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PROFESSOR: OK, let me get going.
00:00:24.950 --> 00:00:29.700
Last time we were talking
about multi-particle states
00:00:29.700 --> 00:00:31.840
and tensor products.
00:00:31.840 --> 00:00:38.110
And for that, we
explained that if we
00:00:38.110 --> 00:00:42.390
have a system, a quantum
mechanical system of one
00:00:42.390 --> 00:00:46.620
particle described
by a vector space V,
00:00:46.620 --> 00:00:51.020
and the quantum mechanical
system of another particle
00:00:51.020 --> 00:00:56.530
described with a vector space
W, the quantum mechanics
00:00:56.530 --> 00:01:02.310
of the total system composed
by the two particles
00:01:02.310 --> 00:01:08.910
is defined on a new vector space
called the space V tensor W.
00:01:08.910 --> 00:01:12.920
And that was a construction that
showed that in particular it
00:01:12.920 --> 00:01:17.230
was not true to say
that, oh if you want
00:01:17.230 --> 00:01:20.770
to know the system
of particle 1 and 2,
00:01:20.770 --> 00:01:23.780
you just tell me what
state particle 1 is
00:01:23.780 --> 00:01:26.930
and what state particle
2 is, and that's
00:01:26.930 --> 00:01:28.050
the end of the story.
00:01:28.050 --> 00:01:32.520
No, the story is really more
sophisticated than that.
00:01:32.520 --> 00:01:38.390
So the typical
elements on this space
00:01:38.390 --> 00:01:47.230
were of the form
aij vi cross wj.
00:01:49.780 --> 00:01:59.790
And it's a sum over i and j
numbers times these vectors.
00:01:59.790 --> 00:02:02.050
So you pick a vector
in the first vector
00:02:02.050 --> 00:02:04.120
space, a vector in the
second vector space,
00:02:04.120 --> 00:02:09.580
you put them in here and take
linear combinations of them.
00:02:09.580 --> 00:02:15.310
So that's the general
state in the system.
00:02:15.310 --> 00:02:19.590
Now we said a few
things about this.
00:02:19.590 --> 00:02:22.860
One thing I didn't
say too much about
00:02:22.860 --> 00:02:29.630
was the issue of the vector
0 in this tensor space.
00:02:29.630 --> 00:02:34.650
And well, vector 0 is some
element of any vector space
00:02:34.650 --> 00:02:36.700
is an important element.
00:02:36.700 --> 00:02:40.360
And we could get a little
confused about how it looks.
00:02:40.360 --> 00:02:51.305
And here's for example,
the vector 0 in v cross w.
00:02:55.120 --> 00:03:04.770
An example of the vector 0
is the vector 0 tensor wi.
00:03:04.770 --> 00:03:12.150
If you put in the first input,
the vector 0, that's it.
00:03:12.150 --> 00:03:18.635
That is also the
vector 0 in here.
00:03:21.680 --> 00:03:28.320
Vi tensor the vector 0 in w.
00:03:28.320 --> 00:03:31.790
Here is 0 in w.
00:03:31.790 --> 00:03:39.240
Here is 0 in v. This is also 0.
00:03:39.240 --> 00:03:41.330
It's maybe a little surprising.
00:03:41.330 --> 00:03:43.070
Now how do we see that?
00:03:43.070 --> 00:03:44.570
Well we had a property.
00:03:44.570 --> 00:03:55.610
For example, this one. av tensor
w is equal to av tensor w,
00:03:55.610 --> 00:03:56.710
where a is a number.
00:03:56.710 --> 00:04:00.660
So pick a equals 0.
00:04:00.660 --> 00:04:05.270
Well 0 times any
vector is the 0 vector.
00:04:05.270 --> 00:04:08.660
0 cross w.
00:04:08.660 --> 00:04:12.910
But 0 times any vector
is also the vector 0.
00:04:12.910 --> 00:04:17.579
So this is the 0 in v cross w.
00:04:17.579 --> 00:04:24.110
So 0 cross w is the vector 0.
00:04:24.110 --> 00:04:28.320
Once you put 0 in one of the
two inputs, you're there.
00:04:28.320 --> 00:04:29.700
You're at 0.
00:04:29.700 --> 00:04:32.570
You don't have more.
00:04:32.570 --> 00:04:37.820
So that's just a
comment on the vector 0.
00:04:37.820 --> 00:04:40.280
Now we did a few things.
00:04:40.280 --> 00:04:43.530
And one thing I
didn't do last time
00:04:43.530 --> 00:04:48.270
was to define an inner product
on the new vector space.
00:04:48.270 --> 00:04:52.570
So let's define a
way to get numbers
00:04:52.570 --> 00:04:55.080
from one vector in
the tensor space
00:04:55.080 --> 00:04:56.980
and another vector
in the tensor space.
00:04:56.980 --> 00:04:58.075
So inner product.
00:05:03.700 --> 00:05:06.470
And again, here you're
supposed to define it
00:05:06.470 --> 00:05:10.511
to your best
understanding and the hope
00:05:10.511 --> 00:05:12.260
that, once you make
the right definitions,
00:05:12.260 --> 00:05:15.950
it has all that axiomatic
properties it should have.
00:05:15.950 --> 00:05:17.970
So let me take the
following thing.
00:05:17.970 --> 00:05:21.650
The inner product
with this thing aij
00:05:21.650 --> 00:05:35.470
vi omega j with bpq vp wq.
00:05:39.560 --> 00:05:46.220
So I will define
this by assuming
00:05:46.220 --> 00:05:53.100
the linearity in the
inputs on the right inputs
00:05:53.100 --> 00:05:56.540
and the anti-linearity
here on the left input.
00:05:56.540 --> 00:05:59.950
So this would be the
sum over inj here.
00:05:59.950 --> 00:06:07.300
So I'll put sum over
inj aij star sum
00:06:07.300 --> 00:06:22.160
over pq bpq and then
vi wj comma vp wq.
00:06:22.160 --> 00:06:24.530
So by declaring that
this is the case,
00:06:24.530 --> 00:06:31.700
I'm saying that the inner
product in the tensor space
00:06:31.700 --> 00:06:36.560
has the-- I'm demanding it has
the properties that we expect.
00:06:36.560 --> 00:06:40.030
If you have a vector plus
another vector here, well
00:06:40.030 --> 00:06:43.350
you get this times the first
plus this times the second.
00:06:43.350 --> 00:06:47.830
So you can take the sums
out and arrange it this way.
00:06:47.830 --> 00:06:50.570
But I still haven't
got a number,
00:06:50.570 --> 00:06:53.230
and the inner product is
supposed to be a number.
00:06:53.230 --> 00:06:56.260
So how do we get a
number at this stage?
00:06:56.260 --> 00:06:58.870
I have this thing,
and nobody has told me
00:06:58.870 --> 00:07:02.130
what this is supposed to be.
00:07:02.130 --> 00:07:04.290
At this stage, the
only thing you can say
00:07:04.290 --> 00:07:07.670
is, well, you know
I suspect that, if I
00:07:07.670 --> 00:07:14.630
had an inner product in V and
I had an inner product in w,
00:07:14.630 --> 00:07:16.910
I must have an
inner product here,
00:07:16.910 --> 00:07:19.390
and somehow I should use that.
00:07:19.390 --> 00:07:29.700
So they still define
to be ij pq aij bpq.
00:07:29.700 --> 00:07:34.050
And then what you do is
use the inner product in v
00:07:34.050 --> 00:07:36.100
to get a number from
these two vectors.
00:07:39.910 --> 00:07:42.460
This is going the
v inner product.
00:07:42.460 --> 00:07:50.556
And use the inner product on w
to get a number from the two w
00:07:50.556 --> 00:07:51.055
vectors.
00:07:57.320 --> 00:07:58.030
And that's it.
00:07:58.030 --> 00:07:59.175
The end of the definition.
00:08:02.900 --> 00:08:08.300
Now here maybe this is the sort
of most interesting step, where
00:08:08.300 --> 00:08:10.785
this part was set equal to this.
00:08:16.890 --> 00:08:22.480
And consistent with what
I was telling you about 0,
00:08:22.480 --> 00:08:27.880
suppose any of this vi was 0.
00:08:27.880 --> 00:08:31.520
If this vi was 0, we
would have 0 with vp.
00:08:31.520 --> 00:08:35.600
That would be 0, so
this whole number is 0.
00:08:35.600 --> 00:08:40.230
So the way this
can happen is one
00:08:40.230 --> 00:08:42.830
of the vectors must be 0 here.
00:08:42.830 --> 00:08:46.560
And well, you have
the 0 vector here,
00:08:46.560 --> 00:08:51.240
and the zero vector inner
product with anything is 0.
00:08:51.240 --> 00:08:54.650
So it's, again,
consistent to think
00:08:54.650 --> 00:08:58.030
that, once you put one
of these entries to 0,
00:08:58.030 --> 00:08:59.570
you've got the 0 vector.
00:09:04.320 --> 00:09:06.490
So where are we going today?
00:09:06.490 --> 00:09:08.610
Well, we have now
the inner product,
00:09:08.610 --> 00:09:12.950
and I want to go back to
a state we had last time.
00:09:12.950 --> 00:09:16.590
What we're going to
do today is define
00:09:16.590 --> 00:09:20.200
what we called an
entangled state.
00:09:20.200 --> 00:09:24.710
Then we will consider
basis of entangled states,
00:09:24.710 --> 00:09:28.830
and we will be able to discuss
this sort of nice example
00:09:28.830 --> 00:09:33.430
of teleportation,
quantum teleportation.
00:09:33.430 --> 00:09:36.050
So that's where
we're going today.
00:09:36.050 --> 00:09:39.035
I wanted to remind
you of a calculation
00:09:39.035 --> 00:09:40.680
we were doing last time.
00:09:40.680 --> 00:09:45.020
We had established that there
was a state in the tensor
00:09:45.020 --> 00:09:47.470
product of 2 spin 1/2 particles.
00:09:53.300 --> 00:10:02.460
And the state was alpha
plus tensor minus minus
00:10:02.460 --> 00:10:06.020
minus tensor plus.
00:10:06.020 --> 00:10:13.250
Now you can sometimes-- this is
an example of a superposition
00:10:13.250 --> 00:10:17.740
of vectors of the
from in the v cross w.
00:10:17.740 --> 00:10:19.790
So here is a vector
of that form.
00:10:19.790 --> 00:10:21.430
There is a vector of this form.
00:10:21.430 --> 00:10:28.140
Sometimes we put here 1 and 2.
00:10:28.140 --> 00:10:33.190
And sometimes it will be
useful to put those labels.
00:10:33.190 --> 00:10:35.230
Because if you don't
put the labels,
00:10:35.230 --> 00:10:37.080
you better make sure
that you're always
00:10:37.080 --> 00:10:40.290
talking that the first ket, is
the one for the first vector
00:10:40.290 --> 00:10:43.200
space, and the second ket is
the one for the second vector
00:10:43.200 --> 00:10:44.320
space.
00:10:44.320 --> 00:10:47.490
There's nothing really
known commutative here.
00:10:47.490 --> 00:10:54.830
So if somebody would
write for you 1 minus 2,
00:10:54.830 --> 00:11:02.900
or they would write
minus 2 1, both of you
00:11:02.900 --> 00:11:04.990
would be talking
about the same state.
00:11:04.990 --> 00:11:07.112
But if you don't
put the labels, you
00:11:07.112 --> 00:11:09.070
know you're not something
about the same state,
00:11:09.070 --> 00:11:11.530
because you assume
always the first one goes
00:11:11.530 --> 00:11:13.390
to the first Hilbert space.
00:11:13.390 --> 00:11:17.130
The second one goes with
the second vector space.
00:11:17.130 --> 00:11:20.520
So we considered an
entangled state of two
00:11:20.520 --> 00:11:22.360
spin 1/2 particles.
