WEBVTT
00:13.710 --> 00:21.430
so we have been discussing in this week an
overall summary of whatever we have read throughout
00:21.430 --> 00:29.570
this course as this is our final week we have
just discussed the basics from the classical
00:29.570 --> 00:36.640
concept of computing to how we get to the
point of quantum computing let us now continue
00:36.640 --> 00:43.010
in that direction and see what we mean when
we have entered the concept of qubits so typically
00:43.010 --> 00:50.070
qubits as we have been discussing are two
level quantum systems so for an un perturbed
00:50.070 --> 00:58.309
quantum system there are two stationary energy
levels denoted by say the x upper level e
00:58.309 --> 01:04.960
a and the lower level e b the wave functions
for these two levels are denoted by state
01:04.960 --> 01:15.550
vectors a upper level and b lower level the
total hamiltonian is h which is the zero th
01:15.550 --> 01:25.630
order one and h i which is the interaction
one so h zero is the unperturbed part and
01:25.630 --> 01:31.520
h i is interaction part so as far as a steady
state goes this is how the system looks like
01:31.520 --> 01:38.130
there is an energy gap which corresponds to
omega a b in terms of energy it would be h
01:38.130 --> 01:48.490
cross omega a b the solution for the two level
system would essentially mean that we time
01:48.490 --> 01:55.549
the coefficient would evolve as has been shown
here for these two cases
01:55.549 --> 01:59.600
the main point to note here is that in the
absence of any perturbation the probabilities
01:59.600 --> 02:08.780
of finding the two level system are independent
of time however when we perturbed system say
02:08.780 --> 02:15.510
for example by providing an external electric
field the total hamiltonian in this case the
02:15.510 --> 02:23.660
interaction hamiltonian is the one which is
going to influence how this evolves and now
02:23.660 --> 02:29.250
we represent this a b is by say one and two
then this is how the interaction picture looks
02:29.250 --> 02:37.850
like the system could either h i is the interaction
hamiltonian and the system would evolve as
02:37.850 --> 02:43.910
a result of the external applied electric
field
02:43.910 --> 02:49.970
the probabilities of finding the two level
systems now vary with time the solutions for
02:49.970 --> 02:56.400
the probabilities are obtained by solving
the differential equations which are based
02:56.400 --> 03:03.700
on how the amplitudes or the probabilities
of each of these components of the states
03:03.700 --> 03:14.920
change and we get the time dependent result
which is shown here so this results in the
03:14.920 --> 03:26.560
rabi oscillations this delta is the detuning
or the difference in the energy from the excitation
03:26.560 --> 03:35.040
to the actual state if it is resonant which
is basically the case where delta is equal
03:35.040 --> 03:44.950
to zero then this is the case where it is
exactly going to have an excitation which
03:44.950 --> 03:57.210
will correspond to the gap so we get the population
oscillate from the ground to the excited and
03:57.210 --> 04:02.650
back and forth and this is known as the rabi
flopping with a frequency which is known as
04:02.650 --> 04:11.960
rabi frequency which is equivalent to two
pi so at every two pi the population is back
04:11.960 --> 04:21.860
to its ground state and it goes on cycling
and that is true for both the ground and the
04:21.860 --> 04:29.280
exiled state the total population always remains
conserved which is one
04:29.280 --> 04:35.629
so this particular picture that we just discussed
is true when we are talking about a single
04:35.629 --> 04:44.880
quantum state in principle we have an ensemble
of states and thats when we need to invoke
04:44.880 --> 04:53.490
something known as a density matrix so a density
matrix includes the statistics of the problem
04:53.490 --> 05:01.590
and as pointed out by richard feynman in statistical
mechanics book when we solve a quantum mechanical
05:01.590 --> 05:07.560
problem what we really do is divide the universe
into two parts the system in which we are
05:07.560 --> 05:14.870
interested and the rest of the universe when
we include the part of the universe outside
05:14.870 --> 05:22.819
the system the motivation of using the density
matrices become clear so what does this mean
05:22.819 --> 05:29.030
it means that when we are talking about a
pure state as for example the quantum state
05:29.030 --> 05:35.469
that we just talked about the vectors or the
states that we discussed in terms of the states
05:35.469 --> 05:44.939
that we looked at in case of an ensemble a
collection of these states would be representing
05:44.939 --> 05:55.479
the density matrix often also known as density
operators so as per definition we have discussed
05:55.479 --> 06:02.919
this earlier the density matrix of a pure
state is the matrix where rho which is the
06:02.919 --> 06:12.979
outer product of the two vectors psi ket and
