else cartoons chambers movies student: sir if quantum are like if entangled
principle how can we prepare oh actually we are not really preparing them
what happens is whenever you have so the question is how do you prepare entangled state so the
idea is whenever you can interact states together ok in in a certain manner then you will be
always creating the entangled state or super position state now it depends on how you are
making the interaction so that's the reason why i through the gate before that because
we wanted to make you appreciate that how these operations are actually giving rise
to the different results that we are getting so one part is measurement but point of measurement
is the once you are making the measurement is the final answer which is like a classical
answer but going before that are all the different steps which were gates so first was definition
of a qubit next was the operations which were our gates
now these operations sort of had different results as a result of interactions some of
them can end up producing only super position or only amplitude enhancement or something
like that nothing very amazing and they can be completely ratified as for all our knowledge
in terms of classical measurements or classical understanding but there are some cases where
the movement you do that you are not able to get back to the individual behaviors or
characteristics that's where we defined entanglement we said that although we had two super positions
of states so generally it starts off like this when you have one qubit which is basically
two states that's the simplest case once i take one qubit mix with another qubit i get
the next level of problem which can be as simple a super position right its only hm
some of the two conditions all possible sums now when i take one super position state and
i have another super position state come together their interaction is going to be defining
as to how they will interact do they choose to remain to be like super position like in
the sense that each of them will behave as independent qubits and they will again only
produce up to super position but that's the rare in quantum system that doesnt happen
they lose their individual identity at that very point they all become something else
when they come together so those four qubits now which i initially started off has now
actually no it was just two plus two right so i had two one which had two conditions
had another one which had two conditions we put them together they already started in
interacting in a different way right that's the bell condition where i can produce entanglement
when i can put another one that's getting even harder and that's where the starting
point of this other conditions starts and then as you when you make it grow is the same
idea mathematically it is to be represented by tensors and before that whatever you do
they are going to be represented as vectors that's the idea
and the biggest advantage we have as a result of all of this one example is quantum teleportation
is a reality by the way this is one thing which i should first say that this part of
quantum computing or quantum information processing is essentially a reality and there are hundred
percent proves that this one axis and it is being used so this is the basic idea that
there are two individual hm communicators let say alice or bob that the common nomenclature
in this area and if they have a single particles so here we start the first one a single particle
from an entangled pair ok then it is possible for alice to teleport a qubit to bob using
only a classical channel the state of the original qubit will be destroyed right because
the movement measurement is made state is destroyed so the point is only using a classical
channel this can be done and therefore its kind of interesting to note that you can do
this when you have an entangle pair and this is this the question of this is the using
the properties of entangled particles this can be done so what is the idea behind this
particular idea where teleportation the first thing to remember in this cases is this is
something which we have to done before in quantum mechanics we cannot create something
we cannot something nothing can be created nothing to that's known which means that it
is impossible to clone or duplicate an unknown quantum state when you know a quantum state
then it is classical anyway right so that part is gone however it is possible to recreate
a quantum state in a different physical location through the process of quantum teleportation
now this is kind of very interesting which means that although so these are the things
why quantum computing in some sense can become a prove reality because once i went if we
go back we saw the cases where we decline this cannot be done that cannot be in let
me quickly go back so let see we started off here ok now although it looked very pale
hm it was very important to realize that in a actual computer in a actual computation
it would be necessary to have these situations to arise also now the very fact in quantum
systems which is supposed which has to be reversible these are not going to happen which
means that i will not be actually able to create a quantum computer in this sense of
how we know about a computer if this is strictly all how it is going to happen but the very
fact now i am telling you when i come here is that it is true that i cannot do this ok
so this is the no cloning i cannot really do that that i already just told that that
was my no cloning but it is also true that since i have teleportation i can actually
do this in a different way at a different location see so i have found a way out of
some of the necessary gates which i could not otherwise i have implemented if i only
go by the logics and laws of quantum mechanics ok this is perfectly allowed through quantum
mechanical system that i can actually do at teleportation quantum teleportation ok
so that's the reason why quantum teleportation is important so the so although opened it
by saying how we are going to do this here is the mathematical point of how we are going
to actually achieve the idea of quantum teleportation so alice wants to teleport the particle one
as i mentioned to bob so in order to do that what is necessarily required is we need two
other particles two and three that are prepared in an entangled state so this is basically
the bell pair as we have done so this is state zero and zero having of of two and three which
is we superpose super imposed with the two and three three or one and one of root
two so this is this is the bells state ok that is created
now particle two is given to alice particle three is given to bob once you upgraded this
in order to teleport particle one alice now entangles it with her particle using the c
not and hadamard gates now you have to look back into your nots in the gates that
you did before so c not you are going to apply on one and two states and hadamard we are
going to apply on the one states so now the particle one is disassembled and combined
with the entangled pair ok so this was the process that i was telling
you about earlier now you have to understand one thing very important in this particular
course is that in a implementation quantum course this is the part which is the extremely
important how is the process happening ok knowing the mathematics as we have been doing
in terms of saying that this is the math this is the way it is fine but we have to somehow
get to a point where we are discussing the scenario as to how these are happening
so once you have done this part where you have manage to put your particle one with
the entangled particle where sorry where you have managed to entangle your particle one
with the particle which is now with alice then you are in a disassembled dis assembled
these assembled state this assembled state of particle one with entangle pair which alice
has now alice measures particle particles one and two producing a classical outcome
which is all these possibilities now depending on the outcome of alices measurement bob applies
a pauli operator to particle three reincarnating the the original qubit now if the outcome
is zero zero bob uses operator i if the outcome is zero one bob uses operator sigma x now
this is the pauli set as you know if it is one one then he uses sigma y if it is one
zero he uses sigma z so based on this measurements bob is able
to produce the original state of particle one so the basic idea behind this is that
alice and bob can perform a sequence of measurements on their qubits to move the quantum state
of the particle from one location to the to the other the actual operations are more involved
than what we have presented here and the actual of operations are something which will actually
be doing in detail here this is just summary so this is just to set you up as to how will
be doing it i think i will be stopping here because will hm we almost come into the point
of here yeah will be stopping here and will be exploring more details on this areas
from the next class ok thank you