We have started to discuss weak measurements, that is, measurements where you don't suddenly

project your quantum system into one state or the other.

And we've learned that the basic idea is to assume that your quantum system interacts

for some time with another quantum system, call it an ancilla, and then you do a projective

measurement on that ancilla.

So that's the idea.

And so just to remind you, the picture we had in mind is probably the simplest possible

model for such a process of weak measurement.

We have an atom somewhere whose state we would like to detect.

And we send in a large number of these ancillas, which we also treat as true-level systems.

They are moving past the atom, and whenever they come close, they will interact for some

given short amount of time.

So here there will be an interaction, then they pass on.

That state has been changed, possibly, by the interaction.

And then finally, they will be detected, possibly after you have applied some unitary transformation.

That just means you can detect them in any basis you like.

So these are the ancillas.

And every ancilla comes in with a state that, in my example, I take to be a sigma x equals

plus 1 state.

So that's just the vertical position of up and down.

In addition, I take the atom to be in the initial state, which is a superposition of

its excited state and its ground state.

And I take the detector to detect in the basis not of sigma x equals plus or minus 1, but

sigma y equals plus or minus 1.

Because we found out this is the best setup in order to get the maximum amount of information

in this example.

I still have to tell you what is the interaction.

I have to remind you.

And we took an interaction, again, that is as simple as possible.

It simply means that if the atom is in the excited state, we impart a relative phase

shift between up and down for the ancilla.

And if the atom is in its ground state, we also impart a phase shift, but with the opposite

sign.

So the interaction Hamiltonian was h bar times some coupling constant g times the projector

onto the up spin state of the ancilla times sigma z, which relates to the item.

So depending on the state of the atom, you get a positive or negative phase shift.

And so then we prescribe a certain interaction time delta t.

This gives rise to a certain unitary operation that is just e to the minus i over h bar h

times g times delta t.

And you have to apply this unitary transformation to which state?

Well, to the combined state of ancilla and atom.

And that we take to be a product state because each incoming ancilla is completely uncorrelated

with the atom initially.

So you would have psi tilde, which is to be the joint state of atom and ancilla after

the interaction is equal to this unitary applied to the initial state, which, as I said, is

just the product state.

OK.

And so this is something which we worked out last time.

You can write it in many different ways, but we chose to write it already in a basis where

we choose to describe the ancilla not in the sigma z eigen basis, not in the sigma x eigen