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U1L6h.txt
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U1L6h.txt
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#
# File: content-mit-8370x-subtitles/U1L6h.txt
#
# Captions for course module
#
# This file has 121 caption lines.
#
# Do not add or delete any lines. If there is text missing at the end, please add it to the last line.
#
#----------------------------------------
So now we can go on and give our strategy.
So I'm going to write down a three-by-three matrix,
with observables on two cubits.
So this is sigma x tensor I, I tensor sigma x,
sigma x tensor sigma x.
I tensor sigma z, sigma z tensor I, sigma z tensor sigma z.
And minus sigma z tensor--
minus sigma x tensor sigma z, minus sigma z tensor sigma x,
and sigma y tensor sigma y.
So we need some facts about this matrix.
Any two entries in same row or column commute.
Well, sigma x tensor I and I tensor sigma z commute.
I tensor z and minus sigma x tensor sigma z commute,
because sigma z's commute and the sigma x's commute.
And sigma x tensor sigma x and sigma z tensor sigma z commute.
And let me proof.
E.g., sigma x tensor sigma x times sigma z tensor sigma z.
So let's put superscript on this.
a, b, a-- no, a and b are the wrong things, here,
because they don't correspond to apples and bob--
1, 2, 1, 2.
Is equal to sigma x sigma z 1, tensor sigma x sigma z 2,
equals sigma z tensor sigma x 1, sigma z--
I'm sorry, sigma z times sigma x 1, tensor sigma z times sigma
x 2.
Good?
So, what are we doing here?
We're taking this tensor product and this tensor product
and multiplying them out, which means
we multiply the first slot with this
and the second slot with this.
So we get sigma x sigma z, in the first slot, tensor sigma x
sigma z in the second slot.
And now, when-- remember, sigma x and sigma z anticommute,
so maybe we should do this in a little more detail.
This is equal to sigma x sigma z minus sigma z tensor sigma z
sigma x.
Because, when we flip these around,
we get a minus sign because they anticommute.
But that is equal to--
Now we want to flip these around,
and we erase this minus sign.
Sigma z sigma x tensor sigma z sigma x.
And that is equal to sigma z tensor sigma x times
sigma z tensor sigma x.
So we take the tensor product of two things
that anticommute, they commute.
Hello?
But z tensor z minus x tensor x?
z tensor z times x tensor x.
Yeah, good.
Thank you.
I'm glad some people are paying attention.
[LAUGH] OK, so--
And there's another fact we need.
Well, so first, because any of them commute,
we can consider the product of these,
because it doesn't matter which order we take the product in.
Product of any row equals the identity.
Product of any column equals minus the identity.
Is that good?
Well, here, take the product of this column, we have sigma
x times sigma x.
That's identity.
Sigma z times sigma z, that's identity.
And we have a minus sign here.
Now, we can take the product of these three,
sigma x tensor sigma x, and sigma z tensor sigma z.
Sigma x tensor sigma z is minus I sigma y.
So this times this is minus I sigma y tensor minus I sigma y.
That's minus sigma y tensor sigma y.
And this is sigma y sigma y.
So I'm going to write that down.
Last column, column, we get sigma x sigma
z sigma y tensor sigma x sigma z sigma y.
And, if you remember how Pauli matrices work,
sigma x times sigma z is minus I sigma y.
So, this times this times this is
minus I times the identity times tensor minus I
times the identity, which is just the minus the identity.
And this is a two-by-two identity,
and this is a four-by-four identity.
And, so, all of these matrices have eigenvalues plus and minus
1, because, well, sigma x has eigenvalues plus or minus 1,
sigma z has eigenvalues plus or minus 1, sigma, y has
eigenvalues plus or minus 1.
And we take the product of two of them,
we just get eigenvalues of plus or minus 1,
because it's a tensor product.
So, for example, sigma z tensor sigma z
is equal to 1 minus 1 minus 1 1.
And when both eigenvalues are minus 1, you get a 1.
When one of these eigenvalues is minus 1 and 1 is 1,
you get a minus 1.
And when they're both 1, you get a 1.
So tensoring two matrices with eigenvalues plus or minus 1
gives you a matrix for the eigenvalues plus or minus 1.
And the same thing for--
you put a minus sign on it, that doesn't
change the fact that eigenvalues are plus or minus 1.
And these matrices all commute, so they're all simultaneously