U1L4k.txt

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So H x H is equal to sigma z.
H sigma to the z H equals sigma-x.

Well, actually, remember that H--
if you're thinking of this as a change of basis,
H just takes 0 to plus and plus to 0 and 1 to minus
and minus to 1.
H sigma-x-- OK.
Changing bases with unitary UM U-dagger equals M tilde.
This is in a different basis.
Or rather in the basis that, you know,
M takes the standard base--
U takes the standard basis to some other basis,
because remember, the unitary is just take
orthogonal unit vectors to orthogonal unit vectors.
So unitaries take bases two bases.
So UM U-dagger is just a change of basis for M. So we change--
H takes the x spaces to the z basis and vice versa.
So h sigma xH equals sigma-z and H sigma-z H equals sigma-x.
And maybe I should--

so that's the easy way to see it.
But you can also write out 1 1 1 minus 1, 1 minus 1,
that's sigma-z.

1 1 1 minus 1, and let's put a factor of 1/2
because each of these H's has a 1 over 2 on it.
And that should equal 1 1.

And let's just do this to make sure that 1 1 1 1,
and then the second.

When we multiply this by that, we
change the sign of the second column, we get 1 1 1 minus 1,
and now it's easy to see that this is equal to 0 1 1 0.

And of course, H squared is the identity,
so you can get this by just applying H
to both sides of that top one.