so we have been discussing in this week an overall summary of whatever we have read throughout
this course as this is our final week we have just discussed the basics from the classical
concept of computing to how we get to the point of quantum computing let us now continue
in that direction and see what we mean when we have entered the concept of qubits so typically
qubits as we have been discussing are two level quantum systems so for an un perturbed
quantum system there are two stationary energy levels denoted by say the x upper level e
a and the lower level e b the wave functions for these two levels are denoted by state
vectors a upper level and b lower level the total hamiltonian is h which is the zero th
order one and h i which is the interaction one so h zero is the unperturbed part and
h i is interaction part so as far as a steady state goes this is how the system looks like
there is an energy gap which corresponds to omega a b in terms of energy it would be h
cross omega a b the solution for the two level system would essentially mean that we time
the coefficient would evolve as has been shown here for these two cases
the main point to note here is that in the absence of any perturbation the probabilities
of finding the two level system are independent of time however when we perturbed system say
for example by providing an external electric field the total hamiltonian in this case the
interaction hamiltonian is the one which is going to influence how this evolves and now
we represent this a b is by say one and two then this is how the interaction picture looks
like the system could either h i is the interaction hamiltonian and the system would evolve as
a result of the external applied electric field
the probabilities of finding the two level systems now vary with time the solutions for
the probabilities are obtained by solving the differential equations which are based
on how the amplitudes or the probabilities of each of these components of the states
change and we get the time dependent result which is shown here so this results in the
rabi oscillations this delta is the detuning or the difference in the energy from the excitation
to the actual state if it is resonant which is basically the case where delta is equal
to zero then this is the case where it is exactly going to have an excitation which
will correspond to the gap so we get the population oscillate from the ground to the excited and
back and forth and this is known as the rabi flopping with a frequency which is known as
rabi frequency which is equivalent to two pi so at every two pi the population is back
to its ground state and it goes on cycling and that is true for both the ground and the
exiled state the total population always remains conserved which is one
so this particular picture that we just discussed is true when we are talking about a single
quantum state in principle we have an ensemble of states and thats when we need to invoke
something known as a density matrix so a density matrix includes the statistics of the problem
and as pointed out by richard feynman in statistical mechanics book 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 when we include the part of the universe outside
the system the motivation of using the density matrices become clear so what does this mean
it means that when we are talking about a pure state as for example the quantum state
that we just talked about the vectors or the states that we discussed in terms of the states
that we looked at in case of an ensemble a collection of these states would be representing
the density matrix often also known as density operators so as per definition we have discussed
this earlier the density matrix of a pure state is the matrix where rho which is the
outer product of the two vectors psi ket and psi bra so this is the representation we have
used and we know that they basically represent the matrices which are row and column and
therefore they can form the matrix that we are looking for
so for example the density matrix of alpha zero and beta one is given by this form where
we get the two by two matrix for this qubit which could be in any of the basis states
so that is the importance of the density matrix which includes all the possible conditions
of the ensemble if we represent psi i ket to be the complete set of vectors in the vector
space describing the system and phi i ket to be the complete set of rest of the universe
the most general way to write the wave function for the total system is to have the wave function
written in this form now if a is an operator that acts only on the system that is a does
not act on the rest of the universe theta i then we can write this out in this form
and these we have studied earlier and this can be simplified to get to the part which
is essentially give raise to the density matrix so this was the motive for defining the density
matrix as we had done so this operator rho is such that this follows
this principle and that rho is going to be hermitian and one of the most important property
of the density matrix that we have learned is the fact that the expectation value of
the operator is essentially given by the trace of the product of the operator with the density
matrix due to the hermitian nature of the density matrix it can be diagnosed with a
complete orthonormal set of eigen functions i with real eigen values w i and so we can
have a final form which looks like this and we have gone ahead to show that the orthonormal
set of eigenvectors and eigenvalues are the ones which give raise to density matrix that
we are interested in so thats the power of the density matrix which incorporates the
elements along the diagonal which gives the probability of presence of the individual
state and this can be explored further here any system described by density matrix rho
which has this form would have these properties one that the set i is complete orthonormal
set of vectors the diagonal elements would be greater than equal to zero are essentially
non negative elements the sum of the diagonals equal to one given an operator a the expectation
value is going to be this trace rho a this essentially ensures that the total probability
of the system is going to be one so w i is the probability that the system is in state
i as i was trying to say if all but one of w i is zero we say that the system is in a
pure state so w i is like the weightage or the contribution of a particular state to
the density matrix so if it is only one state then its a pure state otherwise its a mixed
