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U1L4g.txt
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#
# File: content-mit-8370x-subtitles/U1L4g.txt
#
# Captions for course module
#
# This file has 61 caption lines.
#
# Do not add or delete any lines. If there is text missing at the end, please add it to the last line.
#
#----------------------------------------
Now let's take the tensor product of two matrices.
1 over root 2.
1 1 1 minus 1 one.
That was H. OK, let's do this in steps.
And let's take I equals 1 0 0 1.
And again, let's look at these coordinates.
So this is 0 1 0 1, 0 1 0 1.
H tensor I is equal to, again, we're
going to assume we put the coordinates in the right--
in the lexicographical order.
So this-- if you look at this upper-left corner, 1 0 1 1.
I want to divide this into four pieces.
If you look at this upper-left corner,
you have 0 in the first coordinate
for all of these things.
And then you have a copy, 0 1 0 1, in the second matrix.
So that gives you--
so what we're going to do is take the second matrix
and multiply it by the--
let's make this 1 over root 2, 1 over root 2, 1 over root 2,
minus 1 over root 2.
And multiply it by the 0 0 entry in the first matrix.
So this is going to give us 1 over root 2, 1 over root 2.
Now here, we have that 0 1 entry in the first matrix
times the second matrix.
That is again 1 over root 2 0 0 1 over root 2, 1 over root 2
0 0 1 over root 2, and now the last one is a minus,
so we get a minus 1 over root 2 0 0 minus 1 over root 2.
It's clear what we're doing?
And now what would I tensor H be?
Well?
We want to put an H in here.
1 over root 2, 1 over root 2, 1 over root 2, minus 1
over root 2, 0, 0, 1 over root 2 1 over root 2 1
over root 2 minus 1 over root 2.
Again, 0 0 0 1, 1 0 1.
Can anyone tell me what I tensor I is?
Yeah?
I. Yeah, you take the density, tensor the identity,
you get the identity.
Which is good, because if you apply the identity
to the first qubit and the identity to the second qubit,
nothing should happen.
It should be the same, so it should be the identity.
And now let's compute H tensor H. What is it?
It is, well, we're going take H and multiply it
by 1 over root 2 in here, so you get 1/2 1/2 1/2 minus 1/2.
We're going to do the same thing in the next coordinate.
1/2 1/2 1/2 minus 1/2, minus 1/2 minus 1/2 minus 1/2 and 1/2.
Because H has a minus sign in the bottom right
entry and plus signs in all the others.