psi bra so this is the representation we have
06:12.979 --> 06:20.699
used and we know that they basically represent
the matrices which are row and column and
06:20.699 --> 06:28.199
therefore they can form the matrix that we
are looking for
06:28.199 --> 06:37.150
so for example the density matrix of alpha
zero and beta one is given by this form where
06:37.150 --> 06:48.249
we get the two by two matrix for this qubit
which could be in any of the basis states
06:48.249 --> 06:56.220
so that is the importance of the density matrix
which includes all the possible conditions
06:56.220 --> 07:02.860
of the ensemble if we represent psi i ket
to be the complete set of vectors in the vector
07:02.860 --> 07:09.229
space describing the system and phi i ket
to be the complete set of rest of the universe
07:09.229 --> 07:14.621
the most general way to write the wave function
for the total system is to have the wave function
07:14.621 --> 07:22.580
written in this form now if a is an operator
that acts only on the system that is a does
07:22.580 --> 07:32.199
not act on the rest of the universe theta
i then we can write this out in this form
07:32.199 --> 07:39.639
and these we have studied earlier and this
can be simplified to get to the part which
07:39.639 --> 07:45.660
is essentially give raise to the density matrix
so this was the motive for defining the density
07:45.660 --> 07:54.620
matrix as we had done
so this operator rho is such that this follows
07:54.620 --> 08:03.340
this principle and that rho is going to be
hermitian and one of the most important property
08:03.340 --> 08:10.479
of the density matrix that we have learned
is the fact that the expectation value of
08:10.479 --> 08:20.180
the operator is essentially given by the trace
of the product of the operator with the density
08:20.180 --> 08:26.719
matrix due to the hermitian nature of the
density matrix it can be diagnosed with a
08:26.719 --> 08:35.219
complete orthonormal set of eigen functions
i with real eigen values w i and so we can
08:35.219 --> 08:42.860
have a final form which looks like this and
we have gone ahead to show that the orthonormal
08:42.860 --> 08:52.720
set of eigenvectors and eigenvalues are the
ones which give raise to density matrix that
08:52.720 --> 08:59.620
we are interested in so thats the power of
the density matrix which incorporates the
08:59.620 --> 09:07.850
elements along the diagonal which gives the
probability of presence of the individual
09:07.850 --> 09:17.250
state and this can be explored further here
any system described by density matrix rho
09:17.250 --> 09:24.120
which has this form would have these properties
one that the set i is complete orthonormal
09:24.120 --> 09:31.570
set of vectors the diagonal elements would
be greater than equal to zero are essentially
09:31.570 --> 09:38.210
non negative elements the sum of the diagonals
equal to one given an operator a the expectation
09:38.210 --> 09:47.790
value is going to be this trace rho a this
essentially ensures that the total probability
09:47.790 --> 09:54.790
of the system is going to be one so w i is
the probability that the system is in state
09:54.790 --> 10:02.660
i as i was trying to say if all but one of
w i is zero we say that the system is in a
10:02.660 --> 10:13.720
pure state so w i is like the weightage or
the contribution of a particular state to
10:13.720 --> 10:20.880
the density matrix so if it is only one state
then its a pure state otherwise its a mixed
10:20.880 --> 10:26.580
state so as the most important part about
the density matrix
10:26.580 --> 10:32.190
so when we have a pure state we also have
found out that the trace has discussed before
10:32.190 --> 10:40.760
is anyway going to be equal to one but its
also true that their square of the matrix
10:40.760 --> 10:47.250
is also equal to the matrix itself so the
square of the trace of the matrix is also
10:47.250 --> 10:54.340
equal to one however when it is a mixed state
for instance fifty percent of up and fifty
10:54.340 --> 11:01.930
percent of down then though the trace is going
to be one trace of the matrix because that
11:01.930 --> 11:06.950
has to happen because that ensures the total
probability of the system is one which is
11:06.950 --> 11:19.750
the requirement the trace of rho square is
not equal to one and rho square is also not
11:19.750 --> 11:28.780
equal to rho and so that is the most important
part of mixed state so one of the most important
11:28.780 --> 11:33.280
part of density matrix that we have discussed
is the fact that the diagonal elements represent
11:33.280 --> 11:38.500
the populations whereas the off diagonal elements
represent coherence and therefore so which
11:38.500 --> 11:46.540
is the very important part of understanding
quantum mechanics in an ensemble because most
11:46.540 --> 11:53.320
of the issues that we have delt with in terms
of quantum information relies a lot on coherence
11:53.320 --> 12:06.490
so density matrices have the critical information