state so as the most important part about the density matrix
so when we have a pure state we also have found out that the trace has discussed before
is anyway going to be equal to one but its also true that their square of the matrix
is also equal to the matrix itself so the square of the trace of the matrix is also
equal to one however when it is a mixed state for instance fifty percent of up and fifty
percent of down then though the trace is going to be one trace of the matrix because that
has to happen because that ensures the total probability of the system is one which is
the requirement the trace of rho square is not equal to one and rho square is also not
equal to rho and so that is the most important part of mixed state so one of the most important
part of density matrix that we have discussed is the fact that the diagonal elements represent
the populations whereas the off diagonal elements represent coherence and therefore so which
is the very important part of understanding quantum mechanics in an ensemble because most
of the issues that we have delt with in terms of quantum information relies a lot on coherence
so density matrices have the critical information of coherence also present in its elements
so while the populations are extremely important the coherence which is one of the critical
components of this information processing quantum information processing is also embedded
in the density matrix formalities and when we have mixed states it gets difficult to
get the information on the individual states which could lead to a condition when they
are completely mixed to a case when there is a state of total ignorance unless and until
itss measured the other most important part of the density
matrix which we point out here is the fact that density matrices can be defined for pure
coherent superposition or statistically averaged states the eigen states of density matrix
form a complete basis for subsystem block the eigenvalues if the weight of the state
we can keep the m eigen states corresponding to the m highest eigenvalues the eigen values
of the whole system can thus be given by the sum of the root of the weight of the individual
elements which is often known as schmidt decomposition this is the optimal approximation for this
and for entanglement state the mutual quantum information is possible to be found from the
entropy of the system this was a part of many of the exercise which was given in relation
to the density matrices which is essentially minus trace rho times log rho and that is
a very important part of understanding the system
so the density matrix is the alternative state vector the ket or the bra representation for
a certain set of state vectors appearing with certain probabilities as we have mentioned
giving raise to the form that we have shown before and we have discussed in terms of the
pure state as well as the mixed state mixing give raise to weighting with classical probabilities
superposition is weighting with quantum probability amplitudes which means that the weighting
can be written in terms of square roots so for example a pure state can be a superposition
so that the weightage factor can be written as a square root thats this schmidt decomposition
one important point to realize is that density matrices are not unique so here is an example
which kind of shows the taken a density matrix of this kind how they can be formed from different
conditions and that is that is the price for being able to decompose entangled state because
in no other form of understanding can be look at density matrices density matrices are not
unique this is because by using density matrices we are able to handle or treat entangles states
to some level and that is the price for being able to a certain concepts related to entangled
states however it is still extremely important to be able to have this mathematical formulation
and this non uniqueness has its advantage also
we need to remember three properties of density matrices which were mentioned during the regular
classes also a how it is written b that the trace is always going to be one of the density
matrix that it is going to be hermitian and that the expectation value of the is always
going to be positive definite for any state moreover for any matrix satisfying the above
properties there exists a probabilistic mixture whose density matrix is rho the trace of the
matric has been defined several time before is essentially the some of the elements of
the main diagonal and they can be decomposed into the eigenvalues and the eigenvectors
and the properties of the trace are summarized here these are thing which we are done earlier
again but just been put up for given their importance
the trace of any density matrix is equal to one for a pure state square of the trace is
also one for a mixed state the square of the trace is less than one for a pure entangled
system the trace of this square of the density matrix is one for any mixed subsidy of a e
p r pair therefore is going to be less than one so these are consequences of how we treat
density matrices the evolution of any closed physical system in time can be characterized
by means of unitary transforms and that is true for any quantum systems and that is the
how density matrices are also treated so we can devise the unitary operator which can
operate on the density matrix and any quantum measurement can be described by means of a
of measurement operators m s of m where m stands for the possible results of the measurement
small m the probability of measuring small m if the system is in state v can be calculated
as below which we show here essentially its the projection of the state into the frame
of reference that is being measured so that is how it is the trace of the projection operator
with the density matrix and the system after measurement the state m is left in the state
which is given by this form which would essentially be transforming into a form which has the
projection operator taking the density matrix into its form of this kind
so this can be illustrated based on the measurement basis that we use and there is an example
which has been shown here for clarity so if alpha and beta are the constituent compositions
of zero and one then upon measurement would be basically getting the probabilities of
alpha mod square and beta mod square as the measured values of each of the components
now if you are decomposing a system into a smaller set it can still be done by using
