M1L6e.txt

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Let's measure entanglement.
The basic idea here is that if have a state up down
plus down up, it's a pure state.
But this pure state has a correlation, a correlation
between the two subsystems.
And the idea is now, entanglement
is that there is a correlation between the two subsystems.
And you can now say, well, it would be a good way
to characterize entanglement if I only look at one subsystem.
In this case, you look at one subsystem
and you would just randomly see spin up, spin down.
Spin up, spin down is described by the unity density
matrix, which is therefore the most random state on earth.
So if you have a pure state and you only look at one subsystem,
the more random the subsystem is the more
the pureness of the initial state comes from correlations,
comes from what is entanglement.
So therefore, what I want to introduce now
is a measure of entanglement is that we take the total system
and then we perform the partial trace.
We only look at one subsystem.
And the purer the subsystem is the less entangled it is.
Because in the ultimate limit that our system factorizes
into two pure states, when you look at the subsystem,
we still have a pure state.
So the purity of the subsystem in terms of pure state
is now imagine of entanglement.
The purer the subsystem is, the less entangled it was.

So the basic idea here is entanglement is
related to correlations.
And if you take half of a Bell state--
so we have two particles which are an EPR pair
and we take half of it-- then half of it
is completely random.
So let me illustrate that.
So if you take 1/2 of-- let's just
take one of the Bell states-- phi plus,
which was the superposition of 0 0 plus 1 1.

So let me be specific because we need that for the definition.
We describe this ensemble of this system in a pure EPR state
by a density matrix.
The density matrix is nothing else
than you take your total system-- so this is now
the density matrix-- so in our case, psi AB is just the state.
And now we describe the subsystem
by performing a partial trace on [INAUDIBLE].
The partial trace is over the system B,
and that means we take all eigenfunctions,
kB of state B sum over all case and this is our partial trace.
So therefore, we would take our statistical operator
from the line above and perform the partial trace
where those states are the state 0 and 1 of B.
So these are the states of B.
So when you do that you just insert that.
You'll find that what you get is 1/2, a has been traced out,
so b has been traced out, and what
we obtain for a is just from the two terms above.
This gives that and this gives that.
And you immediately realize that this is 1/2 times the identity
matrix.
So therefore, we have shown that this
is a completely random state.
So now we can characterize the randomness of the partial trace
by-- of the entity operator obtained
by performing the partial trace with the standard for Neumann
entropy.

As a reminder, the von Neumann entropy
for statistical operator rho is defined as the expectation
value of rho log rho where we take the logarithm with respect
to the base 2.
So this is the trace of rho log rho.
Or if we use the eigenvalues of rho,
we multiply eigenvalues with a logarithm of the eigenvalues.

So for pure state, the entropy is 0
because the pure state has one eigenvalue, which is 1,
and the log of 1 is 0.
So we get 0 for pure state.
For completely mixed state, we're
talking about a state which has two dimensions
or it can be up and down.
A completely mixed state has probabilities of 1/2 each.
And then we say the entropy of this state is 1
or we call it one beat.
Yes?
I'm confused about the [INAUDIBLE].