of coherence also present in its elements
12:06.490 --> 12:13.480
so while the populations are extremely important
the coherence which is one of the critical
12:13.480 --> 12:21.600
components of this information processing
quantum information processing is also embedded
12:21.600 --> 12:29.780
in the density matrix formalities and when
we have mixed states it gets difficult to
12:29.780 --> 12:36.820
get the information on the individual states
which could lead to a condition when they
12:36.820 --> 12:45.460
are completely mixed to a case when there
is a state of total ignorance unless and until
12:45.460 --> 12:49.830
itss measured
the other most important part of the density
12:49.830 --> 12:54.650
matrix which we point out here is the fact
that density matrices can be defined for pure
12:54.650 --> 13:02.320
coherent superposition or statistically averaged
states the eigen states of density matrix
13:02.320 --> 13:10.200
form a complete basis for subsystem block
the eigenvalues if the weight of the state
13:10.200 --> 13:17.980
we can keep the m eigen states corresponding
to the m highest eigenvalues the eigen values
13:17.980 --> 13:25.820
of the whole system can thus be given by the
sum of the root of the weight of the individual
13:25.820 --> 13:36.620
elements which is often known as schmidt decomposition
this is the optimal approximation for this
13:36.620 --> 13:43.240
and for entanglement state the mutual quantum
information is possible to be found from the
13:43.240 --> 13:52.660
entropy of the system this was a part of many
of the exercise which was given in relation
13:52.660 --> 14:04.910
to the density matrices which is essentially
minus trace rho times log rho and that is
14:04.910 --> 14:09.040
a very important part of understanding the
system
14:09.040 --> 14:19.140
so the density matrix is the alternative state
vector the ket or the bra representation for
14:19.140 --> 14:25.000
a certain set of state vectors appearing with
certain probabilities as we have mentioned
14:25.000 --> 14:32.910
giving raise to the form that we have shown
before and we have discussed in terms of the
14:32.910 --> 14:41.120
pure state as well as the mixed state mixing
give raise to weighting with classical probabilities
14:41.120 --> 14:46.340
superposition is weighting with quantum probability
amplitudes which means that the weighting
14:46.340 --> 14:52.420
can be written in terms of square roots so
for example a pure state can be a superposition
14:52.420 --> 14:59.680
so that the weightage factor can be written
as a square root thats this schmidt decomposition
14:59.680 --> 15:06.170
one important point to realize is that density
matrices are not unique so here is an example
15:06.170 --> 15:16.610
which kind of shows the taken a density matrix
of this kind how they can be formed from different
15:16.610 --> 15:22.190
conditions and that is that is the price for
being able to decompose entangled state because
15:22.190 --> 15:31.300
in no other form of understanding can be look
at density matrices density matrices are not
15:31.300 --> 15:42.530
unique this is because by using density matrices
we are able to handle or treat entangles states
15:42.530 --> 15:52.240
to some level and that is the price for being
able to a certain concepts related to entangled
15:52.240 --> 15:59.450
states however it is still extremely important
to be able to have this mathematical formulation
15:59.450 --> 16:02.610
and this non uniqueness has its advantage
also
16:02.610 --> 16:07.730
we need to remember three properties of density
matrices which were mentioned during the regular
16:07.730 --> 16:19.000
classes also a how it is written b that the
trace is always going to be one of the density
16:19.000 --> 16:28.740
matrix that it is going to be hermitian and
that the expectation value of the is always
16:28.740 --> 16:38.720
going to be positive definite for any state
moreover for any matrix satisfying the above
16:38.720 --> 16:45.220
properties there exists a probabilistic mixture
whose density matrix is rho the trace of the
16:45.220 --> 16:51.650
matric has been defined several time before
is essentially the some of the elements of
16:51.650 --> 16:58.480
the main diagonal and they can be decomposed
into the eigenvalues and the eigenvectors
16:58.480 --> 17:04.020
and the properties of the trace are summarized
here these are thing which we are done earlier
17:04.020 --> 17:11.650
again but just been put up for given their
importance
17:11.650 --> 17:17.870
the trace of any density matrix is equal to
one for a pure state square of the trace is
17:17.870 --> 17:25.199
also one for a mixed state the square of the
trace is less than one for a pure entangled
17:25.199 --> 17:34.400
system the trace of this square of the density
matrix is one for any mixed subsidy of a e
17:34.400 --> 17:41.519
p r pair therefore is going to be less than
one so these are consequences of how we treat
17:41.519 --> 17:47.059
density matrices the evolution of any closed