a very powerful concept of the partial trace which is unique to density matrices and that
is the part which is extremely important as we have discussed earlier in terms of understanding
mixed states and others and this logic goes by the principle that we can use trace properties
and the operator properties to write the trace of the overall density matrix in a way which
can then be decomposed into a condition which looks like this and since for product state
the trace matrix can be written in this form it is possible to trace out one of the components
of the density matrix say for example trace b of rho a b would essentially give raise
to tracing out the component b to give raise to rho a
so for entangled systems therefore it is important that we can trace out one of the components
to get the results from the other so for example for the particular case that has been shown
here we can find out that you can get the trace to be equivalent with the pure state
if we follow the scheme of discussion that we did until now we can essentially find out
that when we trace out the second component we can get to a point where it is given by
i over two which means that this essentially going to be maximally mixed and so contains
no information about the system we have also done the geometrical interpretation
of density matrices where we utilize the bloch sphere to understand how things are and so
here are the um three poly matrices and the unitary one which is utilized to get to the
density matrix and its form the r is the vector which corresponds to the components in this
spherical axis which is how it is shown here this can be looked at by utilizing a unitary
transform to the entire process and if we do this transformations what we essentially
find out for pure and mixed state is that the density matrix is not unit as we have
if do the same thing for the pure and mixed state we find that the density matrix is not
unique for the pure state is always unique however for the mixed state we can get several
results which would correspond to the same i over two and that is the reason why trace
of the square of the matrix is always less than one
so for two qubits we can take this example case of all the possible states which are
in different conditions and in this case if we measure the first bit or the second bit
the results can be different so for example if we measure the zero which is the first
bit and the one which is the first bit and the second set we will end up getting two
different probabilities just shown here the coefficients would change so that the
ratio is going to remain the same we have done this before i was just bringing it back
to make sure that you understand this important result so there is this case where the sets
of qubits are independent qubits which is a system of two independent qubits where the
two are non interacting particles so we could write them out in such a way so that they
can be put together in case of entangled states however there are no qubits where we in we
could write down the states in such a way so that the states could be represented in
the entire form as we represent here so in this case if we measure the first bit we get
zero in the first case and one in the second case the value of the second bit is can be
measured with hundred percent probability when we measure one of them and that is the
main principle of the entangled states where we keep on saying that if we measure the first
qubit or any one of the two qubits we get a complete information about the other qubit
thats the most important part of entangled state which from density point of view and
partial measurements its hundred percent clear as to how things are going
there are several examples that we are gone through for example we have discussed the
maximally entangled states which are the bells basis and we have also looked at states whether
certain state is entangled or not and finally the most important thing is to realize that
all of this play a critical role in the process of quantum teleportation which we have spent
a lot of time about because quantum computing and quantum information processing (Refer
Time: 26 :00) one of the key relevant parts in this area is the concept of quantum teleportation
where entangled qubits a and b are being shared where entangled qubits are being shared between
a and b which are the kinds out the bell qubits so initially the qubit with unknown state
that alice wants to send to bob makes the a and c entangled then there are some transformations
made and then c is measured by alice that information is being sent say by classical
channel say phone to bob now bob knows the state of b so he makes b into c and so bob
has the qubit c and thats the idea behind this entire concept are good enough once the
communication channel is used to communicate some information which is in this particular
case the qubit a which is good enough to be able to get the value of the unknown qubit
c by bob who already knows the state of b so there are several issues of quantum teleportation
and practicalities as well as discussions which we have earlier presented during the
course work where went through the concepts of quantum teleportation in detail and that
was very important because thats another very important area where the idea of quantum computing
and information has been put to use its an area of strong implementation in terms
of quantum computing where the principle of quantum entanglement in terms of information
secure information transfer has been successful and that is one of the areas where we have
[sown/shown] shown a lot development and we have discussed this as part of this course
so until now what we have done in this week is we have revised the concept of bringing
over the quantum principles for quantum computing and quantum information from classical rules
and then we went ahead to look at one of the critical aspects of quantum computing which
is ensemble principles which are necessary to be understood which is density matrices
and based on that we were able to discuss once more the concept of quantum entanglement
and information quantum information in terms of quantum teleportation this is what we have
managed to finish until now in the next lecture we will go about the principle
of quantum computation implications and their practicalities their implementation issues
which we have done a lot during this course and that way we hope to have a complete summary
over what we have done thank you