WEBVTT
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we are coming towards the end of this series
of lectures for this course we have already
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covered ten weeks of learning about quantum
computing and its implementations and in the
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last week we were going over some of the basics
to make sure that we understand the concept
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of implementation again based on some of the
basics and as we have been interacting more
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and more with the students we have also come
to know that there are certain aspects which
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would help if we relook at them a little bit
more one of the key elements of that happens
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to be density matrices and density operators
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so we start this weeks lecture with this concept
of density matrices and density operators
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we have introduced this earlier and mentioned
certain aspects a bit in relation mainly with
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respect to nmr and here we will be doing a
little bit more understanding a bit so that
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the the missing elements or the implementation
parts that we used density matrices implicitly
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can become clearer
01:18.469 --> 01:27.020
so when we talk about density matrix a quote
comes to mind that again is related to richard
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feynman where he made this statement that
when we solve a quantum mechanical problem
01:33.409 --> 01:39.399
what we really do is divide the universe into
two parts the system in which we are interested
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and the rest of the universe so this was quoted
from his statistical mechanics lectures back
01:49.039 --> 01:56.049
in nineteen seventy two essentially stating
that whenever we are going to talk about density
01:56.049 --> 02:02.530
matrices we are going to discuss about the
statistics and the collective aspect of the
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part
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when we include the part of the universe outside
the system the motivation of using the density
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matrices becomes clear so thats the basic
idea that whenever we are going to include
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the part of the universe outside the system
alone then the motivation of using density
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matrices become very clear and actually this
is very pertinent to quantum computing as
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we will discuss here
02:28.480 --> 02:34.360
so essentially let us look at what these mean
so most cases density matrices and density
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operators are used reversibly and often are
used in one and the same way for both the
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cases so in general the statement that was
made in the last slide is sort of more clearer
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in this term of venn diagram where we have
an ensemble of the states that we are talking
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about with some probability for each of the
states and that essentially forms the quantum
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system and that interaction of the quantum
subsystem as a result of probabilities and
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their interactions is the fundamental point
of view that we are taking
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so let us see what we will be mostly doing
in this lecture and why we are doing it
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most of this lecture will be spent on mastering
density matrices and the density matrix and
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so we would need to master a rather complex
formalism in some sense and that's the reason
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why i thought i will go back on this now where
it has become an important point to understand
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might seem a little strange since the density
matrix is never essential for calculations
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it is a mathematical tool introduced for
convenience and in our particular case the
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convenience meant that we were more inclined
to use it for implementation aspects and
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mostly we have to bother with it as already
we know because we have gone through the implementation
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processes but let me actually point it out
again that the density matrices although
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it seems to be very deep abstraction but once
you have mastered the formalism it becomes
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far easier to understand many other things
including the concepts of quantum entanglement
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and quantum communication which we have already
gone through before so that's the reason why
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we are relooking it and that's the reason
why i had initially introduced it
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but never really gone deeper into it but before
closing our entire series of concepts on
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quantum computing i thought we need to really
look at it once more so mathematically if
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we consider says phi i and theta i to be the
complete set of vectors in vector space describing
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the system so mathematically let phi i be
a complete set of vectors in the vector space
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describing the system and theta i be a complete
for the rest of the universe the most general
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way to write the wave function for the total
system is generally written in this form where
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both phi i and phi theta j are composing the
total system
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now let a be an operator which acts only on
the system that is a does not act on theta
