we have been looking at density matrices and
its implications on the implementation of
quantum computing that we have been looking
at until the last several weeks in this week
we had based our studies from the beginning
of density matrices ah this was prompted as
i mentioned in the very beginning of the week
by some questions raised by the students and
as a result of that we have gone ahead to
discuss the details of density matrix in relation
to the measurements and the importance of
it now that we have come to the point where
we have hopefully understood density matrices
to a point that we can ah use it in a much
more effective manner to these implementations
and understandings that we have been carrying
forward
let us now look at the general quantum operations
which include decoherence partial traces and
measurements based on density matrices so
the general quantum operations are also completely
positive trace preserving maps or admissible
operations as a result we basically choose
several matrices that satisfy the condition
let us consider matrices a one through a one
m which follow the condition as provided here
that i is the identity matrix a j t is the
transpose of a j then the mapping of the density
matrix to this set of matrices can be such
that it becomes a general quantum operator
so the mapping in terms of the density matrix
written in terms of this matrices that we
just defined can be written in terms of this
general quantum operator so for example ah
an unitary operator applying u to rho we already
know gives rise to this result u rho u dagger
let us take the example of decoherence let
us consider a zero matrix which is an outer
product of zero states and a one matrix which
is a outer product of one states this quantum
operation maps rho into this form and now
if we take a state vector psi which is having
super position of zero and one with alpha
and beta coefficients then this mapping would
essentially result in alpha squared and beta
squared formation in this map this would correspond
to measuring rho without looking at the outcome
after looking at the outcome rho would become
zero zero outer product with probability alpha
squared and one one with probability outer
product with probability beta square
another example is the trine state we had
done this state in one of our example problems
ah maybe home assignments where we have these
states which are given in terms of zero and
linear combinations of zero and one in certain
order if we define a naught as two thirds
square root outer product of phi naughts which
will give rise to this matrix whereas a one
has two third of outer product of phi one
and a two has two third outer product of phi
two then we will be getting each of them in
such a way so that their ah transfer products
and their sums are equal to identity which
means that our condition is satisfied so we
can apply the general quantum mapping operator
which is that sum of this is going to give
rise to this operation that we mention
the probability of the state psi k results
in an outcome state a k which is two third
this can be adapted to actually yield the
value of k with this success probability so
that's the reason why this particular principle
that we can have the general quantum mapping
operation work is very important because once
this condition is satisfied we apply the quantum
of mapping operator we can make a measurement
with probabilities that can be the outcome
of the state with the exact values that we
are ah interested in terms of the state so
that's the reason why this is a very important
measurement principle
another generalized quantum operation advantage
is the partial trace which discards the second
of the two qubits and this could be the example
which is of the kind where we discuss here
if we take a matrix a naught which is an identity
times the tensor product to the bra of state
zero then we construct matrix of the form
this and the other one of the form this and
we apply the general quantum mapping operator
as we have discussed before and then the state
becomes hm rho tensor product of sigma gives
rise to the state which will become this particular
form half one along the diagonals is the same
density matrix for zero with half probability
and state one with half probability
it's the operation is the partial trace of
rho so what we have essentially done is basically
we have looked at a way of looking at the
partial trace of an operation an that's an
important operation that is often used so
this is almost like we are discarding the
second of the two qubits and that's why it
is trace two rho considering that as unity
and we get the other part which is a solution
so we are interested in distinguishing mixed
states we have mentioned before that several
mixed states can have the same density matrix
which we cannot distinguish so how to distinguish
by two different density matrices so one of
the options is to try to find an orthonormal
basis phi naught phi one in which both density
matrices are diagonal the best distinguishing
strategy between these two mixed states is
to go with a rotation about the state
so let us consider a state zero with probability
half and super position of zero and one with
probability half giving rise to rho one which
is given here and another wave function another
is mixed state with state zero with probability
half and state one with probability half giving
rise to a density matrix rho two of this kind
rho one can also arise from other orthogonal
mixture where rho phi zero and phi one have
probabilities of cosine theta rho one can
also arise from this orthogonal mixture where
