U2L1e.txt

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So what Alice is going to do is she takes her unknown qubit
and phi plus and measure the first two qubits of this three
qubit state with this basis.
So what happens?
OK, I'm going to compute this in two different ways.
But let's do the more or less straightforward way first.
So let's say zeta is equal to alpha 0 plus beta 1
where alpha and beta can be any complex numbers whose
squares sum to 1.
So state is alpha 0 plus beta 1, 0 0 plus 1 1.

And the first two qubits belong to Alice,
and the second one belongs to Bob.

And now we want to measure this with the basis.

So what we do is we take the inner product of this with 1
over root 2 0 0 plus 1 1 with 1 over root 2--
let's leave the 1 over root 2 in here--
with 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 of course both of these qubits are in Alice's hands.

So what do we get when we take this
I guess partial inner product?
Well, the 0 0 with the 1 and the 1 1 both returns nothing.
So 0 0 with 0, 0 Alice is equal to, well there's a 1/2
here times a 1/2 here.
This is 1/2 0 Bob.
And this 1 1, if you take the inner product
with this one you get 1, you take the inner product
with this 1, you get--
OK, so some of these 1s are qubit states,
and some of these 1s are scalars so it's confusing.
But yeah, so you take the inner product of this and this,
and you get plus alpha 1.
Plus beta 1.
Again, [INAUDIBLE].

Now here what happens, this 0 0, the only term it picks out
is 0 0 0, which has an alpha on it.
So this is equal to 1/2 alpha 0, and this minus 1 1,
the only term it picks out is 1 1 1.
So it's minus beta 1.

OK, same here.
The 0 1 picks out the 0, 1 1.
So that is equal to--
well, the 0 1 picked out the 0 that's in alpha and the 1 1,
so that's one 1/2 alpha 1 plus beta 0.

And the last term we have 2 0 1, the 0 1 picks out 0, 1 1.
So this is equal to 1/2 alpha 1 minus--
because the second term is a minus,
and it picks out 1 0 0 minus beta 0.

So how do we correct?
What do we do to the first--
this first thing, alpha 0 plus beta 1, to correct it?
Anyone?
Nothing.
Nothing.

If we get the outcome phi plus, which is the top thing there,
Bob's unitary is the identity.
How about phi minus?
What do we need to do to correct that one?

Yes?
Multiply by z?
Yeah, multiply by z.
Great.

And any guesses on psi plus?
Sigma x.
Sigma x.
Sigma x equals 1 1.
And we could multiply by sigma y to fix this one,
but that would give us a phase which I am not going to want.
So what do we do to do it without the phase?

Well first I want to say we apply sigma x.
If we apply sigma x, we get alpha 0 minus beta 1.
And then we apply sigma z which gives us alpha 0 plus beta 1.
So when we apply sigma x, we go here, we apply sigma z,
we go up to the top.
So we first apply sigma x then sigma z, so that sigma
z sigma x.

OK.

So that is how we fix it.
So this gives us the teleportation protocol.
Alice measures in this state, and then Bob fixes it.
So what I want to do next is draw a quantum circuit
for teleportation.
So we start with psi and that's the top thing,
and 0 0 plus 1 1, the next two qubits
are going to be in this state.

Well, what is the first thing we do?
First thing we want to do is we want to measure this basis.

So I'm going to go from right to left
so I can leave these up here.
So we're going to apply a z0, and what happens
when we apply a z0 to this?
Well first, why do we apply a z0?

Well, I guess you'll probably see what--
I mean, we're going to get 0 0 plus.
Well, the z0 takes the first qubit, or first bit,
and [INAUDIBLE] it with a second.
So this is 0 0 plus 1 0.

This is 0 0 minus 1 0.
This is 0 1 plus 1 1, and this is 0 1 minus 1 1.

But now we can rewrite these states
because they're separable.
So this is just plus time 0.

This is minus time 0.
This is 0 times plus--
wait, no this is plus times 1, and this is minus times 1.

So you see what I did was I factored out the second qubit
here, and the first qubit became 1
over root 2 0 plus 1, which is just plus,
and similarly for the others.
And now what we want to do is we want
to put these in something that looks like the standard basis.
And you can see that if you do a Hadamard to the first qubit
and nothing to the second, this becomes 0 0, this becomes 1 0,
this becomes 0 1, and this becomes 1 1.

So how do we do this measurement?
Well, what we want to do is we want to major in these basis.
But after we apply this transformation,
that's equivalent to measuring in this basis,
and we know how to do that.

So what was that transformation?
Well, the first thing we do is a c0
from the first qubit to the second qubit.
That's this step.
And the second thing we do is a [? Hadamard ?]
transform on the first qubit at this step.
And now we measure in the standard basis.

OK.

So the next thing we want to do is apply this correction.

And let's go back here.
If for the phi plus our measurement result
was 0 0, for phi minus Alice's measurement result was 1 0,
for psi plus it was 0 1, and psi minus it was 1 1.
So given this measurement result,
we want to apply these transformations.
So you can see if the first qubit is a 1,
we want to apply a sigma z.
And if the second qubit is a 1, we want to apply sigma x.

And we want to apply sigma x first.

So every 1 with a first qubit is a z.
Our first qubit is a 1, we apply sigma z,
that's this guy and this guy.
And where the second qubit 1, we apply sigma x,
and we want to apply sigma x first
because that's what we're doing here.