M1L6d.txt

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Before I introduce several measures for entanglement,
let me talk about entanglement purification.
It's a very nice subject which tells you
that if you have weakly entangled state
you can make them more entangled.
And actually, the effort you have
to spend to make a weakly entangled state more entangled,
can actually act as a measurement how entangled
was your state in first place.
So the purification is introducing us
with one measurement for entanglement,
as I just said, but it also gives you an idea how quantum
state can be manipulated.
The purification is also the first example
we encounter in this course for new insight
into quantum physics.
A lot of people thought quantum physics, at least
nonrelativistic quantum physics, that
was invented in the '20s in the last century
and by now we've understood it all.
But there is-- there was an aspect of quantum physics
which I think nobody understood.
And this is, when you have a quantum state,
which may decohere a quantum state which may no longer be
pure, that you have ways to-- to [INAUDIBLE]--
you have ways to get the pure quantum state back.
And, ultimately, in the early days
when I learned quantum physics, I
thought if you have a quantum state that's nice
but if the quantumness has decayed away
you can't get it back.
But this is something we learn now from the new methods
and new approach from quantum information science
that you can do quantum [? overcorrection, ?] you
can have a state which is decohered
and you can get back to it.
And there is something which-- it's not yet
that what we are discussing today,
but in purification you have states which
have inferior entanglement but you are not--
you don't need to get stuck with it.
You can take several of the purely entangled state
and create the maximally entangled states out of it.
So that's what I want to show you.
So you should look at it as one example
that, wow, it's really cool how we can have states-- quantum
states-- with inferior quality, and by doing quantum operation
on those states we get something which is more entangled
and therefore, if that's our purpose, has a higher quality.
So to introduce purification, I'm
simply mentioning what are, for two qubits,
the standard maximally entangled states.
In 10 minutes or so, we talk about how do we
measure entanglement and indeed those states
will come out as being maximally entangled.
But you already get the idea.
Maximally entangled means they are not factorizable and not
just by a small margin.
There may be two equal parts of the-- you always need
two equal parts of 0 1 and 1 0, and this state
has maximally entangled.
It's maximally non-factorizable.
So the four states, which are actually the Bell
states-- the famous Bell bases-- are 0 1 plus minus 1 0
and 0 0 plus minus 1 1.
Just to remind you, if you have two-- if you have two systems,
each of them has two states.
You can think of spin up, down for particle one,
spin up, down for particles two.
That Hilbert space is four dimensional.
So you need a four dimensional basis.
And the trivial basis is up, up, down, down, up, down, down, up.
But what is relevant for entangled states,
we often use the basis of Bell states.
And this is a new basis which spans the four dimensional
Hilbert space, but each of those spaces function
is maximally entangled.
We should correctly normalize them,
and that state is often called psi plus minus
and this here is phi plus minus.
So now let's get the purely entangled state and let's state
0 0 plus 1 1.
But we will have an example where
B may be very, very small.
So those states-- these states is
entangled for all choices of A and B. So the question is now,
how can we take such an arbitrary state
where A and B-- one of them-- may be small and create
a standard Bell state?

So what we have to assume for that
is that we have a large supply of such states.
So let's assume we have large supply, identical copies.
And now we want to take two such copies.

And what I want to outline you is the following.
You take two of your copies and you do a measurement.

I will tell you what kind of qubit operation we need,
what kind of measurement we perform,
and then when the outcome of the measurement is such and such,
you'll say-- let me be specific.
So if you take two copies, we have
a total-- we have two states with two photons each.
And now we perform a quantum operation on two of the photons
and the other two photons we leave untouched.
So now it depends when the measurement of the two photons
has a good outcome we know those other two photons are maximally
entangled in a Bell state.
If the outcome of the measurement is bad,
it tells us the two photons are not entangled
and we throw them away.
So therefore, we have a finite probability
by performing measurements that a pair of our sample
state with the coefficient A and B
will result into a maximally entangled state in a Bell
state.
And I want to describe you now what
is the protocol, what is the procedure to implement that.
And what we will find-- and this is
what I think you should expect here-- if the initial state has
very bad entanglement in the sense that B is very small,
we will need many, many attempts and many measurements
on our pair of states before we produce a Bell state.
It's probabilistic, but the proper probability
to succeed in preparing a Bell state will depend on-- we
will see-- the product of A and B. And question?
Yes?
Does the assumption of [? large supply ?] the state
violate no cloning in any sense?

The question is, does it violate the no cloning theorem?
No, it doesn't.
Otherwise I would not say we have a large supply
because the no cloning theorem is absolute.
But it simply means we cannot have one state and then clone
and clone more and more copies.
Maybe let me be specific.
If you have an experiment which produces
a certain superposition state, you
can just push the button on your experiment many times
and produce in the identical way as many states you want.
If you have spin up state-- well,
this is now for two photons-- but if you have a spin
up state, you can make as many copies
you want off a state which has been
located by a certain angle.

