U1L6c.txt

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Alice and Bob share 1 over root 2, 0, 0 plus 1, 1.

Fact, if Alice and Bob make a measurement
of form alpha 0 plus beta 1--
what, no, that's not what I wanted--

alpha 0 plus sigma x and sigma z.

So they both make the same measurement, which is on--

well, on this great circle of the Bloch sphere,
they get the same outcome.

So-- and if they measure with the observable sigma y,
they get opposite outcomes.
So I'm not going to--
I mean, you could easily check that.
Proof for alpha equals 0 and alpha equals 1.
And here, I probably should have said alpha squared plus beta
squared is 1, alpha beta real.

I'm not sure it matters that alpha and beta are real, but--
yeah, alpha and beta are real, yeah.
Otherwise, this is not Hermitian.
So it's not an observable.
So proof for alpha equals 0 and alpha equals 1--
alpha equals 1, that's sigma z.
And they're just measuring in the 0, 1 basis, which means 0--
if they either get both 0, or they both get 1.

No, wait, that's alpha equals 0 is easy.
Alpha equals 1, that's 1 over root 2 0, 1, 0, 0 plus 1, 1.

And I'm going to put another 1 over root 2 here.
So this is, say, Alice's state.
And this Alice, Bob, Alice, Bob.
When we multiply this by this, we get 0, Bob.
And we multiply this by this, we get 0, 1, Bob.

And there's a 1/2 in front of here.

And similarly, if we had a minus here and a plus here,
we would get a minus there.
So when we take the projection onto 0 plus 1, we get 0 plus 1.
When we take the projection onto 0 minus 1, we get 0 minus 1--

and in fact--

And the proof for arbitrary alpha is not so hard.
I'm not going to go through it right now for lack of time.
But I want to use this.

So this was what we want.
And I'm going to erase 75%.
But hopefully, you guys can remember that.
And say, what is our strategy?

Well, 0, 1-- no, wait.
I'm trying to draw a piece of a Bloch sphere--
so 0, 1 plus minus.
And these are the points that are 45 degrees.
So this is the slice--
vertical slice through the Bloch sphere.

So Alice chooses 0, 1 or plus minus.
So Alice and Bob are going to measure--
each going to measure their half of the state.
And I want to say, Alice chooses 0, 1 or plus
minus with probability of 50%.

And Bob chooses-- no, not with probably 50%--

depending on what a was.
So a equals 0, and a equals 1.

And Bob chooses-- we need variables.
So let's call this s and this axis t.

So Bob chooses s and the opposite of s, if b equals 0.
And he chooses t and opposite state of t, if b equals 1.