U1L5h.txt

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What we'll look at in the next--
well, next couple of lectures, we
will be using these four states called Bell states.

0 0 plus 1 1.

1 over root 2 0 0 minus 1 1, 1 over root 2 0 1 plus 1 0,
and 1 over root 2 0 1 minus 1 0.

And these are all entangled.

They form a basis for the state space of two qubits.

And I want to say that this state,
remarkable because the spin is 0 in any direction.

Along any axis.

And I think the rest of this period--

well, some of the rest of this period
will go into proving this.

And, you know, for these, the spin is 0 along some axes
and 1 along other axes.

So first calculation.

x-- we'll call this psi.
psi Sx psi equals 0.
psi Sy psi equals 0.
And psi Sz psi equals 0.

Well, we've already--
I mean, this is easy to check because Sz was--
I erased that, didn't I?
Sz was 1 0 0 minus 1, and these--
this state is 0 1 minus 1 0, so it lies in the 0 eigenspace.

And Sx was 1 1 1 1 1 1 1 1, and we can multiply it by 0 1
minus 1 0.

I think I need a normalization vector of some kind here.

Not that it matters.
And this is equal to, well, multiply it
by the first term, top and 0, 0 0 0, so this is 0.

And Sy equals 1/2, OK.
You're going to have to trust me for the calculation here.

Is equal to 0.
So this is in the 0 eigenspace of Sx, Sy, Sz.

So S alpha is equal to sigma alpha tensor 1 plus 1 tensor
sigma alpha.

Well, that's just alpha-x sigma-x plus alpha-y
sigma-y plus alpha-z sigma-z.
So this is just 1/2--
I want to say Sx alpha-x Sx plus alpha-y Sy plus alpha-z Sz.
So something in 0 eigenspace of Sx, Sy, Sz
also is in 0 eigenspace of S alpha.