00:11:22.360 --> 00:11:26.690
I'm not using-- it's not fair
to use the word entangled yet,
00:11:26.690 --> 00:11:29.920
but we'll be able to
say this very soon.
00:11:29.920 --> 00:11:33.990
So the one thing we can do
now given the inner product
00:11:33.990 --> 00:11:37.000
is try to normalize this state.
00:11:37.000 --> 00:11:41.310
So how do we
normalize this state?
00:11:41.310 --> 00:11:44.070
Well, we must take the
inner product of this state
00:11:44.070 --> 00:11:44.970
with itself.
00:11:44.970 --> 00:11:48.110
So psi psi.
00:11:52.440 --> 00:11:54.560
So then what do we do?
00:11:54.560 --> 00:11:57.570
Well, given these
rules, we're supposed
00:11:57.570 --> 00:12:01.480
to take all this vector
here, all that vector there,
00:12:01.480 --> 00:12:04.360
1 alpha-- the alpha
that is on the left
00:12:04.360 --> 00:12:06.440
goes out as an alpha star.
00:12:06.440 --> 00:12:10.320
The alpha that this on the
right goes out as an alpha.
00:12:10.320 --> 00:12:21.900
And we have plus minus minus
minus plus inner product
00:12:21.900 --> 00:12:32.315
with plus minus
minus minus plus.
00:12:40.140 --> 00:12:44.460
Now this is easier than what
it seems from what I'm writing.
00:12:44.460 --> 00:12:47.120
You will be able to do
these things, I think.
00:12:47.120 --> 00:12:52.610
Or you can already maybe
do them by inspection.
00:12:52.610 --> 00:12:54.970
Basically at this
stage, you have
00:12:54.970 --> 00:12:58.640
to do each one
with each one here.
00:12:58.640 --> 00:13:00.210
And let's see what we get.
00:13:00.210 --> 00:13:03.870
Well, what is the inner
product of this with this?
00:13:03.870 --> 00:13:08.720
This works, because the inner
product of plus with plus is 1
00:13:08.720 --> 00:13:11.750
and minus with minus is 1.
00:13:11.750 --> 00:13:16.270
This on the other hand,
doesn't give any contribution,
00:13:16.270 --> 00:13:18.620
because the first
one is a plus has
00:13:18.620 --> 00:13:20.770
0 inner product with a minus.
00:13:20.770 --> 00:13:22.900
A minus has 0 with a plus.
00:13:22.900 --> 00:13:23.810
That doesn't matter.
00:13:23.810 --> 00:13:25.250
It's an overkill.
00:13:25.250 --> 00:13:31.990
So this one couples with this,
and this one couples with that.
00:13:31.990 --> 00:13:34.750
Another way people
would do this is
00:13:34.750 --> 00:13:40.790
to say oh don't worry
just take the bra here.
00:13:40.790 --> 00:13:43.766
So it's plus minus.
00:13:48.140 --> 00:13:50.300
Here is one.
00:13:50.300 --> 00:13:52.060
I'll put the labels too.
00:13:52.060 --> 00:13:56.690
Minus the bra of the minus
is the minus like that.
00:13:56.690 --> 00:14:04.800
1 plus 2.
00:14:04.800 --> 00:14:08.880
And now you do
this with this ket,
00:14:08.880 --> 00:14:16.090
the plus minus 1 2 minus
the minus plus 1 2.
00:14:19.960 --> 00:14:24.020
And bras and kets, you know that
this one goes with this one.
00:14:24.020 --> 00:14:27.450
Plus plus, minus minus,
this one goes with this one.
00:14:27.450 --> 00:14:32.015
And here I put the labels,
because when I form the bra,
00:14:32.015 --> 00:14:35.000
it's not obvious which
one you would put first,
00:14:35.000 --> 00:14:36.630
but it doesn't really matter.
00:14:39.830 --> 00:14:47.070
So back here, we have
norm of alpha squared.
00:14:47.070 --> 00:14:51.290
And this with this is 1.
00:14:51.290 --> 00:14:54.570
And minus is one,
this is another one.
00:14:54.570 --> 00:14:57.360
So this is 2 alpha squared.
00:14:57.360 --> 00:15:03.850
So if I want it to be
normalized, I take alpha 1
00:15:03.850 --> 00:15:05.500
over square root of 2.
00:15:05.500 --> 00:15:09.705
And this is the well
normalized state.
00:15:23.110 --> 00:15:26.220
So this is the unit
normalized state.
00:15:36.250 --> 00:15:40.550
So we have this state.
00:15:40.550 --> 00:15:45.580
This state is something you've
played with over last week.
00:15:45.580 --> 00:15:49.530
Is that state that
we started very fine
00:15:49.530 --> 00:15:54.790
in lecture that had 0 z
component of angular momentum,
00:15:54.790 --> 00:15:58.690
0 x component of
angular momentum,
00:15:58.690 --> 00:16:01.720
and 0 y component
of angular momentum.
00:16:01.720 --> 00:16:05.730
Total angular momentum
as we defined it.
00:16:05.730 --> 00:16:11.290
And this has a state with
absolutely no angular momentum.
00:16:11.290 --> 00:16:14.040
And what you verified
in the homework
00:16:14.040 --> 00:16:19.020
was that that state, in fact,
is rotational invariant.
00:16:19.020 --> 00:16:21.810
You apply a rotation
operator to that state
00:16:21.810 --> 00:16:27.080
by rotating in both spaces,
and out comes the same state.
00:16:27.080 --> 00:16:29.180
The state is not changed.
00:16:29.180 --> 00:16:32.955
So it's a very interesting state
that will be important for us
00:16:32.955 --> 00:16:33.455
later.
00:16:36.370 --> 00:16:41.520
All right, so having taken
care of inner products
00:16:41.520 --> 00:16:46.700
and normalizations, let's talk
a little about entangled states.
00:16:46.700 --> 00:16:49.800
So entangled states.
00:16:57.690 --> 00:17:02.260
So these are precisely those
states in which you cannot say,
00:17:02.260 --> 00:17:06.060
or describe them by saying
particle one is doing this,
00:17:06.060 --> 00:17:09.240
particle two is doing that.
00:17:09.240 --> 00:17:26.819
You've learned that v cross w
includes a state superpositions
00:17:26.819 --> 00:17:33.660
alpha ij vi cross omega j.
00:17:33.660 --> 00:17:40.480
The question is, if somebody
hands you a state like this,
00:17:40.480 --> 00:17:44.910
maybe you could do some
algebra or some trickery.
00:17:44.910 --> 00:17:54.310
And is it equal, you ask, to
some sort of vector u star
00:17:54.310 --> 00:18:02.290
tensor v star times
some vector w star.
00:18:02.290 --> 00:18:02.920
Is it equal?
00:18:05.820 --> 00:18:14.530
Is there vectors v star and
omega star belonging to v
00:18:14.530 --> 00:18:19.630
and belonging to w in such a
way that this thing, the sum,
00:18:19.630 --> 00:18:22.280
can be written as a product
of something and that.
00:18:22.280 --> 00:18:26.690
If you would have that, then
you would be able to say look,
00:18:26.690 --> 00:18:29.470
yes, this is an
interesting state,
00:18:29.470 --> 00:18:32.180
but actually it's
all simple here.
00:18:32.180 --> 00:18:36.420
Particle one is to state v star.
00:18:36.420 --> 00:18:40.830
Particle two is in state w star.
00:18:40.830 --> 00:18:54.790
If this has happened, if so,
this state of the two particles
00:18:54.790 --> 00:18:57.573
is not entangled.
00:19:02.350 --> 00:19:06.750
So if you can really factor
it, it's not entangled.
00:19:06.750 --> 00:19:14.310
If there are no such
vectors v star and w star,
00:19:14.310 --> 00:19:17.420
then it is entangled.
00:19:17.420 --> 00:19:22.060
So you can say, well, it's
a complicated factorization
00:19:22.060 --> 00:19:22.580
problem.
00:19:22.580 --> 00:19:25.360
And indeed, it might
take a little work
00:19:25.360 --> 00:19:28.610
to figure out if a state
is entangled or not.
00:19:28.610 --> 00:19:34.030
It's not a basis
dependence problem.
00:19:34.030 --> 00:19:36.930
It's not like it's entangled
in one basis or not.
00:19:36.930 --> 00:19:40.400
Here is a state, and
you find any two things
00:19:40.400 --> 00:19:43.650
that tensor this way
give you the state.
00:19:43.650 --> 00:19:46.310
So the simplest
example to illustrate
00:19:46.310 --> 00:19:52.680
this is two dimensional
vector spaces, v and w.
00:19:52.680 --> 00:19:57.810
Two dimensional complex.
00:19:57.810 --> 00:20:03.570
So v will have a
basis e1 and e2.
00:20:03.570 --> 00:20:07.615
w will have a basis f1 and f2.
00:20:10.650 --> 00:20:14.600
And the most general
state you could write
00:20:14.600 --> 00:20:26.420
is a state, general state,
is a number a11 e1 f1
00:20:26.420 --> 00:20:41.640
plus a2 e1 f2 plus a21
e2 f1 plus a22 e2 f2.
00:20:44.560 --> 00:20:46.520
That's it.
00:20:46.520 --> 00:20:52.310
There's two basis states
in v, two basis state in w.
00:20:52.310 --> 00:20:58.440
v cross w is dimension for
product of dimensions for basis
00:20:58.440 --> 00:21:01.660
states, the products of
the e's with the f's.
00:21:01.660 --> 00:21:03.410
So that's it.
00:21:03.410 --> 00:21:07.900
That's the general vector.
00:21:07.900 --> 00:21:14.120
The question is if this
is it equal to something
00:21:14.120 --> 00:21:19.670
like a1 e1 plus a2 e2.
00:21:19.670 --> 00:21:26.355
Some general vector, you write
the most general vector in v,
00:21:26.355 --> 00:21:37.620
and you write the most general
vector b1 f1 plus b2 f2 in w.
00:21:37.620 --> 00:21:41.940
And you ask is it equal to
a product, tensor product,
00:21:41.940 --> 00:21:45.390
of some vector in v
with some vector in w.
00:21:45.390 --> 00:21:52.125
So the question is really
are their numbers a1, a2, b1,
00:21:52.125 --> 00:21:56.275
and b2 so that this whole
thing gets factorized.
00:22:00.640 --> 00:22:06.170
So that's happily not
a complicated problem.
00:22:06.170 --> 00:22:11.930
We could see if
those number exist,
00:22:11.930 --> 00:22:18.170
if a1, a2, b1, b2 exist, then
the state is not entangled.
00:22:18.170 --> 00:22:22.120
You've managed to factor it out.
00:22:22.120 --> 00:22:23.650
So let's see.
00:22:23.650 --> 00:22:27.120
Well, we know the
distributive laws apply.
00:22:27.120 --> 00:22:32.830
So actually e1 f1 can only
arise from this product.
00:22:32.830 --> 00:22:42.650
So to have a solution
you must have
00:22:42.650 --> 00:22:48.390
that a11 is equal to a1 b1.
00:22:48.390 --> 00:22:54.470
a12 can only appear from
the product of e1 with f2.
00:22:54.470 --> 00:23:01.428
So a12 must be equal to a1 b2.
00:23:01.428 --> 00:23:05.790
a21 must be equal to a2 b2.
00:23:05.790 --> 00:23:13.050
And a22 must be equal to a2 b2.
00:23:13.050 --> 00:23:16.515
And we must try to solve
for these quantities.
00:23:20.350 --> 00:23:23.075
Actually, there is a
consistency condition.
00:23:26.540 --> 00:23:31.820
You see these quantities
repeat here in a funny way.
00:23:31.820 --> 00:23:43.510
If this holds from this, a11 a22
minus a12 a21 is equal to what?
00:23:43.510 --> 00:23:47.180
a11 a22 would be a1 b1 a2 b2.
00:23:51.680 --> 00:23:55.440
And a12 a21 also
have the same things.