Isn't just the [INAUDIBLE] of an operator [INAUDIBLE] trace
of the operator [INAUDIBLE]?
Thank you.
Yes.
So this is just a reminder how we
measure entropy of density matrix
and now we apply it to entanglement.
We define now that entanglement for the entanglement E
over state psi AB to be the entropy of the density
matrix for system A after tracing out system B.
And for pure state, this is-- it doesn't matter whether we trace
out A or B. If you start with a pure state,
the entanglement, the entropy of the statistical operator rho
A and rho B, are the same.
Tried for a moment to prove it.
I saw it quoted somewhere.
I didn't succeed in a split second
so either I overlooked something or it's a little bit more
involved to show that.
So therefore, to use it inverse, our definition
says that the entropy-- so the entanglement--
is nothing else than the entropy of the reduced density matrix.
And we immediately see, if we have any of the four Bell
states, by performing the partial trace over one qubit,
we obtain the identity matrix, so therefore the entropy
of all the Bell states is 1.
Let me state without proof when we come back
to the purification scheme, the result
is that the probability or the optimum probability-- if you
do stupid measurements on your states,
of course you get nothing-- but the optimum strategy
to create pure Bell states out of your reservoir of poorly
entangled states-- so for an optimum strategy, the success
probability m over n actually turns out
not to be a different measure of entanglement.
It is the entanglement which we have just
defined through the entropy of the partial trace.
So therefore, and I think this is nicely
illustrated with a purification scheme,
entanglement is a real resource.
When you have better entanglement to start with,
then you have-- you can get more copies,
you can get more pure Bell states, out of your supply
of poorly entangled states.
So therefore you lose one-- if your states are not
fully entangled, you lose more of them,
and therefore the success of the purification scheme
makes it clear how entanglement is a resource.
If you have entanglement it's precious.
You had to do something to get it.
But I didn't point out that entanglement is not
something which one number characterize it it's all.
We introduced to you already another measurements
of entanglement called the Schmidt number which was
done in homework number two.
And usually when you have different measures
for entanglement, they're not one to one related.
It seems that, similarly when we measured non-classic light,
we had a G2 function, we had bunching, anti-bunching,
we have negative course of probabilities.
And it's often clear that one system, which
is totally non-classical, fills all the criteria,
but how the quantitative measurements are related
to each other is really subtle.
In the case of entanglement, for a long time
it hasn't even been a question in research,
if you have an arbitrary density matrix
with a complicated many body system how can
you even characterize the entanglement.
We are focusing on pure states where things are fairly simple.
But in the general situation of a many body system,
it can be quite challenging just to define and measure
entanglement.
Any questions?
Yes.
Do you know a general density matrix that's not necessarily
a pure state?
Can you show that it's always the reduced
trace of the [INAUDIBLE]?

Say again.
If I have a general state which is--
[INAUDIBLE]
If you have a subsystem-- let's say you have two [INAUDIBLE]
and then you have a density matrix for the first.
Actually, let's just say that you have one [INAUDIBLE]
and you have a density matrix for that single spin.
It's any matrix not necessarily a pure state.
Let say it's a mixed state.
Then can you introduce a fake second spin
and show that this matrix is the [INAUDIBLE] matrix [INAUDIBLE],
maybe because you can just put that one in a pure state
and do some stuff [INAUDIBLE] calculations [INAUDIBLE].
Yes.

If you have a density matrix, you
can always regard it is a partial trace over bigger
system.
That means you always represent your state as a pure state
but it is entangled with a bigger system.
However, what the system is, is by no means unique.
There are-- I'm using the technical word.
There isn't-- it's called unraveling the density matrix.
You can always represent the density matrix written down
in forms of pure state.
But this unravelling of the density matr-- no,
actually that's-- it's related.

When I said yes I thought about the unravelling
of the density matrix.
You can always write down a density matrix
as a mixture of pure states, but which are the pure states
is not unique.
So when you say, my density matrix
is half of the atoms are spin up and half of the atoms
are spin down, somebody else would say, no, that's not true.
Half of the atoms are spin-- in spin
x and are in spin minus x and there-- [INAUDIBLE]
representation equivalent.
So I was just thinking of that.
It's sort of to write down the density matrix in a pure state
basis.
But you are asking about--
[INAUDIBLE] what he's asking about and it's possibility?
Somehow.
I was thinking of that but I think
the answer to your question is yes.
So you said-- you confirmed that?
Yes.
But I forgetting the name of the theorem.
Yeah.
Isn't it just purification?
Yeah.
It's related to purification.
You can prove that they're using purification.
[INAUDIBLE]
Using purification, yes.
But I'm completely forgetting the technical names
of the [INAUDIBLE].
But, wait, we have talked here about-- just to be clear,
we have talked here about purification of a pure state.
We started with the state and we purified it to be a Bell state
by doing certain measurements.
[INAUDIBLE]
So we have not talked here about density matrices,
but it sounds very plausible that you can always
construct a bigger system but-- I should look it up and see
if there's an exact proof.
Intuitively it sounds correct.
Other questions?