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i then what will happen is the operator
a acting on the system would give rise to
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a result which would be given by this form
of equation so when we apply this then we
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will have the observable would be having the
form as given in this equation where in
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we will find that this particular part can
be represented in terms of the density matrix
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as we call it and that is the definition in
how we start discussing about it
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so once again we have the vector space describing
the system by phi i and the complete set of
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the rest of the universe as theta i then the
most general way to write the wave function
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of the total system is a composite of the
two psi then we use an operator which acts
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only on the system which is how most of the
operators that we choose work and then what
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we find is that we will be having a situation
where if we find its expectation value for
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the wave function it will end up producing
result where we will be having the resultant
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coming from the part where the system is being
interacted by the operator resulting in a
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part corresponding to this kind which is the
density matrix that we are looking at
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so we basically have defined the operator
rho to be such that it is going to be having
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a form of this kind with the understanding
that rho is going to be hermitian so once
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we look at this one again where we have defined
this part then we get doing a little bit of
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math as we show here that the trace of the
operator on the density matrix will give rise
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to the expectation value of the operator that
we were looking at
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so due to the hermitian nature of rho it can
be diagnosed with a complete orthonormal set
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of eigenvectors with real eigen values wi
so that the final density matrix would look
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like this rho so we have essentially stated
that the expectation value would be the trace
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of operator a with rho and rho which is our
density matrix will be of this form and if
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we let our operator to be just identity then
what we will get is the sum of all the eigen
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functions would be equal to just the expectation
value in this case it would be one because
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it's the we have taken the operator to be
one unity
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and if we let a to be of this form then we
will have the eigen functions to follow
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as a result of this trace application a form
which would be of this kind which means that
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our eigen functions are going to be either
equal or greater than zero and the sum total
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of each of the eigen values would be equal
to one so its an orthonormal set of eigenvectors
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and real eigenvalues are the solutions which
are with respect to the density matrix that
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we have defined rho
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so that is the first criteria that we have
to understand that the density matrix would
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always give us orthonormal set of of eigenvectors
and real eigenvalues so that is what we are
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getting any system that is described by density
matrix rho where rho is of the form that we
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have just now defined summation all over
i for the eigen function w i with for the
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bracket notation that we have used the set
ket i is a complete orthonormal set of vectors
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so these are the rules that we finally get
out as the result of this op mathematical
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understanding
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and given an operator the expectation value
of a is given by trace of rho of a and we
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also notice that this can be rewritten in
this format which gives rise to the fact that
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w i is a probability that the system is in
state i if all but one w i are zero then we
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say that the system is in a pure state otherwise
its in a mixed state
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so this definition allows us to find out that
the probability of the system and its weightage
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factor and if it is only one state as we have
just discussed that w i is the probability
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if it if all but one is zero then that particular
state is the only one which represents the
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state and therefore it's a pure state otherwise
its in a mixed state that's the basic idea
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so in that sense density matrices of pure
states are then represented as in terms of
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the vectors that we are doing the ket vectors
psi and all such states are called pure states
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an alternative way of representing quantum
state in terms of density matrix therefore
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are the way that we have just introduced and
it is also as i mentioned reversibly used
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with the term density operators so the
density matrix of a pure state is the matrix