00:23:55.440 --> 00:23:58.570
a1 b2 a2 b1.
00:23:58.570 --> 00:24:06.090
Well, both terms have both a's
and both b's, so this system
00:24:06.090 --> 00:24:13.690
only has a solution
if this product is 0.
00:24:13.690 --> 00:24:18.420
So if you give me four numbers,
if you hope to factorize it,
00:24:18.420 --> 00:24:23.240
you must have the
determinant of this matrix--
00:24:23.240 --> 00:24:30.670
if you collapse it into a
matrix, a11 a12 a21 a22,
00:24:30.670 --> 00:24:35.230
if you encode the information
about this state in a matrix,
00:24:35.230 --> 00:24:41.645
it's necessary that the
determinant of the matrix a
00:24:41.645 --> 00:24:44.920
be equal 0.
00:24:44.920 --> 00:24:53.510
So the determinant
of a is equal to 0
00:24:53.510 --> 00:24:59.210
is certainly necessary for the
factorization to take place.
00:24:59.210 --> 00:25:03.330
But a very small argument
that will be in the notes,
00:25:03.330 --> 00:25:06.530
or you can try to
complete it, shows
00:25:06.530 --> 00:25:09.660
that the determinant
equal to 0, in fact,
00:25:09.660 --> 00:25:13.270
guarantees that then you
can solve this system.
00:25:13.270 --> 00:25:15.170
There's a solution.
00:25:15.170 --> 00:25:17.370
And this is not complicated.
00:25:17.370 --> 00:25:26.390
So determinant equals 0 is
actually the same as not
00:25:26.390 --> 00:25:26.890
entangled.
00:25:35.440 --> 00:25:38.720
We've done not entangled.
00:25:38.720 --> 00:25:42.080
So there's a solution implies
determinant a equals 0,
00:25:42.080 --> 00:25:46.140
but determinant of a equals
0 also implies not entangled.
00:25:46.140 --> 00:25:50.660
You do that by solving this.
00:25:50.660 --> 00:25:53.920
Let's not spend time doing that.
00:25:53.920 --> 00:25:56.700
The basic way to do
it is to assume--
00:25:56.700 --> 00:26:00.320
consider, say, a11
equals 0 and solve it.
00:26:00.320 --> 00:26:03.000
Then a11 different
from 0, and then you
00:26:03.000 --> 00:26:06.690
can show that you can
choose these quantities.
00:26:06.690 --> 00:26:08.540
So it can be factored.
00:26:08.540 --> 00:26:14.220
And you have that,
if these numbers are
00:26:14.220 --> 00:26:18.400
such that the determinant is
0, then the state is entangled.
00:26:18.400 --> 00:26:22.530
And it's very easy to have a
determinant of this non-zero.
00:26:22.530 --> 00:26:26.900
For example, you could have
these two 0 and these two
00:26:26.900 --> 00:26:28.530
non-zero.
00:26:28.530 --> 00:26:32.930
That will be entangled because
the determinant is non-zero.
00:26:32.930 --> 00:26:36.290
You can have this two
that will be entangled.
00:26:36.290 --> 00:26:40.110
There are many ways of
getting entangled states.
00:26:40.110 --> 00:26:44.970
So in fact, there's enough
ways to get entangled states
00:26:44.970 --> 00:26:49.980
that we can construct a basis.
00:26:49.980 --> 00:26:56.290
We had a basis here
of e1 f1 e2 f2.
00:26:56.290 --> 00:26:56.920
This thing.
00:26:56.920 --> 00:27:00.430
This four vector basis.
00:27:00.430 --> 00:27:05.550
We can construct a basis
that is all the states,
00:27:05.550 --> 00:27:09.920
all the basis vectors
are entangled states.
00:27:09.920 --> 00:27:11.430
That's what we're
going to do next.
00:27:11.430 --> 00:27:14.850
But maybe it's about
time for questions,
00:27:14.850 --> 00:27:19.905
things that have become a
little unclear as I went along.
00:27:24.970 --> 00:27:25.620
Yes?
00:27:25.620 --> 00:27:29.500
AUDIENCE: So what exactly
does an entangled state mean?
00:27:29.500 --> 00:27:32.895
What are the [INAUDIBLE] to
give me an entangled state.
00:27:35.750 --> 00:27:39.010
PROFESSOR: Well, the main
thing that it happens
00:27:39.010 --> 00:27:42.730
is that there will be
interesting correlations when
00:27:42.730 --> 00:27:45.730
you have an entangled state.
00:27:45.730 --> 00:27:48.470
If you have an
entangled state and you
00:27:48.470 --> 00:27:51.720
find a state that
is not entangled,
00:27:51.720 --> 00:27:54.220
you can say particle
one is doing this
00:27:54.220 --> 00:27:56.280
and particle two is doing that.
00:27:56.280 --> 00:27:58.860
And particle two is
doing this independent
00:27:58.860 --> 00:28:00.780
of what particle one is doing.
00:28:00.780 --> 00:28:03.850
But when a state is
entangled, whatever
00:28:03.850 --> 00:28:06.180
is happening with
particle one is
00:28:06.180 --> 00:28:07.805
correlated with
what is happening
00:28:07.805 --> 00:28:11.320
in particle two
in a strange way.
00:28:11.320 --> 00:28:14.640
So if particle one
is doing something,
00:28:14.640 --> 00:28:16.420
then particle two is
doing another thing.
00:28:16.420 --> 00:28:18.480
But if particle one is
doing another thing,
00:28:18.480 --> 00:28:20.450
then particle two
is doing something.
00:28:20.450 --> 00:28:23.190
And these particles
can be very far apart,
00:28:23.190 --> 00:28:26.110
and that's when it gets
really interesting.
00:28:26.110 --> 00:28:30.950
So we're going to do a lot of
things with entangled states.
00:28:30.950 --> 00:28:33.320
Today we're doing
this teleportation
00:28:33.320 --> 00:28:37.870
using entangled state, and
you will see how subtle it is.
00:28:37.870 --> 00:28:42.770
Next time we do EPR, these
Einstein Podolsky Rosen
00:28:42.770 --> 00:28:45.320
arguments and the
Bell inequalities
00:28:45.320 --> 00:28:48.360
that answered that
with entangled states.
00:28:48.360 --> 00:28:51.300
There's a couple of problems
in the homework set also
00:28:51.300 --> 00:28:55.770
developing entangled states
in different directions.
00:28:55.770 --> 00:28:57.820
And I think by the
time we're done,
00:28:57.820 --> 00:29:03.200
you'll feel very
comfortable with this.
00:29:03.200 --> 00:29:06.590
So a basis of entangled states.
00:29:06.590 --> 00:29:08.840
Here are those.
00:29:08.840 --> 00:29:10.220
We're going to use spins.
00:29:10.220 --> 00:29:22.110
So we're going to use v is
the state space of spin 1/2.
00:29:22.110 --> 00:29:26.110
And we're going to
consider a v tensor
00:29:26.110 --> 00:29:30.900
v where this refers to the
first particle and this
00:29:30.900 --> 00:29:32.195
this to the second particle.
00:29:35.330 --> 00:29:44.440
So let's take one
state, phi 0, defined
00:29:44.440 --> 00:29:49.820
to be 1 over square root of
2, and I don't put indices.
00:29:49.820 --> 00:29:52.510
And probably at
some stage, you also
00:29:52.510 --> 00:29:56.550
tend to drop the tensor product.
00:29:56.550 --> 00:29:59.680
I don't know if it's
early enough to drop it.
00:29:59.680 --> 00:30:02.940
Probably we could drop it.
00:30:02.940 --> 00:30:09.640
We'll put plus plus minus minus.
00:30:09.640 --> 00:30:13.200
Of course, people eventually
drop even the other ket
00:30:13.200 --> 00:30:16.320
and put it plus plus.
00:30:16.320 --> 00:30:19.630
So those are the
evolutions of notation.
00:30:19.630 --> 00:30:23.970
As you get to more and more
calculations, you write less,
00:30:23.970 --> 00:30:26.490
but hopefully, it's still clear.
00:30:26.490 --> 00:30:28.420
But I will not do this one.
00:30:28.420 --> 00:30:30.940
I will still keep that
because many times,
00:30:30.940 --> 00:30:33.480
I will want to keep labels.
00:30:33.480 --> 00:30:36.110
Otherwise, it's a
little more cumbersome.
00:30:36.110 --> 00:30:41.900
So this state is normalized.
00:30:41.900 --> 00:30:46.610
Phi0 phi0 is equal to 1.
00:30:46.610 --> 00:30:48.330
It's the state we built.
00:30:48.330 --> 00:30:50.280
Oh, in fact, I want
it with a plus.
00:30:50.280 --> 00:30:52.070
Sorry.
00:30:52.070 --> 00:30:55.110
It's similar to the
state we had there.
00:30:55.110 --> 00:30:59.040
And by now, you say, look,
yes, it's normalized.
00:30:59.040 --> 00:31:01.070
Let's take the dual.
00:31:01.070 --> 00:31:03.680
Plus plus with plus
plus will give me 1.
00:31:03.680 --> 00:31:06.780
The minus minus with minus
minus will give me 1.
00:31:06.780 --> 00:31:07.990
This is 2.
00:31:07.990 --> 00:31:11.890
1 over square root
of 2 squared, 1.
00:31:11.890 --> 00:31:14.921
It should become sort
of easy by inspection
00:31:14.921 --> 00:31:15.920
that this is normalized.
00:31:19.310 --> 00:31:25.150
And this is entangled
state because in the matrix
00:31:25.150 --> 00:31:29.190
representation, it's a
1 here and a 1 there.
00:31:29.190 --> 00:31:33.510
You have the 1 1 product
and the 2 2 product.
00:31:33.510 --> 00:31:37.110
So 1 1, the determinant
is non-zero.
00:31:37.110 --> 00:31:39.710
There's no way,
we've proven, you
00:31:39.710 --> 00:31:42.640
can find how to factor this.
00:31:42.640 --> 00:31:45.860
There's no alpha.
00:31:45.860 --> 00:31:50.670
There's no way to write this as
an alpha plus, plus beta minus,
00:31:50.670 --> 00:31:54.490
times a gamma plus,
plus delta minus.
00:31:54.490 --> 00:31:55.660
Just impossible.
00:31:55.660 --> 00:31:56.450
We've proven it.
00:31:56.450 --> 00:31:58.650
It's entangled.
00:31:58.650 --> 00:32:05.340
So this is an entangled
state, but the state space
00:32:05.340 --> 00:32:08.110
is four dimensional.
00:32:08.110 --> 00:32:12.700
So if it's four dimensional, we
need three more basis states.
00:32:12.700 --> 00:32:13.930
So here they are.
00:32:13.930 --> 00:32:17.040
I'm going to write
a formula for them.
00:32:17.040 --> 00:32:23.170
Phi i for i equals
1, 2, and 3 will
00:32:23.170 --> 00:32:26.640
be defined to be
the following thing.
00:32:26.640 --> 00:32:34.820
You will act with the operator
1 tensor sigma i on phi 0.
00:32:42.400 --> 00:32:44.300
So three ways of doing.
00:32:44.300 --> 00:32:47.560
Let's do 1, for example, phi 1.
00:32:47.560 --> 00:32:48.410
What is it?
00:32:51.230 --> 00:32:59.950
Well, you would have 1 times
sigma 1 acting on the state phi
00:32:59.950 --> 00:33:05.770
0, which is 1 over square
root of 2 plus, plus,
00:33:05.770 --> 00:33:09.360
plus minus, minus.
00:33:09.360 --> 00:33:12.910
Well, the 1 acts on the
first ket, the sigma
00:33:12.910 --> 00:33:16.240
acts on the second ket.
00:33:16.240 --> 00:33:18.200
So what do we get here?
00:33:18.200 --> 00:33:22.660
1 over square root of 2-- let
me go a little slow-- plus
00:33:22.660 --> 00:33:32.620
sigma 1 plus, plus,
minus sigma 1 minus.