the states are in phi one and phi zero with
probabilities cosine squared pi over eight
and sine squared pi over eight which is essentially
in the rotated condition
similarly rho two can arise with probabilities
half when they are in other states so if we
can effectively find an orthonormal basis
rho one and rho two that are simultaneously
diagonalizable then we can get the solution
so rotating phi naught phi one to zero and
one the scenario can now be examined using
classical probability theory you can distinguish
between two classical coins whose probabilities
heads are cosine square pi over eight and
half respectively so rotating phi naught and
phi one to zero and one the scenario can now
be examined by using classical probability
theory so the rotation actually allows us
to go from one state to the other so we can
distinguish between the two classical coins
whose probabilities whose heads are now ah
cosines squared pi over eight and half respectively
and possible as we have found the orthonormal
basis where both the density matrices are
diagonal
now the question is what do we do if we are
not so lucky to get two density matrices that
are simultaneously diagnosable we should recheck
the basic properties of the trace as we have
done once before the trace of a square matrix
is defined as as we mentioned trace of the
matrix which is the sum of the diagonal terms
it is easy to check that the trace of two
matrices are essentially the trace of individual
matrices summed together and the product of
the trace of two matrices is cumulative which
means ah trace of m n is equal to trace of
n m the second property implies however that
the trace of m would be equivalent to the
trace of the unitary transform of m which
means that its essentially looking at the
sum of all the eigen values or the diagonals
so the calculation maneuvers which are worth
remembering are the fact that the outer product
of the trace a and b with a matrix m would
give rise to the expectation value of m and
trace of outer product of a b and c d is equivalent
to getting the expectation of expectation
value of ah ah is equivalent to writing out
ah outer product of b c times that of d a
ah trace of this a b outer product of a b
and c d it give rise to inner dot product
of b c and d a together we can all we should
also keep in mind that in general trace of
m of n is not equal to the trace of m and
trace of n that is not quite generally true
you cannot make the trace apply on individually
although they are commutative in terms of
the product of the matrices
so the idea of partial trace that i mentioned
before can be looked at now how do we compute
the probabilities for a partial system for
example we have two states x and y with a
probability of xy which is ah with all their
sums then ah if we look at the probability
of one of them with respect to the rest then
that's the partial measurement of one with
respect to the other so this is the form that
we are looking at this is the partial measurement
where the term which gives rise to the part
which is correlated to only the part summing
over all the y is giving rise to the partial
measurement for that so if the second system
is taken away and never again directly or
indirectly interacting with the first system
then we can treat the first system as the
following mixture
for example we wrote the earlier slide in
the earlier slide we came up with this term
this will be equivalent to the rho trace of
two when we are considering that sum over
all the ys are taking care of all the y component
or the second component of this mixture so
that's why it is the trace two so in terms
of partial trace we derive an important formula
to use which means that we have rho two which
is the trace two of rho is basically sum over
all the ys for the particular probability
of p of y with the outer product of the psis
where the psi y is written in terms of all
the sum of the x vector
the probability of measuring for example w
in the first register depends only on trace
rho now so the sum over all the probabilities
of w y would be given by probability of y
summed over with the weightage factor w y
with respect to root p y square give rise
to this particular set where we get the trace
of the measuring the outer product w times
the trace of rho two
so the partial trace can therefore be calculated
in an arbitrary basis it does not matter in
which orthonormal basis we traced out the
second system for example if we have alpha
zero zero and beta one one and we do the trace
out of the second system then we are left
with alpha squared outer product of zero beta
square outer product of one in the same ah
however in a different basis you can write
this in terms of alpha zero zero beta one
one as in a different basis alpha zero beta
one and root one over root two zero and one
for both and with the with another one which
is in this different basis
now if we do a partial trace can be calculated
on a arbitrary basis then the trace two will
finally again give rise to alpha squared outer
product of zeros and beta square outer product
of one and that's exactly the same as the
earlier one so what it means is the partial
trace is invariant to the orthonormal basis
that we choose and that's actually a very
important ah statement is that's the way how
the properties of the system can be determined
without problem
now methods to calculate partial trace partial
trace is a liner map that takes bipartite
states to a single system states so what we
have essentially done is we have taken a mixture
of two states into a single system state by
tracing out one of them and that's ah through
