M1L5x.txt

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OK, so our next chapter is on entangled photons.
And what we can accomplish in the next 10 or 15 minutes
is mainly the definition and some of the properties.
What we will do on next week is we
will talk about how we measure entanglement--
you had already one homework assignment where you discussed
one way to measure entanglement but there
are others which we want to discuss next week--
and I also want to show you next week how entanglement leads
to the Einstein-Podolsky-Rosen paradox and Bell's
inequalities.
And eventually I want to show you
how you can entangle not only photons
but how you can entangle atoms.
But that's an outlook.
I think, first of all, we want to understand
what is entanglement.
So entanglement is-- let me first [? motivate ?] it.
Many people regard entanglement as the most quantum
essence you can find.

Well, there's always a discussion,
what is clearly quantum?
Some people would say, waves are quantum.
But, well, often we find quantum systems-- really famously,
Bose-Einstein condensates-- which really behave
like electromagnetic waves.
They can be split, they can interfere,
but Bose-Einstein condensates are so big, have so many atoms,
that you hardly ever encounter quantum fluctuations.
You are really in the classical limit of metawaves,
and I would actually say this is maybe not
at the heart of quantum mechanics.
Well, it's nice, it's powerful, it's important,
but this is not really the new features
which quantum mechanics has shown us in nature.
You would say, well, what else is quantum?
Maybe certain quantum fluctuations,
going down to single photons, and I would agree.
That's the quantization of the electromagnetic field.
That's much more quantum than having millions
of condensate, millions of condensed atoms
in one single wave.
But what I would say is even more quantum
than the single photon is the entanglement.
Entanglement is sort of pure quantum mechanics.
With single photons, you can steel your intuition.
But with entanglement, that is really bizarre.
This is really what other-- I hate the word
but let me use it-- this is sort of pure quantum weirdness.
And there are people who have spent most of their career
until now to just show the weirdness of quantum mechanics
in showing to what peculiar phenomena entanglement leads.
So entanglement has really led to a lot
of-- I should be careful with the word "surprising."
Surprising always means you didn't have enough imagination
to think about it.
But, yes, I would say, really truly unanticipated,
and in that sense, surprising developments.
So entanglement is therefore, for me,
the most quantum aspects of quantum mechanics.
It was through the Einstein-Podolsky-Rosen
argument-- it was actually Einstein-Podolsky-Rosen
in the '30s-- who argued-- Einstein-Podolsky-Rosen
were really the first.
This was almost five or 10 years after the development
of quantum mechanics.
But it was Einstein-Podolsky-Rosen
who looked at the properties of entangled states,
this famous experiment where you have
something entangled in position and momentum space
and then Einstein-Podolsky-Rosen found a paradox.
And their conclusion was that the properties
of entangled states implies that quantum mechanic is not
complete, it's not a complete description of the word.
Well, we don't any longer share this opinion.
It was John Bell and others who said that entanglement really
exemplifies that quantum mechanical correlations go
beyond classical mechanics, go beyond the classical picture.
So therefore you can say, if you have a quantum
system like a Bose-Einstein condensate where
you have n particles doing kind of in unison
what one particle does.
This is sort of like the laser where many photons do
what one photon does.
But it fits very, very well into the concept
of classical probabilities and an intuitive understanding
of, well, many particles do what one particle does.
But if the particles are entangled,
that's almost like you turbocharge your system.
Your system has now more oomph, more power in it.
And this has shown that it has extra correlation.
There is something extra in it which you would never
get from any classical limit.
And this extra oomph, this extra power,
turns out to be a real resource.
So entanglement, these extra correlations, is a resource.
And resource means it's good for something.
It enables teleportation.

Remember when we had the-- when we talked
about the teleportation scheme, Alice and Bob could only
teleport a quantum state because they shared entangled photons.
It was this entangled state which
Alice manipulated with a measurement
and Bob used this half of the entangled state
to recreate the original state.
So it's the engine behind teleportation.
In the world of quantum computation,
the exponential speedup of quantum computers
versus classical computers.
In quantum algorithms, it's due to the entanglement which we
can put into a quantum system.
And eventually, next week, we will see that,
if we have an atom interferometer
with entangled states, we can operate it
at a precision which is better than shot noise.
So in other words, you use light-- laser light--
to measure something and you're limited
by the fundamental noise limit of classical physics.
Now you entangle your laser beam and you get higher precision
out of it.
So this shows that entanglement is a very special resource.
It's very precious, very powerful,
and can extend what physics can do.
So let's define it.
I'm not describing to you entanglement
in the most general situation.
If you talk about many modes, if you
talk about not just pure states but statistical operators,
it can become quite involved.
I rather start with the simple situation,
which for conceptual discussions is also the most important one.
I restrict our focus now on two modes.
So I want to ask, if you have two modes--
actually, two modes means two subsystems here--
if you have two subsystems, A and B-- maybe
let me tell you what entanglement is not
and then I make it the definition.
There wouldn't be anything special
if you have a system consisting--
you have a system which has two parts, A and B.
The global system is psi AB.
But if your system would simply factorize
into a wave function of subsystem
A direct product with subsystem B,
and this here is a tensor product,
then there would be nothing special
going on between subsystem A and subsystem B.
And this is non-entangled.
Everything else is entangled.
So this is our definition.
When we have two separate systems, A and B,
we call the situation-- the total system-- bi-partite.
So a bi-partite state, which has half of it
in system A, half of it in system B.
We also need-- we need those two different systems.
It's a composite system, A plus B. And this state is entangled,
if with two F's-- if and only if-- you cannot find two
states, psi A and psi B, such that the state can be
factorized.
We need a few examples to fully comprehend this discussion,
what systems A and B qualify.
What does it mean if there not exist any combination?
So let me give you some examples now and then I think we stop.
If we have the 0 0 state-- photons in two mode--
we can factorize it.
And this is not entangled.
If we have a state which is 0 0 plus 1 1, well,
you cannot write it as a product of one state times another
state.
This is entangled.
Well, if you take the following state,
is that entangled or not?
I've written it as the sum of four states.
But if you just stare at it for a split second,
you realize you can just write it as a product of two states.
It's just a product of 0 1 with 0 1.
So a state is not entangled if you
can write it as a linear super position of those states.
The definition of entanglement is,
if you try hard and hard and hard
and there doesn't exist any product state decomposition,
then it is entangled.
So that was easy.
If you use a state like-- let me just leave the 1 1 out-- what
about this one?
Well, you have to-- you can try as hard as you want,
you will not find a decomposition.
So eventually, when I say you have
to try hard to find decomposition,
maybe what we want is we want to apply
some operator or some procedure to this state
and get the answer yes no.
This naturally asks for question, how in general can
we measure if a state is entangled or not?
But this is something to discuss next week.
So let me just conclude with the following.
I'm focusing here-- and for pretty much
all of this course-- when we talk about
entanglement about pure states.

If you have no decoherence, if you have the transition
from pure states to density matrices,
it gets much more involved to talk about entanglement
and measure entanglement.
But at least the basic definition
can still be maintained for statistical operators.
If you describe this system-- the total system-- by one
statistical operator, the system is not entangled
if the statistical operator can be broken up into direct tensor
product of statistical operator for system A and system B.
Otherwise, the system is entangled.
So the basic definition can be generalized
to density matrices, but a lot of things
become much, much more messy if you don't have pure states.
OK.