00:33:36.690 --> 00:33:46.150
And this is phi 1 equals sigma
1 plus is the minus state,
00:33:46.150 --> 00:33:51.560
and sigma 1 minus
is the plus state.
00:33:51.560 --> 00:33:53.480
1 over square root of 2.
00:33:53.480 --> 00:33:57.360
Those are things
that you may just
00:33:57.360 --> 00:34:01.830
remember sigma 1 is this matrix.
00:34:01.830 --> 00:34:05.730
So you get 1 over
square root of 2 plus,
00:34:05.730 --> 00:34:11.620
minus, plus, minus, plus.
00:34:11.620 --> 00:34:13.100
So that's phi 1.
00:34:18.580 --> 00:34:24.540
And phi 1 is
orthogonal to phi 0.
00:34:24.540 --> 00:34:29.010
You can see that because plus
minus cannot have an overlap
00:34:29.010 --> 00:34:32.130
with plus plus, nor
with minus minus.
00:34:32.130 --> 00:34:34.050
Here minus plus, no.
00:34:34.050 --> 00:34:35.659
In order to get
something, you would
00:34:35.659 --> 00:34:40.429
have to have the same label
here and the same label here
00:34:40.429 --> 00:34:41.599
so that something matches.
00:34:45.469 --> 00:34:49.510
Well, we can do the
other ones as well.
00:34:49.510 --> 00:34:55.090
I will not bother you too
much writing them out.
00:35:01.540 --> 00:35:03.070
So what do they look like?
00:35:03.070 --> 00:35:13.520
Well, you have phi 2 would
be 1 tensor sigma 2 on phi 0.
00:35:13.520 --> 00:35:23.310
And that would give you--
I will just copy it-- an i
00:35:23.310 --> 00:35:25.660
because sigma 2 has i's there.
00:35:25.660 --> 00:35:33.345
So i over square root of 2
plus, minus, minus, minus, plus.
00:35:36.850 --> 00:35:44.720
Finally, phi 3 is 1
tensor sigma 3 phi 0.
00:35:44.720 --> 00:35:48.340
And it's 1 over
square root of 2 plus,
00:35:48.340 --> 00:35:52.815
plus, minus, minus, minus.
00:35:59.620 --> 00:36:02.400
We got the states here.
00:36:02.400 --> 00:36:04.712
Let's just check
they're orthonormal.
00:36:08.140 --> 00:36:10.735
Well, here's one thing.
00:36:13.960 --> 00:36:27.370
If you take phi 0 with
1 tensor sigma i phi 0,
00:36:27.370 --> 00:36:32.230
which is phi 0 with phi i.
00:36:36.500 --> 00:36:39.550
Well, this is 0.
00:36:39.550 --> 00:36:42.120
You could say, well,
how do you know?
00:36:42.120 --> 00:36:44.010
How do you prove it easily?
00:36:44.010 --> 00:36:50.320
Well, I think the best
way is just inspection,
00:36:50.320 --> 00:36:52.140
so let's look at that.
00:36:52.140 --> 00:36:55.660
Phi 1, we said, is
orthogonal to phi 0
00:36:55.660 --> 00:36:59.490
because it has plus
minus and minus plus,
00:36:59.490 --> 00:37:02.440
and that can never do
anything with that.
00:37:02.440 --> 00:37:06.490
Phi 2 also has plus
minuses and minus pluses,
00:37:06.490 --> 00:37:09.980
so we can never have
anything to do with phi 0.
00:37:09.980 --> 00:37:12.480
The only one that
has a chance to have
00:37:12.480 --> 00:37:15.730
an inner product
with phi 0 is phi 2
00:37:15.730 --> 00:37:19.290
because it has a plus
plus and a minus minus.
00:37:19.290 --> 00:37:21.890
On the other hand,
when you flip them,
00:37:21.890 --> 00:37:25.870
this term with a plus plus
of phi 0 will give you 1,
00:37:25.870 --> 00:37:27.710
but here's a difference of sign.
00:37:27.710 --> 00:37:32.770
So this with the second term of
phi 00 will give you a minus,
00:37:32.770 --> 00:37:35.370
and therefore, it will be 0.
00:37:35.370 --> 00:37:41.315
So these things are
all 0 by inspection.
00:37:46.460 --> 00:37:50.070
You don't really have to
do a calculation there.
00:37:50.070 --> 00:37:52.530
The one that takes
a little more work
00:37:52.530 --> 00:37:58.550
is to try to understand what
is the inner product of phi i
00:37:58.550 --> 00:38:00.750
with phi j.
00:38:00.750 --> 00:38:03.850
Now, you could say, OK, I'm
going to do them by inspection.
00:38:03.850 --> 00:38:14.170
After all, there's just
six things to check.
00:38:14.170 --> 00:38:19.250
But let's just do it a
little more intelligently.
00:38:19.250 --> 00:38:24.470
Let's try to calculate this by
saying, well, this is phi 0.
00:38:24.470 --> 00:38:27.770
Since the Pauli
matrices are Hermitian,
00:38:27.770 --> 00:38:33.020
this phi i is also
1 tensor sigma i.
00:38:33.020 --> 00:38:37.440
They're Hermitian, so
acting on the left,
00:38:37.440 --> 00:38:40.260
they're doing the right thing.
00:38:40.260 --> 00:38:44.090
Given our definition, here
is a definition as well.
00:38:47.530 --> 00:38:52.560
So you take the bra
and that's what it is.
00:38:52.560 --> 00:38:56.630
It would have been dagger
here but it's not necessary.
00:38:56.630 --> 00:39:04.240
And then you have the phi j,
which is 1 tensor sigma j.
00:39:04.240 --> 00:39:05.760
And that's phi 0 here.
00:39:08.640 --> 00:39:10.500
That sounds like
the kind of thing
00:39:10.500 --> 00:39:15.450
that we can make progress
using our Pauli identities.
00:39:15.450 --> 00:39:19.990
Indeed, first thing is that
the product of operators, they
00:39:19.990 --> 00:39:24.120
multiply just in that order
in the tensor product.
00:39:24.120 --> 00:39:32.470
So phi 0, you have 1 times 1,
which is 1 tensor sigma i sigma
00:39:32.470 --> 00:39:34.340
j phi 0.
00:39:39.060 --> 00:39:47.090
And this is equal
to phi 0 1 tensor.
00:39:47.090 --> 00:39:49.790
Now, the product of
two Pauli matrices
00:39:49.790 --> 00:39:54.300
gives you an identity
plus a Pauli matrix.
00:39:54.300 --> 00:39:58.700
You may or may not remember
this formula, but it's 1 times
00:39:58.700 --> 00:40:09.635
delta ij plus i epsilon
ijk sigma k phi 0.
00:40:17.420 --> 00:40:21.930
Now, what do we get?
00:40:21.930 --> 00:40:26.580
Look, the second term
has a sigma k on phi 0,
00:40:26.580 --> 00:40:32.430
so it's some number
with a psi k here,
00:40:32.430 --> 00:40:34.950
while the first
term is very simple.
00:40:34.950 --> 00:40:37.270
What do we get from
the first term?
00:40:37.270 --> 00:40:42.256
From the first term,
we get-- well, 1 tensor
00:40:42.256 --> 00:40:46.520
1 between any two things
is nothing because the 1
00:40:46.520 --> 00:40:50.640
acting on things and the 1
acting on another thing is 0.
00:40:50.640 --> 00:40:56.425
So the unit operator in the
tensor product is 1 tensor 1.
00:40:56.425 --> 00:40:58.770
That's nothing whatsoever.
00:40:58.770 --> 00:41:00.590
So what do you get here?
00:41:00.590 --> 00:41:19.866
Delta ij times phi 0 phi 0
plus i epsilon ijk phi 0 phi k.
00:41:22.700 --> 00:41:24.190
But that is 0.
00:41:24.190 --> 00:41:32.140
We already showed that
any phi i with phi 0 is 0.
00:41:32.140 --> 00:41:34.210
And this is 1.
00:41:34.210 --> 00:41:36.040
So what have we learned?
00:41:36.040 --> 00:41:40.040
That this whole
thing is delta ij.
00:41:43.310 --> 00:41:47.810
And therefore, the
basis is orthonormal.
00:41:47.810 --> 00:41:53.180
So we've got a basis
of orthonormal states
00:41:53.180 --> 00:41:58.720
in the tensor product of
two spin 1/2 particles.
00:41:58.720 --> 00:42:01.490
And the nice thing
about this basis
00:42:01.490 --> 00:42:07.890
is that all of these basis
states are entangled states.
00:42:07.890 --> 00:42:09.770
They're entangled
because they fill
00:42:09.770 --> 00:42:11.360
different parts of the matrix.
00:42:11.360 --> 00:42:16.690
Here you have 1 and
1 and minus 1 here.
00:42:16.690 --> 00:42:23.210
This would be plus minus, would
be an i here and a minus i
00:42:23.210 --> 00:42:23.730
there.
00:42:23.730 --> 00:42:28.040
The determinants are
non-zero for all of them,
00:42:28.040 --> 00:42:30.100
and therefore, they
can't be factored,
00:42:30.100 --> 00:42:32.590
and therefore,
they're entangled.
00:42:32.590 --> 00:42:35.920
So the last thing I
want to do with this
00:42:35.920 --> 00:42:39.510
is to record a
formula for you, which
00:42:39.510 --> 00:42:46.550
is a formula of the basis
states in the conventional way,
00:42:46.550 --> 00:42:50.900
written as superposition
of entangled states.
00:42:50.900 --> 00:42:56.668
So for example, you
say, what is plus plus?
00:42:56.668 --> 00:43:01.620
Well, plus plus, looking
there, how would you solve it?
00:43:01.620 --> 00:43:05.420
You would solve it
from phi 0 and phi 3.
00:43:05.420 --> 00:43:11.410
You would take the sum so that
the minus minus states cancel.
00:43:11.410 --> 00:43:15.200
Phi 0 and phi 3, and
therefore, this state
00:43:15.200 --> 00:43:24.160
must be 1 over square root
of 2, phi 0 plus phi 3.
00:43:24.160 --> 00:43:25.720
A useful relation.
00:43:25.720 --> 00:43:28.150
Then we have plus minus.
00:43:28.150 --> 00:43:31.510
Then we have minus plus.
00:43:31.510 --> 00:43:36.440
And finally, minus minus.
00:43:36.440 --> 00:43:40.700
Well, minus minus
would be done by 1
00:43:40.700 --> 00:43:46.440
over square root of
2 phi 0 minus phi 3.
00:43:52.360 --> 00:43:56.320
The other ones, well, they
just leave complex numbers.
00:43:56.320 --> 00:44:05.180
Phi 1 has this plus minus, and
this has a plus minus in phi 2.
00:44:05.180 --> 00:44:07.670
The only problem is
it has an i, so you
00:44:07.670 --> 00:44:12.470
must take this state
minus i times this state
00:44:12.470 --> 00:44:18.510
will produce this state twice
and will cancel this term.
00:44:18.510 --> 00:44:19.780
That's what you want.
00:44:19.780 --> 00:44:29.880
So phi 1, this should be 1 over
square root of 2 phi 1 minus i
00:44:29.880 --> 00:44:33.090
phi 2.
00:44:33.090 --> 00:44:40.870
And this one should
be phi 1 plus i phi 2.
00:44:44.080 --> 00:44:47.190
And if this was a
little quick, it's
00:44:47.190 --> 00:44:50.100
just algebra, one more line.
00:44:50.100 --> 00:44:54.070
You do it with
patience in private.
00:44:57.100 --> 00:44:59.780
So here it is.
00:45:02.710 --> 00:45:08.025
It's the normal product,
simple product basis expressed
00:45:08.025 --> 00:45:11.640
as a superposition
of entangled states.
00:45:11.640 --> 00:45:18.330
This is called the bell
basis, this phi 1 up to phi 4,
00:45:18.330 --> 00:45:19.350
the bell basis.
