M1L5s.txt

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Good.
So what I just said, that we want to restrict the Hilbert
space to exactly one photon-- this
is so important for implementations and discussions
of quantum information protocols that it has its own name.
It's called the dual-rail photon state, the dual-rail photon
state space-- exactly one photon,
but the photon can either be in mode a or in mode b.
So this part, this dual-rail photon state space,
is a Hilbert space which has a basis which is 0, 1 and 1, 0.
And this is, of course, a two-level system.
In this so defined Hilbert space,
we can have an arbitrary state psi
which is, well, as in any two-dimensional Hilbert space,
it's a linear superposition of your two basis
states with coefficients alpha and beta.

And the theorem for which we have prepared right now
is that any possible state of this Hilbert space
can be created from any of the basis state--
let's say from 0, 1-- simply by beam splitters and phase
shifters.
When I taught this unit the last time, I went through the proof.
But I want to make room for more in-class discussion
like we had on Wednesday with the clicker question.
So what I decided is I give you the idea behind the proof,
and the formal parts you can simply
fill in by reading about it on the wiki page.
So let me just focus on the idea.
So the proof is that if I use some column vector for alpha,
beta, that we can identify, with the beam splitter and the phase
shifter, we can identify them as rotation operators.
This is immediate obvious if I use the transformation on psi
by a beam splitter, it is by just-- we learned further--
we learned just few minutes ago how the beam splitter operator
acts on the basis state.
So now we know how it acts on an arbitrary linear combination.
And the matrix here is nothing else
than the rotation matrix with the minus
sign around the y-axis.

So that's a beam splitter.
The phase shifter is not changing any state.
It shifts one of the states by e to the minus i phi.
But I can sort of [? summarize ?] it
by taking half of it out as a prefactor
and then adding here e to the i phi over 2.
And this here is immediately recognized as the rotation
matrix around the z-axis.
Now, you may ask, what about this?
Well, this is an irrelevant global phase factor.
If you have any state in Hilbert space
and you change the global phase, it doesn't matter.
So therefore, we are realizing that the beam splitter is
a rotation around the y-axis.
It's now a definitional thing, but we put a factor of 2
here by an angle 2 theta.
And the phase shifter is a rotation
around the z-axis by an angle phi.
Now, with that, we can formulate what
is called Bloch's theorem that any unitary operation
of a two-dimensional Hilbert space
can be written as a product of rotations.
Now, let's first count an arbitrary unitary operation,
valid two-by-two matrix, in complex space
has eight numbers for real parts, for imaginary parts.
But if the matrix is unitary, there
are only four independent element-- alpha, beta, gamma,
delta.
And if you parametrize the unitary matrix
with alpha, beta, gamma, delta, which
is shown in the wiki notes, then we
can write an arbitrary, this arbitrary, unitary matrix
by a combination of a phase factor, an overall phase
factor, which is irrelevant because it's a global phase
factor, and then a rotation around z, a rotation around y,
and a rotation around the z-axis.