physical system in time can be characterized
17:47.059 --> 17:51.950
by means of unitary transforms and that is
true for any quantum systems and that is the
17:51.950 --> 18:01.740
how density matrices are also treated so we
can devise the unitary operator which can
18:01.740 --> 18:10.230
operate on the density matrix and any quantum
measurement can be described by means of a
18:10.230 --> 18:18.809
of measurement operators m s of m where m
stands for the possible results of the measurement
18:18.809 --> 18:28.769
small m the probability of measuring small
m if the system is in state v can be calculated
18:28.769 --> 18:38.029
as below which we show here essentially its
the projection of the state into the frame
18:38.029 --> 18:45.460
of reference that is being measured so that
is how it is the trace of the projection operator
18:45.460 --> 18:52.610
with the density matrix and the system after
measurement the state m is left in the state
18:52.610 --> 19:00.340
which is given by this form which would essentially
be transforming into a form which has the
19:00.340 --> 19:08.090
projection operator taking the density matrix
into its form of this kind
19:08.090 --> 19:15.799
so this can be illustrated based on the measurement
basis that we use and there is an example
19:15.799 --> 19:31.309
which has been shown here for clarity so if
alpha and beta are the constituent compositions
19:31.309 --> 19:39.610
of zero and one then upon measurement would
be basically getting the probabilities of
19:39.610 --> 19:48.120
alpha mod square and beta mod square as the
measured values of each of the components
19:48.120 --> 19:59.419
now if you are decomposing a system into a
smaller set it can still be done by using
19:59.419 --> 20:04.899
a very powerful concept of the partial trace
which is unique to density matrices and that
20:04.899 --> 20:10.789
is the part which is extremely important as
we have discussed earlier in terms of understanding
20:10.789 --> 20:21.690
mixed states and others and this logic goes
by the principle that we can use trace properties
20:21.690 --> 20:35.529
and the operator properties to write the trace
of the overall density matrix in a way which
20:35.529 --> 20:43.820
can then be decomposed into a condition which
looks like this and since for product state
20:43.820 --> 20:55.580
the trace matrix can be written in this form
it is possible to trace out one of the components
20:55.580 --> 21:02.129
of the density matrix say for example trace
b of rho a b would essentially give raise
21:02.129 --> 21:10.220
to tracing out the component b to give raise
to rho a
21:10.220 --> 21:20.269
so for entangled systems therefore it is important
that we can trace out one of the components
21:20.269 --> 21:26.330
to get the results from the other so for example
for the particular case that has been shown
21:26.330 --> 21:36.139
here we can find out that you can get the
trace to be equivalent with the pure state
21:36.139 --> 21:41.980
if we follow the scheme of discussion that
we did until now we can essentially find out
21:41.980 --> 21:48.759
that when we trace out the second component
we can get to a point where it is given by
21:48.759 --> 21:54.639
i over two which means that this essentially
going to be maximally mixed and so contains
21:54.639 --> 22:00.640
no information about the system
we have also done the geometrical interpretation
22:00.640 --> 22:06.970
of density matrices where we utilize the bloch
sphere to understand how things are and so
22:06.970 --> 22:14.659
here are the um three poly matrices and the
unitary one which is utilized to get to the
22:14.659 --> 22:26.200
density matrix and its form the r is the vector
which corresponds to the components in this
22:26.200 --> 22:33.279
spherical axis which is how it is shown here
this can be looked at by utilizing a unitary
22:33.279 --> 22:41.330
transform to the entire process and if we
do this transformations what we essentially
22:41.330 --> 22:46.570
find out for pure and mixed state is that
the density matrix is not unit as we have
22:46.570 --> 22:52.840
if do the same thing for the pure and mixed
state we find that the density matrix is not
22:52.840 --> 22:58.240
unique for the pure state is always unique
however for the mixed state we can get several
22:58.240 --> 23:13.710
results which would correspond to the same
i over two and that is the reason why trace
23:13.710 --> 23:17.440
of the square of the matrix is always less
than one
23:17.440 --> 23:26.980
so for two qubits we can take this example
case of all the possible states which are
23:26.980 --> 23:35.409
in different conditions and in this case if
we measure the first bit or the second bit
23:35.409 --> 23:42.190
the results can be different so for example
if we measure the zero which is the first
23:42.190 --> 23:49.720
bit and the one which is the first bit and
the second set we will end up getting two
23:49.720 --> 24:01.090
different probabilities just shown here
the coefficients would change so that the