00:45:29.080 --> 00:45:33.620
And now, I have to say
a couple more things
00:45:33.620 --> 00:45:38.230
and we're on our way to begin
the teleportation thing.
00:45:38.230 --> 00:45:40.330
Are there questions?
00:45:40.330 --> 00:45:46.670
Any questions about bell basis
or the basis we've introduced?
00:45:46.670 --> 00:45:47.910
Any confusion?
00:45:47.910 --> 00:45:49.245
Errors on the blackboard?
00:46:00.290 --> 00:46:08.510
So we have a basis, and I want
to make two remarks before we
00:46:08.510 --> 00:46:13.840
get started with
the teleportation.
00:46:13.840 --> 00:46:16.260
It's one remark
about measurement
00:46:16.260 --> 00:46:20.960
and one remark about
evolution of states.
00:46:20.960 --> 00:46:21.620
Two facts.
00:46:29.230 --> 00:46:32.010
The first fact has to
do with measurement
00:46:32.010 --> 00:46:33.730
in orthonormal basis.
00:46:33.730 --> 00:46:38.360
If you have an
orthonormal basis,
00:46:38.360 --> 00:46:42.320
the postulate of measurement
of quantum mechanics
00:46:42.320 --> 00:46:44.640
can be stated as
saying that you can
00:46:44.640 --> 00:46:47.550
do an experiment
in which you find
00:46:47.550 --> 00:46:52.750
the probability of your state
being along any of these basis
00:46:52.750 --> 00:46:56.320
states of the orthonormal basis.
00:46:56.320 --> 00:46:59.890
So you can do an
experiment to detect
00:46:59.890 --> 00:47:03.590
in which of the basis
states the state is.
00:47:03.590 --> 00:47:06.240
Now, the state, of course,
is in a superposition
00:47:06.240 --> 00:47:11.230
of basis states, but it will
collapse into one of them
00:47:11.230 --> 00:47:12.720
with some probability.
00:47:12.720 --> 00:47:17.010
So the Stern-Gerlach
experiment was an example
00:47:17.010 --> 00:47:20.232
in which you pick two
basis states, orthogonal,
00:47:20.232 --> 00:47:22.170
and there was a
device that allowed
00:47:22.170 --> 00:47:26.190
you to collapse the state
into one or the other.
00:47:26.190 --> 00:47:28.920
So this is a little
more general, not just
00:47:28.920 --> 00:47:30.810
for two state systems.
00:47:30.810 --> 00:47:33.770
If there would be a
particle with three states,
00:47:33.770 --> 00:47:36.310
well, orthonormal
states, then there
00:47:36.310 --> 00:47:39.270
is in principle an operator
in quantum mechanics
00:47:39.270 --> 00:47:43.640
that allows it to measure
which of these basis states
00:47:43.640 --> 00:47:45.380
you go into.
00:47:45.380 --> 00:48:00.530
So let me state this as saying,
given an orthonormal basis, e1
00:48:00.530 --> 00:48:16.225
up to en, we can
measure a state, psi,
00:48:16.225 --> 00:48:31.640
and we get that the probability
to be in ei is, as you know,
00:48:31.640 --> 00:48:36.336
ei overlapped with
a state squared.
00:48:36.336 --> 00:48:42.090
And if you measure
that this probability,
00:48:42.090 --> 00:48:45.150
the state will collapse
into one of these states.
00:48:45.150 --> 00:49:05.500
So after the measurement,
the state goes into some ek.
00:49:05.500 --> 00:49:07.050
There are different
probabilities
00:49:07.050 --> 00:49:09.980
to be in each one of
those basis states,
00:49:09.980 --> 00:49:12.000
but the particle
will choose one.
00:49:16.580 --> 00:49:18.870
Now, the other thing
I want to mention
00:49:18.870 --> 00:49:25.950
is that a fact that has
seemed always a gift,
00:49:25.950 --> 00:49:30.820
the Pauli matrices are
not only Hermitian,
00:49:30.820 --> 00:49:36.520
but they square to one, and
therefore they're also unitary.
00:49:36.520 --> 00:49:40.910
So the Pauli
matrices are unitary.
00:49:40.910 --> 00:49:47.480
So actually, they can be
realized as time evolution.
00:49:47.480 --> 00:49:53.340
So you have a state and you
want to multiply it by sigma 1.
00:49:53.340 --> 00:49:57.700
You say, OK, well, that's
a very mathematical thing.
00:49:57.700 --> 00:50:01.720
Not so mathematical because
it's a unitary operator,
00:50:01.720 --> 00:50:04.880
so it could respond to
some time evolution.
00:50:04.880 --> 00:50:07.190
So we claim there
is a Hamiltonian
00:50:07.190 --> 00:50:10.830
that you can construct
that will evolve the state
00:50:10.830 --> 00:50:14.390
and multiply it by sigma 1.
00:50:14.390 --> 00:50:23.330
So all these Pauli matrices,
sigma 1, sigma 2, and sigma 3
00:50:23.330 --> 00:50:32.370
are unitary as operators.
00:50:32.370 --> 00:50:45.590
They can be realized
by time evolution
00:50:45.590 --> 00:50:51.400
with a suitable Hamiltonian.
00:50:51.400 --> 00:50:55.020
So if you're
talking spin states,
00:50:55.020 --> 00:50:58.160
some magnetic field
that lifts for some few
00:50:58.160 --> 00:51:01.890
picoseconds according to
the dipole, and that's it.
00:51:01.890 --> 00:51:04.250
It will implement sigma one.
00:51:04.250 --> 00:51:06.990
Just in fact, you can
check, for example,
00:51:06.990 --> 00:51:13.670
that e to the i pi over
2 minus 1 plus sigma i.
00:51:16.350 --> 00:51:19.260
This is i this, and
this is Hermitian.
00:51:22.920 --> 00:51:25.350
Well, this is 1 and sigma i.
00:51:25.350 --> 00:51:28.040
1 and sigma i
commute, so this is
00:51:28.040 --> 00:51:32.090
equal to e to the
minus i pi over 2 times
00:51:32.090 --> 00:51:37.330
e to the i pi sigma 1 over 2.
00:51:37.330 --> 00:51:42.640
The first factor is a minus
i, and the second factor
00:51:42.640 --> 00:51:54.990
is 1 times cosine of pi
over 2 plus i sigma 1 sine
00:51:54.990 --> 00:51:57.800
of pi over 2.
00:51:57.800 --> 00:52:04.000
So this is minus i times--
this is 0-- times i sigma 1.
00:52:04.000 --> 00:52:06.960
So this is sigma 1.
00:52:06.960 --> 00:52:10.172
So we've written sigma
1 as the exponential
00:52:10.172 --> 00:52:13.730
of i times the
Hermitian operator.
00:52:13.730 --> 00:52:16.960
And therefore, you
could say that this
00:52:16.960 --> 00:52:23.500
must be equal to some time times
some Hamiltonian over h bar.
00:52:23.500 --> 00:52:28.120
And you decide, you
put the magnetic field
00:52:28.120 --> 00:52:30.260
in the x, y, z direction.
00:52:30.260 --> 00:52:33.270
You realize it.
00:52:33.270 --> 00:52:36.360
So sigmas can be
realized by a machine.
00:52:39.480 --> 00:52:46.960
We're all done with our
preliminary remarks,
00:52:46.960 --> 00:52:51.530
and it's now time to do
the teleportation stuff.
00:53:00.020 --> 00:53:04.600
Quantum teleportation.
00:53:12.200 --> 00:53:17.040
So we all know
this teleportation
00:53:17.040 --> 00:53:20.890
is the stuff of science
fiction and movies
00:53:20.890 --> 00:53:28.180
and kind of stuff like that, and
it's pretty much something that
00:53:28.180 --> 00:53:32.180
was, classically,
essentially impossible.
00:53:32.180 --> 00:53:36.410
You have an object, you sort of
dematerialize it and create it
00:53:36.410 --> 00:53:37.820
somewhere else.
00:53:37.820 --> 00:53:39.740
No basis for doing that.
00:53:39.740 --> 00:53:43.150
The interesting thing is
that quantum mechanically,
00:53:43.150 --> 00:53:46.680
you seem to be able
to do much better,
00:53:46.680 --> 00:53:49.720
and that's the idea that
we want to explain now.
00:53:49.720 --> 00:53:52.400
So this is also
not something that
00:53:52.400 --> 00:53:56.240
has been known for a long time.
00:53:56.240 --> 00:54:00.960
The big discovery that this
could be done is from 1993.
00:54:00.960 --> 00:54:05.870
So it's just 20 years ago
people realized finally
00:54:05.870 --> 00:54:08.536
that you could do
something like that.
00:54:08.536 --> 00:54:12.330
In that way, quantum
mechanics is, in a sense,
00:54:12.330 --> 00:54:14.960
having a renaissance
because there's
00:54:14.960 --> 00:54:19.310
all kinds of marvelous
experiments-- teleportation,
00:54:19.310 --> 00:54:23.060
entanglement, ideas that you
could build one day a quantum
00:54:23.060 --> 00:54:23.750
computer.
00:54:23.750 --> 00:54:26.330
It's all stimulating
thinking better
00:54:26.330 --> 00:54:31.170
about quantum mechanics
more precisely,
00:54:31.170 --> 00:54:33.255
and the experiments
are just amazing.
00:54:36.920 --> 00:54:39.540
This thing was done by
the following people.
00:54:39.540 --> 00:54:41.330
We should mention them.
00:54:41.330 --> 00:54:57.780
Bennett at IBM,
Brassard, Crepeau--
00:54:57.780 --> 00:55:05.650
can't pronounce that-- Jozsa,
all these people in Montreal.
00:55:12.700 --> 00:55:25.840
Peres, at Technion, and
Wootters at Williams College.
00:55:31.600 --> 00:55:35.800
1993.
00:55:35.800 --> 00:55:41.580
So big collaboration
all over the world.
00:55:41.580 --> 00:55:45.790
So what is the question
that we want to discuss?
00:55:48.480 --> 00:55:53.900
In this game, always
there's two people involved,
00:55:53.900 --> 00:55:57.680
and the canonical names
are Alice and Bob.
00:55:57.680 --> 00:56:00.370
Everybody calls Alice and Bob.
00:56:00.370 --> 00:56:02.330
It's been lots of
years that people
00:56:02.330 --> 00:56:04.670
talk about Alice and Bob.
00:56:04.670 --> 00:56:08.390
They use it also for
black hole experiments.
00:56:08.390 --> 00:56:11.750
Depending on your
taste, Alice stays out
00:56:11.750 --> 00:56:16.230
and Bob is sucked into the
black hole, or Bob stays out,
00:56:16.230 --> 00:56:18.270
Alice goes down.
00:56:18.270 --> 00:56:21.010
But it's Alice and
Bob all the time.
00:56:21.010 --> 00:56:23.700
So this time, the way
we're going to do it,
00:56:23.700 --> 00:56:26.145
Alice has a quantum state.
00:56:36.650 --> 00:56:40.510
It has been handed
to her, and it's
00:56:40.510 --> 00:56:44.670
a state of a spin 1/2 particle.
00:56:44.670 --> 00:56:48.050
Spin 1/2 is nice because
you have discrete labels.
00:56:48.050 --> 00:56:50.460
So she has this state.
00:56:50.460 --> 00:56:55.010
It's alpha plus beta minus.
00:56:58.200 --> 00:57:02.340
And she has it carefully
there in a box, just
00:57:02.340 --> 00:57:05.570
hoping that the state doesn't
get entangled with anything
00:57:05.570 --> 00:57:08.940
and disappear, or
doesn't get measured.
00:57:08.940 --> 00:57:15.870
And her goal is to send this
state to Bob, who's far away.
00:57:18.860 --> 00:57:32.650
So Alice is sitting
here and has this state,
00:57:32.650 --> 00:57:38.240
and Bob is sitting somewhere
here and has no state.