So therefore, in state preparation,
by going through the exact procedure,
we can just create as many copies of a state we want.
The no cloning theorem meant the following.
I'll give you one state which may
be a spin which has been rotated at a certain angle-- you know
nothing about the state-- and now you
should try to make a copy out of it.
And the answer is, you can't because any measurement you do
is if the particle were spin up and you would measure spin
up and spin down, you could say, hey, I could spin up,
now I produce 10 spin up particles.
But if-- but you don't know along which axis
the spin has been prepared.
So unless you know which axis the spin has been oriented,
if you choose another axis, you have irreversibly
retrieved-- you have irreversibly lost information
which can not be retrieved.

I don't know if it helps you, but if you
have a certain eigen, if you have a certain state
and you want to measure it without destroying it,
you need a quantum non demolition measurement.
If you're in energy eigenstate you can measure energy.
If you're in a spin eigenstate which points along Z,
you can measure the direction of the spin--
the direction of the spin in the Z direction
without destroying it.
So if you can do a quantum non demolition measurement
on a state, you could clone it.
But that violates the assumption that if I give you an arbitrary
state you do not know, by definition,
what kind of measurement is a quantum non demolition
measurement.
You just take your chance, take a [INAUDIBLE] experiment,
separate the spins in the Z component,
and then it turns out I gave you a state which
was polarized along X.
So that's a subtle but important difference.
Other questions?
So we take two copies, and let's bring in Alice and Bob.
So the first photon we associate with Alice
and the second photon is associated with Bob.
So if you take two copies, we have
now, as in Hilbert space, four dimensional Hilbert
space, a direct product.
And if you just take the state and calculate
the direct product, you get four terms, 0 0 1 1, 0 0 1 1, 1 1
0 0, and 1 1 1 1.
And the coefficients are given here.
Let me just underline that if you
think we have an entangled-- we have
some state with some entanglement
and we separate the system, one goes to Alice
and one goes to Bob.
So the one which Alice has is the first part of those states
and Bob has the other part.
So the protocol is now that Alice and Bob first-- they're
not doing a measurement, they're not
reading out the system-- they perform a unitary operation.
And what Alice and Bob perform is the controlled NOT operation
on the two qubits.
Let me just write it down and then explain it.
Perform what is called the controlled NOT, or CNOT, gate.
And I have explained you before in the last section how
the controlled NOT can be performed using a nonlinear
[INAUDIBLE].
So Bob and Alice both run their two photons
with a nonlinear [INAUDIBLE].
So what is a controlled NOT?
Let me remind you.

If you have two qubits, the first one is the control.
If the control is 1, you flip the second one.
If the control is 0, you do nothing to the second one.
So they control it-- the controlled, the CNOT,
transforms.
So since the first qubit is the control qubit, out
of those four combinations, the controlled NOT
only does something if Alice's controls
and Bob's controls, Alice's controls and Bob controls is 1.
And then the second beat is flipped.
If Alice's and Bob's controls are 0,
they do nothing to the second beat.
So therefore, the 1 1 0 is transformed into 1 1 1,
and the 1 1 1 1 is transformed into 1 1 0 0.
So that's the first operation.
Let Was just indicate it so we have trans-- let me just
use the color here.
So what has happened here, the 0 0 has been flipped into 1 1
and the 1 1 has been flipped into 0 0.
That's the first step.
The second step is that now Alice and Bob
measure their target qubit.

What does it mean?
We have a controlled NOT operation,
and the controlled NOT operation we have the first one
is the control qubit, the second one
is the target of the operation.
So in our-- the way how we write it,
Alice's target is the third in line
and Bob is the fourth in line.
So now Alice and Bob measure the target qubits.
Let me just be specific.
So Alice has number one and number three,
Bob has number two and four, and the target qubits
are the third and the fourth.
So what is the probability that Alice and Bob measure both 1 1?
That they both find a target qubit of 1?
Well, it is this one and it is this one
where the controlled NOT has flipped it.
So one has probability a square b square,
the other one has probability a square b square.
So with probability, 2a square b square-- well, let's
allow a and b to be complex.
They obtain 1 and 1.
So in this case, what is left is here, 0 0.
What is left is here, 1 1.
You may just need an intermediate line
to write down what the state is after the measurement,
but if you read it off here, you find
that when they obtain 1 1 1, that that implies
the post-measurement state is then 0 0 plus 1 1
divided by square of 2.
And this is one of our Bell states.
So what we had is we had a system of four photons.
After a qubit operation, the controlled NOT,
Alice and Bob do a measurement together on two of the photons.
And if the outcome of this measurement is 1 1,
the rest of the system is in the Bell state.
So we assume that Alice and Bob have
a large supply of those copies.
So let's assume that they start with n copies of psi
and then, because their probability is m over n,
they successfully obtain m copies of the Bell state.
And the question is, what is the probability in the limit
of a large ensemble?
And we will see in a few moments that this is actually
a measurement of how entangled the original state is.

Any questions about purification?