a linear mapping as we just did we can also
trace out the first system if necessary is
the choice how we like to do this we can compute
the partial trace directly from the density
matrix description its another important part
of this exercise
so partial trace using matrices tracing out
the second system hm would be of this kind
where you can set up the traces by individual
parts and then we can write it out in these
kinds of forms so here are some examples on
partial traces two quantum resisters example
two qubits are in states sigma and mu respectively
are independent if they can be come if the
combined system is in the state ah rho which
is a tensor product of the two in such circumstances
if the second register say is discarded then
the state of the first register remains in
sigma in general the state of a two register
system may not be of the form sigma tensor
product with mu it may contain entanglement
or correlations we can define the partial
trace two rho as the unique linear operator
satisfying the identity at trace two sigma
tensor product mu is equal to mu the second
the index two here essentially means that
the second system is being traced out second
system is mu
so for example it turns out that trace two
of this particular form that we have written
here is half one one diagonal element it turns
out that if we take this particular set of
sigma and mu then we get a result which is
of this kind is the half
we have already seen this defined in the case
of two qubit systems discarding the second
of the two qubits let a equal to i times the
tensor product of bra zero state then we have
this and a one is the tensor product of i
times one for the resulting quantum operation
state which is the tensor product of these
two becomes sigma for d dimensional registers
the operators are a k times i tensor product
of bra phi k where phi naught phi one all
the way up to phi d minus one are an orthonormal
basis as we saw in the last slide partial
trace is a matrix how to calculate this matrix
of partial trace
so calculating matrices of partial trace is
for two qubit system the partial trace is
explicitly ah given in terms of the traces
of each individual parts which add up as we
had shown before and we trace them out and
we get this and as a result we can sum them
up as we show here into these individual traces
and we can write them out in this form
the unitary transformation don't change the
local density matrix that's a very important
property a unitary transformation on the system
that is traced out does not have an effect
on the result of the partial trace which means
that if we take a system which is sort of
defined in this form ah and we take the partial
trace over two the second system then the
unitary transform in the process does not
affect the state the state also distant transformations
dont change the local density matrix also
in fact any legal quantum transformation on
the traced out system including measurement
without communicating back to the answer does
not affect the partial trace also so here
is an here is the meaning of that that if
we mention it in this format then the partial
trace will remain the way it is
now these are because of the fact that operations
on the second system should not affect the
statistics of any outcomes of the measurement
of the first system otherwise the parity of
the control of the second system would instantaneously
communicate information to controlling the
first system which would be violation of ah
information control and thats why ah individual
operations that we showed until now essentially
has no effect hm on the other operations as
we were discussing principle of these implicit
measurements ah lie on the fact that if some
qubits are in a computation are never used
again you can assume if you would like that
they have been measured and the result ignored
the reduced density matrix of the remaining
qubits is the same
so these are the very important aspects of
ah implementation that we have already used
while we were ah doing the different operations
that we looked at in the earlier weeks and
so these measurements are ratified in terms
of the density matrices and their developments
and their behaviors as we are discussing here
one of the most important ah measurement aspect
which we have utilized all the time is the
positive operator valued measurements p o
v m if we had matrices which satisfy the form
that the adjoined of that products of the
adjoineds are identity then the corresponding
positive operator valued measurement p o v
m is a stochastic operation on density matrix
that with probability produces the outcome
which is the trace of the matrix trace of
the ah matrix and its products with the density
matrix and its so basically ah the trace of
the observable of the particular matrix and
so it leads to the classical information and
the collapsed state is the one where ah its
the product of the jth matrix with the density
matrix and its ah dagger normalize with respect
to its states
so for example if we have a of j which is
an outer matrix outer dot product of phi j
which orthonormal projectors this reduces
to our previously defined measurements when
a of j is phi of j is outer product are orthonormal
projectors and rho is the outer product of
psi's then the trace would be the probability
of the outcome which has been show here and
similarly the collapsed state would essentially
be the state phi j outer product and thats
the collapsed ah quantum state as expected
because we are basically finding the projection
of the collapsed quantum state and its probability