00:57:38.240 --> 00:57:43.010
And she wants to
send this state.
00:57:43.010 --> 00:57:46.270
This is the state
to be teleported.
00:57:46.270 --> 00:57:49.540
Now, there's a
couple of things you
00:57:49.540 --> 00:57:53.040
could try to do before even
trying to teleport this.
00:57:53.040 --> 00:57:55.490
Why teleport it?
00:57:55.490 --> 00:58:01.600
Why don't you create a
copy of this state and just
00:58:01.600 --> 00:58:07.950
put it in FedEx and send
it to Bob, and he gets it?
00:58:07.950 --> 00:58:12.300
The problem is that there's
something in quantum mechanics,
00:58:12.300 --> 00:58:14.550
something called no
cloning, that you
00:58:14.550 --> 00:58:17.030
can't create a copy
of a state, actually,
00:58:17.030 --> 00:58:19.720
with a quantum
mechanical process.
00:58:19.720 --> 00:58:21.430
It's really a funny thing.
00:58:21.430 --> 00:58:27.560
You've got a qubit-- this is
called a qubit-- a quantum bit.
00:58:27.560 --> 00:58:30.730
Bit is something
that can be 0 or 1.
00:58:30.730 --> 00:58:32.630
Quantum, it can be two things.
00:58:32.630 --> 00:58:35.380
So instead of calling
it a spin state,
00:58:35.380 --> 00:58:38.170
sometimes people
call it a qubit.
00:58:38.170 --> 00:58:40.390
For us, it's a spin state.
00:58:40.390 --> 00:58:43.220
It has two numbers.
00:58:43.220 --> 00:58:46.040
And there's no cloning.
00:58:46.040 --> 00:58:47.700
We will not discuss it here.
00:58:47.700 --> 00:58:49.640
It's a nice topic
for a recitation.
00:58:49.640 --> 00:58:51.130
It's a simple matter.
00:58:51.130 --> 00:58:52.540
You can't make a copy.
00:58:52.540 --> 00:58:57.410
So given that you can't make
a copy, let's avoid that idea,
00:58:57.410 --> 00:59:05.340
save ourselves $15 of FedEx and
just try to do something else.
00:59:05.340 --> 00:59:08.170
So the one thing
Alice could do is
00:59:08.170 --> 00:59:10.590
that she could say, all right.
00:59:10.590 --> 00:59:12.810
Well here is alpha and beta.
00:59:12.810 --> 00:59:15.290
Let me measure the state.
00:59:15.290 --> 00:59:17.960
Find alpha and beta.
00:59:17.960 --> 00:59:22.460
And then I'll of send
that information to Bob.
00:59:22.460 --> 00:59:23.680
OK.
00:59:23.680 --> 00:59:26.180
But she has one
copy of the state.
00:59:26.180 --> 00:59:28.580
How is she going to
measure alpha and beta
00:59:28.580 --> 00:59:31.130
with one copy of the state.
00:59:31.130 --> 00:59:34.130
She puts it through a
Stern-Gerlach experiment,
00:59:34.130 --> 00:59:37.280
and the particle comes
out the plus side.
00:59:37.280 --> 00:59:40.340
Now what?
00:59:40.340 --> 00:59:42.900
The probability that
it went to the plus.
00:59:42.900 --> 00:59:45.370
You've got some information
about the alpha squared.
00:59:45.370 --> 00:59:48.160
Not even because you just
did the experiment once
00:59:48.160 --> 00:59:49.465
and your cubit is gone.
00:59:52.010 --> 01:00:00.760
So Alice actually can't
figure out alpha and beta.
01:00:00.760 --> 01:00:06.950
So if she's handed the qubit,
she better not measure it.
01:00:06.950 --> 01:00:10.360
Because if she measures
it, she destroys the state,
01:00:10.360 --> 01:00:15.130
goes into a plus or a
minus, and it's all over.
01:00:15.130 --> 01:00:19.450
The state is gone before
she could do anything.
01:00:19.450 --> 01:00:23.090
So that doesn't work either.
01:00:23.090 --> 01:00:25.440
Now there's the third option.
01:00:25.440 --> 01:00:31.460
Maybe Alice cannot talk to Bob,
and Alice created that state
01:00:31.460 --> 01:00:32.520
with some Hamiltonian.
01:00:32.520 --> 01:00:34.990
And she knows because
she created it
01:00:34.990 --> 01:00:38.650
what alpha and beta is.
01:00:38.650 --> 01:00:43.750
So she could in
principle tell Bob, OK.
01:00:43.750 --> 01:00:45.290
Here is alpha and here is beta.
01:00:45.290 --> 01:00:47.950
Create it again.
01:00:47.950 --> 01:00:51.680
That would be a fine
strategy, but actually there's
01:00:51.680 --> 01:00:55.970
even plausibly a
problem with that.
01:00:55.970 --> 01:01:02.100
Because maybe she knows this
state, but alpha is a number.
01:01:02.100 --> 01:01:10.490
It is 0.53782106, never ends.
01:01:10.490 --> 01:01:11.670
Doesn't repeat.
01:01:11.670 --> 01:01:16.350
And she has to send that
infinite string of information
01:01:16.350 --> 01:01:20.120
to Bob, which is not
a good idea either.
01:01:20.120 --> 01:01:23.950
She's not going to manage
to send the right state.
01:01:23.950 --> 01:01:27.750
So these are the things
we speculate about
01:01:27.750 --> 01:01:29.940
because it's a natural
thing to one wonder.
01:01:29.940 --> 01:01:33.900
So what we're going to
try to do is somehow
01:01:33.900 --> 01:01:38.810
produce an experiment in
which she'll take this state,
01:01:38.810 --> 01:01:42.700
get it in, and
somehow Bob is going
01:01:42.700 --> 01:01:46.680
to create that state
on his other side.
01:01:46.680 --> 01:01:51.340
That's the teleportation
thing that we'll try to do.
01:01:51.340 --> 01:01:58.210
So let's do a little diagram
of how we're going to do this.
01:01:58.210 --> 01:02:01.560
So here is going to be the state
that is going to be teleported.
01:02:01.560 --> 01:02:07.240
We'll call it the
state C. So I'll
01:02:07.240 --> 01:02:19.770
write it as psi alpha plus in
the state space C sub particle
01:02:19.770 --> 01:02:24.700
plus beta minus in this
state space C. And C is
01:02:24.700 --> 01:02:27.825
the state she is going
to try to teleport.
01:02:31.720 --> 01:02:36.580
But now they're not
going to be able to do it
01:02:36.580 --> 01:02:43.030
unless they use
something different.
01:02:43.030 --> 01:02:44.810
They try something different.
01:02:44.810 --> 01:02:51.110
And the whole idea is going to
be to use an entangled state.
01:02:51.110 --> 01:02:55.780
So basically what
we're going to do
01:02:55.780 --> 01:03:08.615
is we're going to put the source
here, entangled state source.
01:03:11.430 --> 01:03:13.940
And we're going to
produce and an entangled
01:03:13.940 --> 01:03:18.180
state of two particles.
01:03:18.180 --> 01:03:25.110
And one particle is going
to be given to A, to Alice.
01:03:25.110 --> 01:03:29.050
And one particle is
going to be given to Bob.
01:03:29.050 --> 01:03:35.320
So particle B for Bob is
going to be given to Bob.
01:03:35.320 --> 01:03:42.070
And particle A is going
to be given to Alice.
01:03:45.760 --> 01:03:47.855
And this is an entangled pair.
01:03:54.050 --> 01:03:57.950
So there it is.
01:03:57.950 --> 01:03:59.750
Now what's going to happen?
01:03:59.750 --> 01:04:02.690
What are we going to do?
01:04:02.690 --> 01:04:06.140
Entanglement really
correlates what
01:04:06.140 --> 01:04:09.420
goes here with
what goes in there.
01:04:09.420 --> 01:04:14.220
Now entanglement
happens instantaneously,
01:04:14.220 --> 01:04:16.820
and we can discuss this.
01:04:16.820 --> 01:04:19.450
You have no way of
sending information
01:04:19.450 --> 01:04:22.570
through entanglement in general.
01:04:22.570 --> 01:04:26.950
There's no such thing as
learning something about A when
01:04:26.950 --> 01:04:32.260
B doesn't measure, learning
anything nontrivial about A.
01:04:32.260 --> 01:04:36.150
So the entangled state
is there, and that's
01:04:36.150 --> 01:04:41.160
what we're going to try to use
in order to do the teleporting.
01:04:41.160 --> 01:04:47.070
Now morally speaking, suppose
I wanted to teleport myself
01:04:47.070 --> 01:04:49.860
from one place in
this room to another.
01:04:49.860 --> 01:04:55.820
What I would have to do is
create an enormous reservoir
01:04:55.820 --> 01:04:57.290
of entangled states.
01:04:57.290 --> 01:05:00.710
So here's my
generator, and I create
01:05:00.710 --> 01:05:03.680
billions of entangled pairs.
01:05:03.680 --> 01:05:08.470
And I put them all
here, all the ones here
01:05:08.470 --> 01:05:12.040
and all the corresponding
pairs over there.
01:05:12.040 --> 01:05:17.650
And then I sort
of-- somebody takes
01:05:17.650 --> 01:05:24.030
me and these billions of
entangled pairs, one side
01:05:24.030 --> 01:05:26.480
of the pair, and
does a measurement
01:05:26.480 --> 01:05:30.910
in which every atom or every
quantum state in my body
01:05:30.910 --> 01:05:33.840
is measured with
some entangled state.
01:05:33.840 --> 01:05:36.255
They've done the
measurement, and boom.
01:05:36.255 --> 01:05:38.140
I reappear on the other side.
01:05:38.140 --> 01:05:39.840
That's what's going to happen.
01:05:39.840 --> 01:05:42.580
So we're going to do this.
01:05:42.580 --> 01:05:44.570
We're going to have this
state, and now we're
01:05:44.570 --> 01:05:47.980
going to a measurement between
this state and this state.
01:05:47.980 --> 01:05:50.030
Alice is going to
do a measurement.
01:05:50.030 --> 01:05:53.370
That's going to
force this particle
01:05:53.370 --> 01:05:56.480
to actually pretty much
become the state you
01:05:56.480 --> 01:05:59.320
wanted to teleport.
01:05:59.320 --> 01:06:02.240
So that's the goal.
01:06:02.240 --> 01:06:07.150
So let me say a
couple more things.
01:06:07.150 --> 01:06:10.260
Alice will have to send
some information actually.
01:06:10.260 --> 01:06:13.660
Because she is going to have
to do a measurement, and she
01:06:13.660 --> 01:06:22.140
has a console with four lights,
zero, one, two, and three.
01:06:22.140 --> 01:06:24.960
Four lights.
01:06:24.960 --> 01:06:30.830
And when she will do her
measurement, one of the lights
01:06:30.830 --> 01:06:33.790
will blink.
01:06:33.790 --> 01:06:38.280
And she will have to tell
Bob which one blinked.
01:06:38.280 --> 01:06:42.680
So she will have to send the
number and information of two
01:06:42.680 --> 01:06:43.370
bits.
01:06:43.370 --> 01:06:46.500
Because with two bits,
you can represent
01:06:46.500 --> 01:06:50.390
any of four numbers,
binary code.
01:06:50.390 --> 01:06:53.735
So she will send information
of which clicked.
01:07:00.890 --> 01:07:08.100
And then Bob will have a
machine with four entries here.
01:07:16.010 --> 01:07:20.240
And according to the
information that he gets,
01:07:20.240 --> 01:07:27.000
he will make the state
to go through one
01:07:27.000 --> 01:07:32.590
of those machines, the zero,
the one, the two, or the three.
01:07:32.590 --> 01:07:37.600
So he will push B
into one of them out,
01:07:37.600 --> 01:07:42.210
we claim, will come
this teleported state.
01:07:52.260 --> 01:07:55.510
So that's the set up.