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written in this form which as its summation
can be written in terms of the weightage factor
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for the states which are composing the psi
state this is a particular way of representing
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density matrix which we will get into more
in detail
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now as an example the density matrix of
say alpha zero plus beta one can be written
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simply in this terms and which will then be
the coefficient squares along the diagonals
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and their products along the off diagonals
so the summation part of the statement that
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we have shown at the very beginning can also
be summarized in the matrix notation as we
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show here
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for a pure state for example we have conditions
where we can have wave function which is
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a composition of one over root two normalization
factor upward spin and downward spin if
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we take this wave function then the density
matrix can be written in this form and we
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find that the square of the density matrix
is equal to itself and its trace is equal
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to one trace is by the way the sum of the
diagonals
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if we consider a mixed state of fifty percent
upward spin and fifty percent downward spin
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then we have a case which looks like this
and in this case the the matrices will
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turn out to be in this fashion such that the
final result of the density matrix will come
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out as this and in this case the rhos square
is not equal to rho although the trace rho
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is equal to one as before
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again if you take a fifty fifty mixture
of ket psi as we had shown before of
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of spin going one direction versus the
other direction and another wave function
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get phi as one over root two spin going up
versus spin going down and the negative sign
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between them then if we take the if we look
at the density matrix we once again find it
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comes out to be a density matrix as given
in the last case which is a mixed case
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and we find that rho square is going to be
not equal to rho again and the trace is again
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going to be equal
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so now the diagonal elements are always equal
to one essentially that is a reflection
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of the fact that this diagonal elements are
essentially giving the populations of the
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state so the total population irrespective
of how we look at it which is the trace is
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always going to be one and the off diagonal
elements are the ones which will indicate
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something else and as we come to this is known
as the coherence how much they are correlated
14:56.310 --> 15:01.180
to each other the correlation between these
states because they are essentially corresponding
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to the coefficients of the two different states
and their complex conjugate product and therefore
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when they have some correlations then that
will resp represent the coherence of the states
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its important to note that in both the
cases we describe a system which we know nothing
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about is total of total ignorance because
what we have done is we have put mixed states
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in case of the second and the third case whereas
in the first case where it was a pure state
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where rho square in the first case this
was a case where we only had one state to
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look at is so opposition state but its essentially
one state of spin going up or down but
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this is a state which is a pure state
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however in the cases where we looked at later
on they were all mixed states and therefore
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we know nothing about those states it's a
state of total ignorance and that's why it
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is important to get into the concepts of
density matrices because for pure states
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it perhaps would not matter much as to
what we talk about their properties and things
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like that but for mixed states the difficulty
is that we cannot really come up with a way
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as to how to understand them so density matrices
are ways where we can get to know about mixed
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states first of all identify them because
whenever we find that the rho is equal to
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rho square which we saw in density matrix
we know that it's a pure state whereas whenever
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it is rho equ not equal to rho square then
we know that it's a mixed state and so that
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gives us some indication as to how to look
at these states
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now in the case of a matrix which we have
been using a lot i just wanted to say hm just
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keep the clarification as i mentioned before
it's the sum of the diagonal elements which
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for example in this case has been identified
as the diagonal elements and it has some interesting
