U1L3o.txt

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What are the number of degrees of freedom,
and what are the number of degrees
of freedom of 2 by 2 unitary?

Well, we can factor out a global phase.
And, well, the first thing is going
to be 1 or rather it's going to be cosine theta.
The first column has to look like cosine theta.
OK, let's call this alpha.
So in e to the minus alpha cosine--
we can always assume this is real, because we factored out
the global phase.
So if this is cosine theta, this is e to the i phi sine theta,
because You know that this first column has to be a unit vector.
And this is e to the i phi 2 sine theta.
And this-- you know, that this has to be orthogonal to this.
So that means this has to be e to the minus i
by 1 plus by 2 cosine theta.
So there's three degrees of freedom.

And the last thing I want to say is that you
can start with a Bloch sphere.

And you can take any point to any other point
by applying rotation around the z-axis
times a rotation around the x-axis times another rotation
around the z-axis.
So any unitary transformation on a qubit modulo,
the global phase, can be decomposed this way.

Yeah, that's what I wanted to say.
And how do you do that?
Well, let's start-- let's worry about where
the North Pole is taken.
So if we have the North Pole here,
we can take it to any latitude by just rotating
around the x-axis.

So I guess if this is--

if this angle is theta we just rotate it theta degrees
around the x-axis.
That brings us down.
And now we can rotate around the z-axis.

And now it's at the same latitude,
but a different longitude.
OK, I've got my latitude and longitude mixed up now,
same latitude, but a different longitude.

So now we've got the North Pole right,
but we don't know whether we have
rotated it the right amount around the North Pole.
Because if you fix the North Pole, there's another rotation
you can do around the new place on the North Pole,
but we could have done that by starting here and rotating it
the right amount.

I want to say we can always fix this rotation around the North
Pole by going back to the original thing
and rotating this around the North Pole.
And then we rotate the North Pole down,
and we rotate the North Pole across.
And if we figure out the right amount of this rotation,
we'll get this one taken care of.
So you can always get--
you can always rotate a sphere by rotating
around the z-axis, the x-axis, and then the z-axis again.
And of course, z and x are not unique here.
You could pick any two of x, y, and z and do it like this.
So there were three degrees of freedom of 2 by 2 unitaries.
And here we have three degrees--
no, we have three parameters, theta--
oh, I guess, phi 1, theta 1, and theta 2, call them.

OK, good.
Are there any questions?