01:07:55.510 --> 01:07:58.480
You have to get a
feel for the set up.
01:07:58.480 --> 01:08:02.170
So are there questions
on what we're doing?
01:08:02.170 --> 01:08:03.998
AUDIENCE: So after
teleportation would
01:08:03.998 --> 01:08:07.147
have some kind of
copy [INAUDIBLE]?
01:08:07.147 --> 01:08:07.730
PROFESSOR: No.
01:08:07.730 --> 01:08:11.050
After the replication,
this state
01:08:11.050 --> 01:08:15.330
will be destroyed beyond
repair as you will see.
01:08:15.330 --> 01:08:19.520
So there will not be a copy
created by this procedure.
01:08:19.520 --> 01:08:20.330
You destroy.
01:08:20.330 --> 01:08:23.319
It's really what teleportation
was supposed to be.
01:08:23.319 --> 01:08:25.740
Not to create another
copy of you there,
01:08:25.740 --> 01:08:28.075
but to take you there.
01:08:28.075 --> 01:08:31.130
Destroy you here and
recreate you there.
01:08:31.130 --> 01:08:34.270
So no other copy.
01:08:34.270 --> 01:08:35.380
Other questions?
01:08:35.380 --> 01:08:36.270
Yes.
01:08:36.270 --> 01:08:39.488
AUDIENCE: Does this also work
if C is an entangled state?
01:08:39.488 --> 01:08:40.279
PROFESSOR: If what?
01:08:40.279 --> 01:08:43.570
AUDIENCE: If C say itself
contains different parts which
01:08:43.570 --> 01:08:45.460
are entangled with each other?
01:08:45.460 --> 01:08:47.500
PROFESSOR: Well, it's a
more complicated thing.
01:08:47.500 --> 01:08:50.399
I'm pretty sure it would work.
01:08:50.399 --> 01:08:54.560
Maybe you would need more
than one entangled pair here.
01:08:54.560 --> 01:08:59.511
You would need a source
that is more complicated.
01:08:59.511 --> 01:09:00.135
More questions.
01:09:00.135 --> 01:09:03.458
AUDIENCE: What do you mean
about pushes the state
01:09:03.458 --> 01:09:05.930
into one of the [INAUDIBLE]?
01:09:05.930 --> 01:09:08.470
PROFESSOR: What do I mean by
pushes it through one of them?
01:09:08.470 --> 01:09:10.380
Well you know, Hamiltonians.
01:09:10.380 --> 01:09:11.600
You get your state.
01:09:11.600 --> 01:09:14.350
You can put them in
a magnetic field.
01:09:14.350 --> 01:09:16.080
Let them evolve a little bit.
01:09:16.080 --> 01:09:17.640
Those are machines.
01:09:17.640 --> 01:09:22.880
So any of these machines are
some unitary time evolution.
01:09:22.880 --> 01:09:25.141
It does something to the state.
01:09:25.141 --> 01:09:28.417
AUDIENCE: But one [INAUDIBLE]
01:09:28.417 --> 01:09:29.125
PROFESSOR: Sorry.
01:09:29.125 --> 01:09:30.896
AUDIENCE: Are there
Hamiltonians that
01:09:30.896 --> 01:09:33.670
are based off of
what Alice measures?
01:09:33.670 --> 01:09:34.800
PROFESSOR: Yes.
01:09:34.800 --> 01:09:37.420
So they will be correlated
as you will see.
01:09:37.420 --> 01:09:41.700
So if Alice measures that
the light zero beeps,
01:09:41.700 --> 01:09:45.279
the instruction for Bob is to
send the state through the zero
01:09:45.279 --> 01:09:49.979
Hamiltonian, and one, two,
and three Hamiltonian.
01:09:49.979 --> 01:09:52.420
More questions?
01:09:52.420 --> 01:09:57.070
It's good to really have
a good feeling of this
01:09:57.070 --> 01:09:59.850
or what we're trying to do
and why it's nontrivial.
01:09:59.850 --> 01:10:02.270
Yes.
01:10:02.270 --> 01:10:04.240
AUDIENCE: This might be
a little too intuitive,
01:10:04.240 --> 01:10:08.190
but in a state which-- Can a
Hamiltonian which Bob needs
01:10:08.190 --> 01:10:14.050
to send B through in order to
yield the same state that Alice
01:10:14.050 --> 01:10:15.920
had, can that also be
transmitted quantumly
01:10:15.920 --> 01:10:16.460
through qubits?
01:10:16.460 --> 01:10:18.672
Or would you just get like an
infinite line of qubits needing
01:10:18.672 --> 01:10:19.171
to--
01:10:19.171 --> 01:10:20.300
PROFESSOR: No no.
01:10:20.300 --> 01:10:24.210
You know, this is a devise that
they can build by themselves.
01:10:24.210 --> 01:10:27.340
As you will see once
we do the calculation,
01:10:27.340 --> 01:10:31.500
Alice will construct a device
that has these four lights
01:10:31.500 --> 01:10:33.360
and she knows what they mean.
01:10:33.360 --> 01:10:36.940
And Bob will construct a
device that has these things,
01:10:36.940 --> 01:10:39.950
and they can use it to
transport any state.
01:10:39.950 --> 01:10:43.710
So these machines are
independent of the state
01:10:43.710 --> 01:10:44.990
you want to teleport.
01:10:44.990 --> 01:10:48.880
You teleported this, you want
to teleport another state
01:10:48.880 --> 01:10:51.570
with alpha prime and beta prime?
01:10:51.570 --> 01:10:52.170
Sure.
01:10:52.170 --> 01:10:57.560
Use exactly the same machines,
give me another entangled pair,
01:10:57.560 --> 01:10:58.470
and do it.
01:10:58.470 --> 01:11:00.355
AUDIENCE: Well, I
think what I meant
01:11:00.355 --> 01:11:02.860
is that the information
between the two machines,
01:11:02.860 --> 01:11:04.840
does that have to be
transmitted classically,
01:11:04.840 --> 01:11:06.490
or is there some
way to transmit--
01:11:06.490 --> 01:11:08.490
PROFESSOR: There's
no real information.
01:11:08.490 --> 01:11:13.810
The machines were built, say,
in the same laboratory of IBM.
01:11:13.810 --> 01:11:18.180
And then they're built,
and we will tell you
01:11:18.180 --> 01:11:20.540
how to build each
of these machines.
01:11:20.540 --> 01:11:24.310
And then just put aside, taken
away by these two people,
01:11:24.310 --> 01:11:28.320
and then we'll do it.
01:11:28.320 --> 01:11:33.690
There's no mystery of
sending information about it.
01:11:33.690 --> 01:11:36.670
That probably will become
clear with the computation,
01:11:36.670 --> 01:11:39.400
which I better start doing soon.
01:11:39.400 --> 01:11:41.016
Yes.
01:11:41.016 --> 01:11:42.450
AUDIENCE: The difference--
01:11:42.450 --> 01:11:43.340
PROFESSOR: Louder.
01:11:43.340 --> 01:11:45.590
AUDIENCE: Just a question
about the first part
01:11:45.590 --> 01:11:47.340
on the left side of the board.
01:11:47.340 --> 01:11:49.230
So, when we first
do a measurement,
01:11:49.230 --> 01:11:50.730
does that mean it's
something that's
01:11:50.730 --> 01:11:55.077
like a microscopic quantity,
like an energy or something?
01:11:55.077 --> 01:11:57.285
Or does it just refer to any?
01:11:57.285 --> 01:12:00.070
PROFESSOR: When we refer
to measurements and quantum
01:12:00.070 --> 01:12:02.690
mechanics, we talk--
Let me give you
01:12:02.690 --> 01:12:04.660
just a little bit
of intuition here.
01:12:04.660 --> 01:12:08.389
We typically talk about
measuring permission operators,
01:12:08.389 --> 01:12:09.930
because they have
[INAUDIBLE] values.
01:12:09.930 --> 01:12:14.490
So we don't have to say what
they are-- energy, momentum.
01:12:14.490 --> 01:12:17.080
It's a permission
operator you measure.
01:12:17.080 --> 01:12:20.570
And projector operators
into basic states
01:12:20.570 --> 01:12:22.160
of permission operators.
01:12:22.160 --> 01:12:25.560
So you could imagine
that's one way
01:12:25.560 --> 01:12:28.790
of thinking about
these measurements.
01:12:28.790 --> 01:12:29.910
OK.
01:12:29.910 --> 01:12:32.430
So let's do this.
01:12:32.430 --> 01:12:33.110
All right.
01:12:33.110 --> 01:12:37.860
The state to be
teleported is this one,
01:12:37.860 --> 01:12:59.190
and the A B pair is
an entangled state.
01:12:59.190 --> 01:13:03.115
So it will be one
of the bell states,
01:13:03.115 --> 01:13:12.260
psi zero AB 1 over square
root of 2 plus A plus
01:13:12.260 --> 01:13:23.750
b plus minus A plus minus B. So
this is the state they share.
01:13:23.750 --> 01:13:29.610
Of course, Alic only has
a handle on particle A,
01:13:29.610 --> 01:13:35.050
and Bob only has a handle
on particle B. Nevertheless
01:13:35.050 --> 01:13:37.420
the state is entangled
even though this
01:13:37.420 --> 01:13:42.010
could be 200 kilometers apart.
01:13:42.010 --> 01:13:51.880
So the total state-- well,
we've been tensoring two things.
01:13:51.880 --> 01:13:54.990
Well, tensoring three
is three particles.
01:13:54.990 --> 01:14:00.170
So I don't think you will be
too unhappy to just tensor
01:14:00.170 --> 01:14:01.050
the whole thing.
01:14:01.050 --> 01:14:20.710
So psi zero AB tensor alpha
plus C plus beta minus C.
01:14:20.710 --> 01:14:25.100
So here comes the
interesting point.
01:14:25.100 --> 01:14:37.110
Alice has available the state A.
The particle A is not the state
01:14:37.110 --> 01:14:40.115
A because A is in a funny thing.
01:14:40.115 --> 01:14:40.740
It's entangled.
01:14:40.740 --> 01:14:43.925
But it has a
particle A available,
01:14:43.925 --> 01:14:47.210
and it has a
particle C available.
01:14:47.210 --> 01:14:51.510
So Alice is going
to do a measurement,
01:14:51.510 --> 01:14:53.310
and it's going to be
a sneaky measurement.
01:14:53.310 --> 01:14:55.270
It's going to use a bases.
01:14:55.270 --> 01:14:57.550
Since she has two
particles, she can
01:14:57.550 --> 01:15:01.210
choose a basis of
two particle states.
01:15:01.210 --> 01:15:04.502
Any orthonormal basis
will do well by the idea
01:15:04.502 --> 01:15:08.000
that we can measure with
any orthonormal basis.
01:15:08.000 --> 01:15:10.640
So what she's going
to try to do is use
01:15:10.640 --> 01:15:17.510
the bell basis for A and C.
01:15:17.510 --> 01:15:21.080
So let's try to think
of what that means.
01:15:21.080 --> 01:15:24.550
That requires a small
calculation here.
01:15:24.550 --> 01:15:31.530
So this is equal to 1
over square root of 2
01:15:31.530 --> 01:15:38.040
plus-- so I anticipate
that this will become clear
01:15:38.040 --> 01:15:42.860
in a second, what that
measurement means-- plus minus
01:15:42.860 --> 01:15:55.950
A minus b Tensor alpha plus plus
beta minus C. So I just wrote
01:15:55.950 --> 01:15:56.740
what this is.
01:16:00.330 --> 01:16:00.830
OK.
01:16:04.040 --> 01:16:05.350
Some algebra.
01:16:05.350 --> 01:16:08.040
This is the total
state, psi total.
01:16:12.460 --> 01:16:18.550
Let's multiply these things
out, and I will keep the labels
01:16:18.550 --> 01:16:20.390
all the time
because I don't want
01:16:20.390 --> 01:16:23.160
there to be any confusion
about what's happening.