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properties which is that its associative it
has so a times b and b times a all these
17:10.410 --> 17:19.459
laws for multiplication addition these things
work and also the unitary laws work
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with this so unitary operator acting on the
state inside the trace would make it remain
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the same and the trace of a matrix essentially
can also be represented as the expectation
17:36.620 --> 17:45.230
value of the wave function as long as this
is an orthonormal basis phi i for instance
17:45.230 --> 17:50.370
so in terms of notation of density matrices
and traces this is how we go about doing it
17:50.370 --> 17:56.520
we generally represent as we have been saying
the ket wave function as a super position
17:56.520 --> 18:04.990
of two other states with with their coefficients
and we can have them written as we have shown
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here that these coefficients are nothing but
the which can be written in these terms
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because they are orthonormal sets so the probability
of getting zero when measuring this can be
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found out by using this kind of a mathematical
form and what we find is that in this form
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also when we arrive at we get that the density
matrix is the form that we are get arriving
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at where we are getting that the density matrix
is what we are using when we use the trace
18:40.330 --> 18:48.790
so whenever we are going to make a measurement
of a particular state that's where the
18:48.790 --> 18:54.720
density matrix appears we we have this
is a definition which we had already used
18:54.720 --> 19:00.220
that the probability of a particular measure
is always the square of the coefficient and
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that's what we are utilizing and we can always
find that in order to get the value of that
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we can always use the density matrix and that's
one of the ways of saying that the density
19:12.169 --> 19:19.140
matrix would automatically help us in finding
the probability of the or the contribution
19:19.140 --> 19:23.670
of a particular state to the composite state
19:23.670 --> 19:30.690
so the eigenstates of density matrices form
complete basis for subsystem block that we
19:30.690 --> 19:35.390
have already seen the eigenvalues give the
weight of the state that is what we had
19:35.390 --> 19:41.730
discussed earlier and we can keep the m eigenstates
corresponding to the m highest eigenvalues
19:41.730 --> 19:47.250
m is just the running number here running
digit eigenstates of the whole system can
19:47.250 --> 19:53.150
thus be given by for instance the wave
function which is equivalent to the root since
19:53.150 --> 19:57.600
the weightage factors or the weightage of
the state are essentially the probability
19:57.600 --> 20:04.289
the amplitude of it would then be square root
of this weightage times the individual states
20:04.289 --> 20:09.640
that they are made of and this is the famous
schmidt decomposition that has been sometimes
20:09.640 --> 20:18.529
eluded to when we were doing our steps or
processes in quantum computing or quantum
20:18.529 --> 20:20.190
information processing
20:20.190 --> 20:27.740
the this is an optimal approximation however
because we are making a statement that we
20:27.740 --> 20:37.240
understand that they can be made to be decomposed
in this kind of a state in case of entangled
20:37.240 --> 20:45.860
states the mutual quantum information is carried
in terms of the entropy of the state which
20:45.860 --> 20:53.779
is often written in terms of the way how
entropy is often defined and since trace allows
20:53.779 --> 20:58.700
the logarithm to go through this kind of a
process so we can this is the property
20:58.700 --> 21:04.240
of the trace that we looked at earlier so
we can instead write the trace as in this
21:04.240 --> 21:15.299
form minus rho log rho and this will be
equivalent to the weightage factor terms
21:15.299 --> 21:19.340
the log of that
21:19.340 --> 21:23.940
so density matrix can be defined for pure
coherence to a position of statistical average
21:23.940 --> 21:31.240
states and so it's a very powerful approach
whereas in case of looking at the state alone
21:31.240 --> 21:36.980
it might often be difficult to find out
what to do with them so the idea that these
21:36.980 --> 21:41.659
kinds of definitions work for all the cases
whether it is pure whether it is having a
21:41.659 --> 21:47.649
coherence super position or whether they are
a statistical average of many states it gives
21:47.649 --> 21:51.799
us the advantage of using the density matrix
formalism
21:51.799 --> 21:58.899
now when we have the concept of pure states
it is much easier to understand because
21:58.899 --> 22:03.730
as we said that they basically represent one
particular state however when we have mixture
22:03.730 --> 22:11.039
of pure states the state is described by a
state vector and is called a pure state
22:11.039 --> 22:16.860
if we have a qubit which is known to be in
pure state psi one let say with probability
22:16.860 --> 22:22.659
p one and in psi two with probability p two
what will happen so that's the question to
22:22.659 --> 22:31.181
ask when we have a state which is described
by a state vector but can be in the the
22:31.181 --> 22:32.440
mixture of pure states
22:32.440 --> 22:37.490
so more generally we considered the probabilistic
picture of pure states called mixed states
22:37.490 --> 22:44.659
which is then given in terms of the overall
wave function psi which is now going to have
22:44.659 --> 22:50.200
the probability of each wave function in terms
of the particular composition so for example
22:50.200 --> 22:59.480
psi one with probability p one psi two with
probability p two and so on and so forth
22:59.480 --> 23:07.450
so we we have these contributions which make
up the final mixed state which is the psi
23:07.450 --> 23:14.419
state so that density matrices of this mixed
state would be having a probability distribution
23:14.419 --> 23:18.600
on pure states and its called a mixed state
the density matrix associated with such mixed
23:18.600 --> 23:27.570
state would then be given as sum total of
the probability times the inner product
23:27.570 --> 23:35.029
of the two wave functions the the ket and
the bra
23:35.029 --> 23:42.170
so the density matrix for say this particular
mixed state zero with probability half
23:42.170 --> 23:47.279
and one with probability half can be written
in this form so the density matrix of the
23:47.279 --> 23:53.190
mixed state would then the probability of
measuring the zero would be given by the conditional
