Okay, hello, we are just discussing weak measurements where you slowly gain information about the

system and we ended last lecture by saying that while you are slowly gaining information

about the system, there's also decoherence.

So if the system has initially been in a superposition of two states, the relative phase between

the two states will become more and more uncertain.

So let's continue with that discussion.

We still have in mind this example where we have an atom with two levels excited and ground

state that is coupled to an ancilla.

It gets entangled with the ancilla at least a little bit and subsequently you would measure

the state of the ancilla to learn a bit about the atom.

So the state after interaction between atom and ancilla is simply this one.

We have written it in many different ways but this is the way that is the good one to

discuss decoherence, which is you would have the excited state and is associated with some

state chi e of the ancilla plus the ground state is associated with another state chi

e of the ancilla.

And so now if you look only at the atom, you take the reduced state of the atom, that is

you take the reduced density matrix by tracing out the ancilla.

And we discussed how this looks like.

That would contain, that would be a two by two matrix because the atom is a two level

system in our example and it would contain on the diagonal the populations like the probability

to find the atom in the excited state or in the ground state.

But on the off diagonal it would not only contain the product of the amplitude for the

atom to be in the excited and ground state like psi g star psi e but in addition it would

contain the overlap between the two different states of the ancilla.

So that is psi g psi e in this case and the lower entry is just the complex conjugate

because it is a mission matrix.

Okay and so what we learn from this is that precisely the off diagonal entry of the density

matrix that depends on the relative phase between these two amplitudes it will get suppressed

because the overlap between psi g and psi e is in magnitude never larger than one of

course because these are normalized states and usually it will be smaller than one if

we are learning anything about the state of the atom.

And so there are these decouples.

So the geometric picture we have is simply that I have two vectors in Hilbert space psi

g and psi e and they have a finite angle so the scalar product is less than one and in

our particular example you can work out the scalar product and it is of course related

to this theta which measured the interaction between the atom and the ancilla and it happens

to be just cosine theta.

So theta is really the angle between these two vectors in Hilbert space.

I should point out we can also represent the ancilla as a spin that was always our idea

and the interaction with the atom then turns the spin into one or the other direction and

afterwards we do a clever measurement in the direction where we gain the most of information.

If you want to draw it in terms of spin you can do that.

We started from a spin in the x direction and depending on the state of the atom it

is turned a little bit into the plus y direction or a little bit into the minus y direction

and the angle by lecture terms in these two cases is theta or minus theta.

So actually the angle between the two different spin directions is two times theta.

So there is a slight difference between the angle in Hilbert space and the angle between

these two spin one are.

That just has to be kept in mind.

It is always like this because for example if I were to turn by, if I want to have these