so this is the reason why ah this particular
line of development is very essential because
this is essentially connected to measurements
the measurement postulate formulated in terms
of observables are exactly in the form that
we have been discussing now our form a measurement
is described by a complete set of projectors
p of j onto the orthogonal subspace the outcome
j occurs with probability probability of j
which is the ah inner product of psi with
respect to p of j the corresponding post measurement
state is projector of the state with respect
to its
so this is the projector matrix that we have
been discussing earlier the measurement is
described by a complete set of projectors
p of j onto the orthogonal subspace the outcome
j occurs with probability j which is given
as this form the corresponding post measurement
stat is of this form the old form of measurement
which was being discussed is the measurement
is described by an observable a hemitian operator
m with spectral decomposition m which is equivalent
to the sum of the diagonal element times the
projector
the possible measurement outcomes correspond
to the eigen value lambda j and the outcome
lambda j occurs with probability probability
of lambda j psi p of j psi the corresponding
post measurement state is essentially the
same so either we look at it in the hermitian
operator principle state by using eigen state
properties and eigen values or in the projector
way of looking at it they both essentially
give rise to the same obs same result as is
expected either from the density matrix formalism
or from the ah schrodingers formalism of solving
a schrodingers equation
so in terms of quantum mechanics that we have
been utilizing for making the quantum computers
and information quantum information come computer
science can inspire fundamental questions
about some of these we can take a informatic
approach to physics compare the physical approaches
to information what measurements can be performed
in quantum mechanics that are of interest
ah traditional approaches to quantum measurements
is a quantum measurement is described by an
observable m m is a hermitian operator acting
on the state space of the system measuring
a system prepared in an eigenstate of m gives
the corresponding eigenvalue of m as the measurement
outcome the question now present itself can
every observable be measured the answer theoretically
is yes in practice it may be very awkward
or perhaps even beyond the ingenuity of the
experimenter to devise an apparatus which
could measure some particular observable but
the theory allows one to imagine that the
measurement could be made now this is a statement
made long back by dirac essentially understanding
essentially pointing out the importance of
the concept of measurement and to the real
measurement aspect
so one of the most important work in this
area has been the von nuemann measurement
aspect thats because that has been related
to the idea of entropy and all the states
put together sos in that respect if we have
a universal set of quantum gates and the ability
to measure each qubit on the basis zero one
if you measure say phi state we get b with
probability alpha b squared as it is expected
we have the projection operators p naught
which is the outer product of state zero and
p one which is the outer product of state
one satisfying the sum total is one we consider
that the projector operator or observable
to be m which could be written in these terms
and we note that zero and one are the eigenvalues
when we measure these observable m the probability
of getting the eigenvalue b is probability
b is equivalent to psi bra times the probability
of b is essentially given by this form in
is called quantum mechanics and we get ah
it equivalent to the alpha b mod square and
we are in that case left with the state which
is the p of b times psi over the probability
of the process so essentially we are left
in the state which is given by b
the expected value of an observable therefore
can be associated if we associate with the
outcome b the eigenvalue b then the expected
outcome is given by this summation which can
be then related to the trace of trace of the
matrix and the outer product of the phi as
been done before and the von neumann measurements
can therefore give us a universal set of suppose
we have a universal set of quantum gates and
the ability to measure each qubit in the basis
zero one say we have the state alpha x x where
x goes anywhere x is an element of zero and
one if you measure all n qubits then we obtain
hm the state x with probability alpha x mod
square we have to notice that this means that
our probability of measuring a zero state
in the first qubit equals sum of alpha x mod
square for all the values of x or n bits
if we measure only the first qubit and leave
this rest alone then we still get zero with
probability of ah the state zero with probability
p zero ah which is a sum of all this states
the remaining n minus n bits the remaining
n minus qubits are then in the renormalized
state
the most general measurements of these kinds
can be looked at using a simple ah circuit
where there is a unitary operator which has
the inputs coming from the states and then
one of them is making the measurement this
partial measurement corresponds to measuring
the observable m which is of this form and
a von neumann measurement is of a type of
projective measurements given an orthonormal
basis if we can perform a von neumann measurement
with respect to phi k of the state with respect
to psi of the state phi which is given as
alpha k psi then we measure psi with probability