23:53.190 --> 24:00.400
probability such that it is probability
of measuring zero given the pure state psi
24:00.400 --> 24:05.740
i so then we can go ahead and find that this
will be basically as we did this exercise
24:05.740 --> 24:13.190
before the probability of measuring the state
zero given the pure state psi i would be
24:13.190 --> 24:20.250
the trace of the state zero that the inner
product of the two and the density matrix
24:20.250 --> 24:24.890
so the density matrix is again coming into
this picture whenever we want to measure the
24:24.890 --> 24:30.890
probability of the state whenever we have
bunch of other states available and we can
24:30.890 --> 24:37.330
always come back to this picture where the
wave function is a composition of the probability
24:37.330 --> 24:46.190
of the inner product of the composite wave
functions to give rise to the solution so
24:46.190 --> 24:51.440
density matrices contain all the information
about an arbitrary quantum state
24:51.440 --> 24:56.690
now this is one of the most important part
of studying density matrices because as i
24:56.690 --> 25:03.240
mentioned earlier we have limited information
on the states as soon as they are going to
25:03.240 --> 25:09.179
become mixed states so for a pure states it
is possible to get enough information about
25:09.179 --> 25:15.260
the state and we can do operations to get
the results out of it however for mixed
25:15.260 --> 25:20.320
state the only way to get information about
it is as we have been showing is through the
25:20.320 --> 25:23.809
density matrices
25:23.809 --> 25:32.390
so now let's look at what it me implies
it implies that we can now work with operationally
25:32.390 --> 25:38.429
indistinguishable states which are these are
expressions in terms of density matrices alone
25:38.429 --> 25:44.380
independent of any specific probabilistic
mixtures states with identical density matrices
25:44.380 --> 25:48.769
are operationally indistinguishable now this
is the corollary of the fact that the density
25:48.769 --> 25:54.450
matrices essentially carry all the information
because in other words of saying that those
25:54.450 --> 26:01.539
states where the density matrices are going
to be the same we will have very little to
26:01.539 --> 26:08.760
do with their particular understanding because
we if we are going to rely on the concept
26:08.760 --> 26:16.890
of density matrices to understand them so
all the states where they have identical density
26:16.890 --> 26:23.919
matrices then they are operationally indistinguishable
for us to start off with their studies
26:23.919 --> 26:31.000
if we when we apply unitary operator to
a density matrix of a pure state the resulting
26:31.000 --> 26:39.370
state is essentially the result of the
unitary matrix with the density matrix so
26:39.370 --> 26:47.600
when we apply this we basically get the
unitary on operation on the density matrix
26:47.600 --> 26:53.960
when we apply unitary operator to a density
matrix of a mixed state say for example a
26:53.960 --> 27:02.890
state which has this kind of a behavior
the resulting state is going to have the density
27:02.890 --> 27:09.600
matrices which will have the property which
are going to be given by dependency on
27:09.600 --> 27:11.590
the density matrix alone
27:11.590 --> 27:17.029
so if you notice the difference in the in
the earlier case when we applied the unitary
27:17.029 --> 27:25.110
operation to the density matrix of a pure
state the pure state is essentially the only
27:25.110 --> 27:34.260
state that existed in the wave function so
the unitary operator essentially resulted
27:34.260 --> 27:41.841
in only providing the information about the
pure state directly the entire wave function
27:41.841 --> 27:47.500
could be operated on by the unitary operator
however in this in this case when its a
27:47.500 --> 27:52.690
mixed case so with the application of unitary
operation we always get the result which is
27:52.690 --> 27:58.650
expected that the density matrix is being
applied the unitary operator works on the
27:58.650 --> 28:02.100
density operator in this form
28:02.100 --> 28:09.080
now when we do this this is always true and
let us look at how this works the operational
28:09.080 --> 28:17.720
density matrices for mixed state essentially
results in unitary operator working on the
28:17.720 --> 28:23.279
density matrices and which is still true
as a result of that the effect of measurement
28:23.279 --> 28:31.450
on a density matrix would end up measuring
the state rho with respect to the basis say
28:31.450 --> 28:37.320
phi one all the way to phi d still we will
yield the kth outcome with probability
28:37.320 --> 28:51.380
phi of k rho phi of k ket because the effect
of measuring measurement on a density matrix
28:51.380 --> 28:57.070
measures the state rho with respect to
the basis yielding the kth outcome with probability
28:57.070 --> 29:06.460
this this is because the the operation
of the density matrix essentially means that
29:06.460 --> 29:11.789
we are applying this particular wave function
as a result of that which is means that i
29:11.789 --> 29:18.570
have basically ended up producing this
particular state and the state is collapsing
29:18.570 --> 29:27.120
to this particular kth state and therefore
it is always measuring the measuring the
29:27.120 --> 29:33.530
probability of the kth state collapsing into
that final state and would give the kth outcome
29:33.530 --> 29:37.830
with the probability and this is the reason
why this is a universal measurement which
29:37.830 --> 29:44.840
works with the density matrices whenever we
use a quantum operations using of density
29:44.840 --> 29:45.840
matrices
29:45.840 --> 29:51.679
and this is something which would work universally
whether it is a i pure state or a mixed state
29:51.679 --> 29:58.330
and that's the basic idea this is collapses
to this particular state so with this i
29:58.330 --> 30:04.620
would like to end today's lecture because
we have we were planning to just give you
30:04.620 --> 30:10.809
the introduction to the density matrices its
necessity to be one of the important parameters
30:10.809 --> 30:17.960
that we have been using and time whenever
we talk about collapse whether we are doing
30:17.960 --> 30:24.400
a pure state or mixed state and in terms of
communications and in many times of the
30:24.400 --> 30:29.669
computing it has been very important that
the overall property or the overall result
30:29.669 --> 30:35.029
which we have looked at are the important
properties that we looked at and in many cases
30:35.029 --> 30:41.769
it was possible to have a solution only because
we were able to use density matrices and so
30:41.769 --> 30:49.049
that's why i wanted to give you some important
ideas about the density matrices before
30:49.049 --> 31:25.029
we go further we will meet you more on
these issues in the next lecture