U1L3e.txt

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We'll be working a lot with qubits.
So it really helps to understand the geometry of qubits
and how they transform.
So the geometry can be represented
by the Bloch sphere, which--

so a qubit is a two-dimensional complex vector space.
And it turns out our unit vector in a two-dimensional convex
vector space-- and this is equivalent
to a three-dimensional real vector space.
So we have up--
which I'm going to call 0 at the top of the sphere--
down-- which I will call 1 at the bottom of the sphere--

right-- which, remember, was 0 plus 1 over root
two equals right was here.
And let's label the axis.
So the-- OK, this is getting too crowded.
So z-axis is going up.
The x-axis is going to the right.
And the y-axis is going into the board.

And so this should be 0 plus i1 equals
into the board back here and 0 minus by 1
coming out of the board.
I'm not doing a very good job at this, am I?
And let's put the normalization constants in.
OK, so that's the Bloch sphere.