alpha k squared mod squared over this and
that can be written in terms of the trace
of the entire process
for example if you consider von neumann measurement
of the state phi with zero one with alpha
beta probabilities with respect to the orthonormal
basis which is super position of zero and
one plus and minus we note that we get phi
in the basis as which can be transformed to
writ write in this form we therefore get zero
one one root two in that basis the probability
of ah alpha plus beta whole squared over two
and this is an exercise which we had looked
at earlier in one of the examples that we
had done in earlier lectures
so we note that this projective cases can
be measured in this form and here is just
the way of showing how this can be measured
ah here is the little math which you might
have done as a result of the problem that
i had given you in your one of the exercises
so i think the last point we would like to
point out is how do we implement von neumann
measurements which is what we have essentially
done when we were doing all our quantum computing
implementations if we have access to a universal
setup of gates and bit wise measurements in
the computational basis we can implement von
neumann measurements with respect to an arbitrary
orthonormal basis psi k as follows
we can construct a quantum network that implements
a unitary operation psi ah operating on psi
k to give rise to a k ket vector then conjugate
the measurement operation with the operation
u which is a unitary operation so i will get
a probability of alpha k mod square another
approach would be to have the circuit that
we had shown in partial earlier to have both
the inputs in one sen have the entangled state
come in and have a final measurement of this
type both of them will have probabilities
of alphas k and these two approaches will
be ah discussed in terms so both of them have
the same probability of alpha k squared so
these approaches can be immediately connected
to the bells inequality and the bless states
that we have it is one of the cases where
we are able to connect these to the ah studies
that we have done earlier in terms of ah entanglement
and measurement of teleportation and states
like that
so in the next slide if you just observe that
when we take the orthonormal basis consisting
of the bell states and we apply the same ah
formalism that we have been ah discussing
we have discussed bell basis lectures and
other things in the earlier lectures ah we
can have destructive and non destructive measurements
and we will finally end up getting the results
that will give rise to in this terminology
that we have developed or discussed in this
entire week about density matrices tracing
of partial nature which will give rise to
the most general nature of results and this
would give rise to the general cooperate quantum
operation that can be simultaneously be used
ah by applying a unitary operator of larger
quantum system which helps us in terms of
discarding the parts which can give rise to
difficulties in terms of the decoherence and
yet we can get to the results that we are
interested in and get the right solutions
and so this is a much more powerful way of
generalized process where density matrices
can play a major role
another very important aspect is the p o v
m measurements which we introduced here ah
any p o v m can also be simulated by applying
unitary operation on a larger quantum system
and then measuring it and whatever we get
results which have their difficulties can
be ah coupled with the classical inputs which
are not going to be utilized and we can keep
the preserve the quantum nature of the story
by this fashion
ah any bipartite system for instance what
we have done can be looked at in the mon manner
that we have discussed if they are separable
then their probabilities are going to be such
that we will be having a probabilistic mixture
of the probabilities of the product state
as we present it here
how the nature of ah quantum computing goes
we have looked at entanglement where the last
part that we just discussed the separability
and others in terms of density matrices have
been very ah importantly looked at earlier
and we were able to know which are the states
which can be separable and which are which
cannot be separable that are the entangled
states or the for example bell states and
so we have looked into all these aspects and
in this week we basically presented it in
terms of density matrices because that is
one of the most important ways of looking
at all the realistic quantum problems that
we have looked at in terms of implementations
because ah the way ah the realistic measurements
goes the aspects of decoherence and other
asp other issues are inherently present and
they need to incorporate density matrices
to be able to address that
so i hope it has been a good learning experience
and bridging the gap of the parts which we
had earlier not looked into in detail so i
thank the students who had prompted me to
go back to the section and revise it in the
last week which will be upcoming we will be
doing ah a summary and revision of all the
aspects that we have looked into are covered
in this course and i suppose it will take
quite a bit of time or hours but i have left
sufficient time for the final week to cover
almost all the discussions at least in ah
summary for all the implementations of quantum
computing that we have been doing until now
and we will also look into all the problem
sets that we have been giving you as exercises
at the at that last final lecture to make
sure that everything is complete as far as
the course goes
thank you see you next week
