we are going to look at the generalized control
gate for example here so any quantum gate
u ah 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 ah 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 ah the measurement in a quantum circuit
is drawn in this fashion that i have the ah
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
ah 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 ah 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 ah 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 ah 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 ah 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 ah 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
ah 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 ah 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 ah 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 ah 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 ah
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 ah 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 ah 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 ah 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 ah quantum systems behave very different
from classical system because this doesnt
happen in classical systems ok right
so in this period the most important ah 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 ah path way and unless and
until you know how to look back at it or you
have the exact code to ah 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 ah securely so this is the place where
quantum teleportation case where bell states
are very important 
ok ah 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 ah so these are
the three so here we had ah 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 ah 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 ah 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 ah zero raise to the power
the ah 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 ah 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 ah so if you compare
them ah 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 ah the rocket which is
going to ah 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
ah say moon and you can do thought experiments
where you can say that the ah 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 ah paradox is ah 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 ah 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 ah 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 ah 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 ah 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 ah we decline
this cannot be done that cannot be in ah let
me quickly go back ah 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 ah 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 ah
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 ah 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 ah yeah will be stopping here and
will be exploring more details on this areas
from the next class ok
thank you
