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