U1L3h.txt

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A unitary transformation is one that takes complex unit vectors
to complex unit vectors.

So it takes 0.
So let's pretend.
OK.
So d dimensional quantum state space with bases 0, 1,
2, d minus 1--
OK.
So let's work in the d dimensional quantum state space
and see what we need for a transformation to be unitary.

So I times U is equal to--
let's call it C sub i.

C sub i is unit length because--
I should say that I is just the vector 0, 0, 1, 0, 0, where
1 is in the i-th position.

So U times I is a complex vector C sub i.
And if you think about it, C sub i is i-th column of U.
So U looks like C1, C2, through C sub i--
C sub d minus 1.

OK.
I want to say we can write this as summation C sub i
I. So this is a nice thing about Dirac notation.
So this is just [? phi. ?]
So what does this mean?
Well, it means that if you apply U
to j, you get I equals 0 through d minus 1 C sub i Ij, which,
remember, I--
1 [INAUDIBLE] through d minus 1 is an orthonormal basis
of the space.
So that's just equal to C sub i times the I
equals 0 through d minus 1, C sub i times
the Kronecker delta function, delta sub Ij, where this is 1
if I equals j and 0 otherwise.
And that is just equal to C sub i.

So now we have this notation.

Let's see.
Let's erase this.

Apply U to I plus j over root 2.
So that tells you C sub i--

I times-- OK, let's make this j plus k over root 2
so we don't reuse indices.
I equals 0 through d minus 1.
j plus k over root 2 is equal to Cj plus Ck over root 2.
And this needs to be a unit vector
because for a unitary transformation-- needs
to take this unit vector to another unit vector.
So we know Cj plus Ck over root 2 times Cj plus Ck over root 2
is equal to Cj, Cj plus Ck, Ck plus Cj, Ck plus Ck, Cj,
all over 2.

So what is this?
Well, Cj, Cj is 1.
And Ck, Ck--
I made a typo.
No.
I didn't make a typo.
And Ck, Ck is 1.
So this is equal to 1 plus 1/2 Cj, Ck--
that's the inner product of Cj and Ck--
plus-- I'm going to rewrite this as Cj, Ck star that says Cj,
Ck is imaginary.

OK.

Now, if you apply U to j plus Ik over root 2,
you can go through exactly the same calculation.
And you get Cj, Ck is real.

So that tells you that Cj, Ck is equal to 0.

So we go back.
Here, U is a vector made up of columns, all of which
have length 1, and any two of which are orthogonal.
So that is one, another, definition of a unitary matrix
because you can--

what does U do?
Well, it takes one basis to a different basis.

If you start out with a unit vector and one basis,
all you're doing is changing the unit vector
to a unit vector in a different basis.
So for example, summation alpha sub
i I goes to summation alpha sub i C sub i.
But sum of the alpha i squared is 1.
And that means this is a unit vector.
Sum of the alpha sub i squared is also--
is still 1 after the transformation.
So this is also a unit vector.
So a unitary is--

definition two of a unitary is a matrix
with all unit vectors and its columns
and any pair of columns orthogonal.
And there's a third definition of a unitary, which I will--