01:16:23.160 --> 01:16:25.420
So what do we get first?
01:16:25.420 --> 01:16:32.310
Alpha multiplying plus of A.
I should write in plus of B,
01:16:32.310 --> 01:16:35.960
but the order doesn't really
matter if I keep the labels.
01:16:35.960 --> 01:16:46.450
So I'll put plus of
C times plus of B.
01:16:46.450 --> 01:16:48.390
Then keep multiplying.
01:16:48.390 --> 01:16:53.050
So we have plus beta,
from this with that.
01:16:53.050 --> 01:17:05.570
So I'll have plus of A minus
of C and plus of B. Maybe
01:17:05.570 --> 01:17:08.300
it's easier to read
if I use another line.
01:17:08.300 --> 01:17:13.010
So I now must multiply the
second state times this.
01:17:13.010 --> 01:17:30.730
So I get plus alpha minus of A
with plus of C and minus of B.
01:17:30.730 --> 01:17:40.120
So this is this times that,
minus of A plus of C minus of B
01:17:40.120 --> 01:17:55.850
plus beta minus of A
minus of C minus of B.
01:17:55.850 --> 01:17:57.520
OK.
01:17:57.520 --> 01:17:58.775
So there here my state.
01:18:01.860 --> 01:18:08.696
But now I have written it in a
way that I have here A and C A
01:18:08.696 --> 01:18:14.700
and C A and C and
A and C. So I could
01:18:14.700 --> 01:18:17.330
decide to measure in this basis.
01:18:17.330 --> 01:18:23.740
This is an orthonormal
basis for A and C.
01:18:23.740 --> 01:18:28.710
But it's not a very smart basis
because it's not entangled.
01:18:28.710 --> 01:18:32.030
So let's go to the
entangled base.
01:18:32.030 --> 01:18:35.780
So let's rewrite this
state, this total state.
01:18:35.780 --> 01:18:38.390
Nothing has been done
yet to the state.
01:18:38.390 --> 01:18:43.720
We're just mathematically
rewriting it, nothing else.
01:18:43.720 --> 01:18:46.420
We have this, this,
this, and that.
01:18:46.420 --> 01:18:53.380
And I want you now to use
these formulas to do this.
01:18:53.380 --> 01:18:56.106
So I'll do this on
this blackboard.
01:18:58.790 --> 01:19:02.185
We'll have to erase
those important names.
01:19:11.110 --> 01:19:14.060
So what do we get?
01:19:14.060 --> 01:19:16.200
Well a little of algebra.
01:19:16.200 --> 01:19:19.910
Let's do it.
01:19:19.910 --> 01:19:24.470
A with C plus plus
would be that.
01:19:24.470 --> 01:19:27.830
So I'll write it with
one over square root of 2
01:19:27.830 --> 01:19:30.530
becomes one half.
01:19:30.530 --> 01:19:44.450
A with C would be psi zero AC
plus psi three AC multiplying
01:19:44.450 --> 01:19:50.060
alpha plus on B. So I took
care of the first term.
01:19:50.060 --> 01:19:51.800
The alpha is there.
01:19:51.800 --> 01:19:53.020
The B is there.
01:19:53.020 --> 01:19:58.620
And AC is there,
in which, you know,
01:19:58.620 --> 01:20:00.930
you can put any labels
you want to here.
01:20:00.930 --> 01:20:04.950
AB, this is the AB state.
01:20:04.950 --> 01:20:07.900
The entangled AB state.
01:20:07.900 --> 01:20:08.660
We used AC.
01:20:11.890 --> 01:20:17.270
Second term plus one half.
01:20:17.270 --> 01:20:22.690
Now we have plus A minus C. So
it's the second line in there.
01:20:22.690 --> 01:20:39.490
So it would be psi one AC minus
I psi 2 AC beta plus B. Next
01:20:39.490 --> 01:20:43.100
line, I'll just
copy it, one half.
01:20:43.100 --> 01:20:44.160
Well not.
01:20:44.160 --> 01:20:52.110
Alpha minus B and here
you'll have the minus plus
01:20:52.110 --> 01:21:02.280
which is the same thing,
psi 1 AC plus I psi 2 AC.
01:21:02.280 --> 01:21:16.154
And the last term is plus one
half psi 0 AC minus psi 3 AC.
01:21:16.154 --> 01:21:25.810
And we get beta minus B.
01:21:25.810 --> 01:21:29.120
OK, almost was there.
01:21:29.120 --> 01:21:36.790
Let's rewrite this as-- let's
collect the psi zeroes, psi 0
01:21:36.790 --> 01:21:37.830
and psi 0.
01:21:37.830 --> 01:21:41.380
You see we're do nothing yet.
01:21:41.380 --> 01:21:43.870
We're just
mathematically rewriting
01:21:43.870 --> 01:21:48.000
the states in a different
basis, the total states.
01:21:48.000 --> 01:21:56.950
So it is equal to
one half psi 0 AC.
01:21:56.950 --> 01:21:59.240
and look what you get
here, very curiously.
01:21:59.240 --> 01:22:11.410
You get alpha plus B plus beta
minus B. Very curious, that
01:22:11.410 --> 01:22:15.400
was precisely the state
we wanted to teleport.
01:22:15.400 --> 01:22:18.840
Alpha plus plus beta minus.
01:22:18.840 --> 01:22:19.630
All right.
01:22:19.630 --> 01:22:22.090
Let's see what else happens.
01:22:22.090 --> 01:22:26.680
Here we get plus
one half psi-- which
01:22:26.680 --> 01:22:28.040
other one do I want to copy?
01:22:28.040 --> 01:22:30.560
Psi 1 AC.
01:22:39.080 --> 01:22:42.640
You see this is the state
we wanted to teleport.
01:22:42.640 --> 01:22:43.310
It's here.
01:22:46.470 --> 01:22:50.840
And it sort of has
appeared in the B space.
01:22:50.840 --> 01:22:56.670
Psi 1 AC, well this time I
have this term and this term.
01:22:56.670 --> 01:22:58.960
So actually it seems
a little different.
01:22:58.960 --> 01:23:10.720
Now we get beta plus
B plus alpha minus B.
01:23:10.720 --> 01:23:12.520
Then we go to the next.
01:23:12.520 --> 01:23:18.005
One half of psi 2 AC.
01:23:21.880 --> 01:23:23.260
So psi 2 is here.
01:23:25.970 --> 01:23:46.980
So you get I alpha minus
B minus I beta plus B. OK.
01:23:46.980 --> 01:23:48.810
Finally linear combinations.
01:23:48.810 --> 01:23:50.740
And finally psi 3.
01:23:50.740 --> 01:23:52.370
What is psi 3?
01:23:56.430 --> 01:23:59.370
Well two terms also for psi 3.
01:23:59.370 --> 01:24:00.820
This one and this one.
01:24:00.820 --> 01:24:09.270
So you get alpha
plus B minus beta
01:24:09.270 --> 01:24:16.970
minus B. Kind of the
end of math by now.
01:24:16.970 --> 01:24:21.650
You've proven a funny identity
actually in doing this.
01:24:21.650 --> 01:24:25.880
And maybe this
blackboard should--
01:24:25.880 --> 01:24:27.520
to make sure you understand.
01:24:27.520 --> 01:24:30.923
This is the calculation
of total state.
01:24:36.520 --> 01:24:37.470
And here we go.
01:24:37.470 --> 01:24:40.110
So let me show you one thing.
01:24:40.110 --> 01:24:45.300
This is actually
the state we wanted.
01:24:45.300 --> 01:24:52.100
So this will be called psi in
the B basis, in the B space.
01:24:52.100 --> 01:24:54.730
The state that you
wanted to teleport
01:24:54.730 --> 01:25:00.920
that was psi in the C basis,
now it's psi in the B basis.
01:25:00.920 --> 01:25:06.250
Those ones look a little
funny, but this one actually
01:25:06.250 --> 01:25:15.700
looks like this thing, looks
like sigma 3 times psi.
01:25:15.700 --> 01:25:18.520
Because if you have
sigma 3 on this state,
01:25:18.520 --> 01:25:23.490
it gives you a plus 1 here
and a minus [INAUDIBLE] value.
01:25:23.490 --> 01:25:26.300
So that's sigma 3 psi.
01:25:26.300 --> 01:25:29.400
This actually has flipped
the plus and the minus.
01:25:29.400 --> 01:25:36.230
So that actually is sigma 1 psi.
01:25:36.230 --> 01:25:42.880
And this state is
actually sigma 2 psi.
01:25:42.880 --> 01:25:46.830
OK everything is in place now.
01:25:46.830 --> 01:25:51.580
We've just done math, but
now comes the physics.
01:25:51.580 --> 01:25:59.580
Alice is going to measure in
the bell space of A and C.
01:25:59.580 --> 01:26:05.500
So these are the
four bases states.
01:26:05.500 --> 01:26:09.690
So she's going to measure in
one of these bases states.
01:26:09.690 --> 01:26:14.690
And as see measures, she
falls and the wave function
01:26:14.690 --> 01:26:17.690
of her collapses
into one of them.
01:26:17.690 --> 01:26:24.890
So when she gets the zero
basis state, this light blanks.
01:26:24.890 --> 01:26:28.780
If doing the measurement
on AC, because she
01:26:28.780 --> 01:26:31.880
has both particles
A and C, she gets
01:26:31.880 --> 01:26:36.850
this basis state-- recall the
postulate of measurement--
01:26:36.850 --> 01:26:39.210
light one blinks.
01:26:39.210 --> 01:26:46.460
If she gets the third like
2 and the fourth here.
01:26:46.460 --> 01:26:52.260
Suppose the state
light zero shines.
01:26:52.260 --> 01:26:55.260
Well the state
collapsed into this.
01:26:55.260 --> 01:26:58.980
She is now sitting
with psi 0 AC that
01:26:58.980 --> 01:27:05.380
has no memory whatsoever
of the original state C,
01:27:05.380 --> 01:27:08.120
but B is sitting
with this state,
01:27:08.120 --> 01:27:10.770
the state we wanted to teleport.
01:27:10.770 --> 01:27:15.070
So if light zero
shines, she tells
01:27:15.070 --> 01:27:18.220
Bob, let it go to
machine zero where
01:27:18.220 --> 01:27:20.910
there's no magnetic
field, nothing.
01:27:20.910 --> 01:27:25.500
So actually the
same state goes out.
01:27:25.500 --> 01:27:32.510
If she gets psi 1 as
the measured state,
01:27:32.510 --> 01:27:37.670
again no memory in this
state about alpha and beta.
01:27:37.670 --> 01:27:41.886
But Bob gets sigma 1 psi 1.
01:27:41.886 --> 01:27:46.770
So he puts it into the first
Hamiltonian for a picosecond,
01:27:46.770 --> 01:27:48.430
produces a sigma 1.
01:27:48.430 --> 01:27:53.060
This Hamiltonian, this
box I takes a state
01:27:53.060 --> 01:27:54.880
into sigma I state.
01:27:54.880 --> 01:27:56.440
It's a unitary operation.
01:27:56.440 --> 01:28:00.320
So puts a sigma 1 and gets psi.
01:28:00.320 --> 01:28:05.910
If light two shines,
goes to the machine two,
01:28:05.910 --> 01:28:09.570
which produces a sigma 2,
and so he gets the state.
01:28:09.570 --> 01:28:13.530
Light four shines, the third
Hamiltonian, he gets the state.
01:28:13.530 --> 01:28:16.925
Any of the four options,
he gets the precise state.
01:28:16.925 --> 01:28:19.350
The state has been teleported.
01:28:19.350 --> 01:28:23.060
You needed to send only the
information of which light
01:28:23.060 --> 01:28:26.770
shone, and the state is on
the other side of the ocean.
01:28:26.770 --> 01:28:27.420
All right.
01:28:27.420 --> 01:28:29.610
That's it for today.