24:01.090 --> 24:06.129
ratio is going to remain the same we have
done this before i was just bringing it back
24:06.129 --> 24:13.110
to make sure that you understand this important
result so there is this case where the sets
24:13.110 --> 24:20.340
of qubits are independent qubits which is
a system of two independent qubits where the
24:20.340 --> 24:26.520
two are non interacting particles so we could
write them out in such a way so that they
24:26.520 --> 24:35.759
can be put together in case of entangled states
however there are no qubits where we in we
24:35.759 --> 24:43.720
could write down the states in such a way
so that the states could be represented in
24:43.720 --> 24:54.289
the entire form as we represent here so in
this case if we measure the first bit we get
24:54.289 --> 25:00.489
zero in the first case and one in the second
case the value of the second bit is can be
25:00.489 --> 25:07.720
measured with hundred percent probability
when we measure one of them and that is the
25:07.720 --> 25:12.980
main principle of the entangled states where
we keep on saying that if we measure the first
25:12.980 --> 25:18.900
qubit or any one of the two qubits we get
a complete information about the other qubit
25:18.900 --> 25:24.070
thats the most important part of entangled
state which from density point of view and
25:24.070 --> 25:28.299
partial measurements its hundred percent clear
as to how things are going
25:28.299 --> 25:33.900
there are several examples that we are gone
through for example we have discussed the
25:33.900 --> 25:41.960
maximally entangled states which are the bells
basis and we have also looked at states whether
25:41.960 --> 25:50.190
certain state is entangled or not and finally
the most important thing is to realize that
25:50.190 --> 25:55.509
all of this play a critical role in the process
of quantum teleportation which we have spent
25:55.509 --> 26:00.230
a lot of time about because quantum computing
and quantum information processing (Refer
26:00.230 --> 26:08.999
Time: 26 :00) one of the key relevant parts
in this area is the concept of quantum teleportation
26:08.999 --> 26:16.820
where entangled qubits a and b are being shared
where entangled qubits are being shared between
26:16.820 --> 26:24.179
a and b which are the kinds out the bell qubits
so initially the qubit with unknown state
26:24.179 --> 26:33.190
that alice wants to send to bob makes the
a and c entangled then there are some transformations
26:33.190 --> 26:47.789
made and then c is measured by alice that
information is being sent say by classical
26:47.789 --> 27:01.249
channel say phone to bob now bob knows the
state of b so he makes b into c and so bob
27:01.249 --> 27:14.320
has the qubit c and thats the idea behind
this entire concept are good enough once the
27:14.320 --> 27:23.640
communication channel is used to communicate
some information which is in this particular
27:23.640 --> 27:36.299
case the qubit a which is good enough to be
able to get the value of the unknown qubit
27:36.299 --> 27:45.399
c by bob who already knows the state of b
so there are several issues of quantum teleportation
27:45.399 --> 27:53.320
and practicalities as well as discussions
which we have earlier presented during the
27:53.320 --> 27:59.299
course work where went through the concepts
of quantum teleportation in detail and that
27:59.299 --> 28:05.869
was very important because thats another very
important area where the idea of quantum computing
28:05.869 --> 28:11.230
and information has been put to use its an
area of strong implementation in terms
28:11.230 --> 28:20.330
of quantum computing where the principle of
quantum entanglement in terms of information
28:20.330 --> 28:26.080
secure information transfer has been successful
and that is one of the areas where we have
28:26.080 --> 28:30.830
[sown/shown] shown a lot development and we
have discussed this as part of this course
28:30.830 --> 28:38.580
so until now what we have done in this
week is we have revised the concept of bringing
28:38.580 --> 28:46.570
over the quantum principles for quantum computing
and quantum information from classical rules
28:46.570 --> 28:52.519
and then we went ahead to look at one of the
critical aspects of quantum computing which
28:52.519 --> 29:00.619
is ensemble principles which are necessary
to be understood which is density matrices
29:00.619 --> 29:07.950
and based on that we were able to discuss
once more the concept of quantum entanglement
29:07.950 --> 29:15.169
and information quantum information in terms
of quantum teleportation this is what we have
29:15.169 --> 29:20.649
managed to finish until now
in the next lecture we will go about the principle
29:20.649 --> 29:29.480
of quantum computation implications and their
practicalities their implementation issues
29:29.480 --> 29:35.629
which we have done a lot during this course
and that way we hope to have a complete summary
29:35.629 --> 29:36.869
over what we have done
thank you