M1L6c.txt

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Let me illustrate with examples.
For instance, in the helium atom, you have two electrons.
And the ground state, if I use spin notation, it's a singlet.
So does the [INAUDIBLE] of helium
fulfill our definition of an entangled state?

Well, if you find a way to switch off the Coulomb
potential and then the electrons separate from each other,
they still maintain their spin singlet character,
then you can take them, measure them, take measurements,
manipulate them.
But since nobody has come up with a good idea
how to switch off the Coulomb potential in an atom,
you can never separate the states.
And it is physically impossible to address the two electrons,
see them separately and such.
So this state is not entangled because the two systems cannot
be separated.

What is usually a good choice for entanglement,
and this is why we discuss it here with photons,
is you have the photon states.
Photons always fly away.
Photons are in a certain superposition state.
You can always separate them and detect them individually.
So let's assume we have two photons, two modes.
We have two photons.
Actually, the two photons can be in one spatial mode.
But we are now playing with the polarization
always on the vertical.
So if we have a state with horizontal/vertical
polarization, this is a nice entangled state of two photons.
So what it means is that we have one photon each.
One is vertical, and one is horizontal.
But you don't know which one.
If you would just look at one photon,
it would be 50% horizontal, 50% vertical.
It would be completely unpolarized,
which would be a random state which would require a density
matrix for its description.
But if one photon is horizontal, the other one
is vertical and vice versa.
So it's a pure state, but all the pureness of the state
comes from the entanglement and not
from what one photon does by itself.
So these are two photons in one mode,
and they are polarization entangled.
Or this brings us back to our dual real single photon states.
We have two photons, two qubits, and each is in two modes.
So our 0 state-- and just to make sure that you do not
confuse it with no photon-- our logic 0
state-- so L means logic here-- is that the photon is
in the second mode.
And the 1 state of our logic state
means that the single photon is in the other mode.
So we can now have an entangled state, which is now this 01,
10 state.
But these are now logic states, which means
that the 0 has one photon.
It has a photon in one of the modes,
and the 1 has the photon in the other mode.
So each state here has two photons.
But then the two photons have-- 01 and 10 are switched into two
parts of the wave function.
And we actually saw in the last unit
how a Kerr medium and then interferometer
can generate this state, so interferometer
with Kerr medium.

So-- yes?
[INAUDIBLE]

There are two such states.
I mean, if you have [INAUDIBLE]--
I will later tell you what the four famous Bell states are.
There is one which has a plus sign and one
which has a minus sign.
So when we talk about spins in singlet state,
it's often more natural the minus sign.
Here what naturally emerged was the plus sign.
But they are both sort of Bell states.
And therefore, they have what Einstein, [INAUDIBLE]
introduced into it.
So tolerate both signs.
They are two different states.
But for the purpose of the common discussion,
they have the same property.
They are maximally entangled.
Other questions?
OK.
So let me point out that the properties of entangled state
always involve two qualities.
One is the non-local character.
Because we have correlation between two subsystems, which
may be together when they interact, but then
they can be separated.
So we have correlations which happen between the two systems
which are a distance apart.
And we'll later come back when we
talk about Bell's inequalities that we know that physics
has non-local aspects.
And secondly, if you can separate two parts
and they interact with the environment,
the environment may interact with them differently
in those two different parts.
And therefore, entangled states are always regarded as fragile
against decoherence.
And it's a technical challenge.
How do you find states?
How do you implement entangled states which are robust?
Just to give you one example, if have an entangled state which
is based on electron spin, you may
be more than a thousand times more sensitive
to magnetic field fluctuations in your laboratory
than if you have qubits which contain the states which
are based on nuclear spin.
So that's a big research area, to find
states which are less sensitive or even immune
against decoherence.
So I've mentioned to you that entangled states are states
which you cannot factorize.
But now we can sort of start playing with that definition.
And we say, OK, if you have an entangled state which
is up/down plus down/up, but now the contribution of down/up
has only a tiny little amplitude.
So it's almost a pure state which
can be factorized with a little bit of an extra configuration,
which prevents us from factorizing that.
I mean, that doesn't look like good entanglement.
It looks at the whole entanglement
of the state depends on a very small admixture to the wave
function.
And so what you want to address now
is, how can we quantify that?
How can we look at a state and say, hey,
this is sort of not a strongly entangled state.
It has only a weak amount of entanglement.
So let's not forget, entanglement is a resource.
Entanglement allows you to do teleportation.
Entanglement allows you to do more precise measurements.
And what I want to sort of convey you,
if a state is only weakly entangled,
it doesn't help you much to achieve precision
beyond the standard limit-- effective teleportation
and such things.