U1L4e.txt

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So now, we're ready for the no-cloning theorem.
So what is the no-cloning theorem?
Well, it says you cannot take an unknown qubit and turn it
into two copies of this unknown qubit.

So this does not work.
But what does this mean?
Well, here we have an unknown qubit.
We have a state space of dimension 2.
Here, we have a state space dimension 4.
You can't take a state space of dimension 2 and turn it into--
and do anything to put it in a state space of dimension 4.
We need to start with the state space of dimension 4.
So how do we do that?
Well, Psi 0.
So if we need another register, which
starts in some definite state.
So we take our unknown qubit, in say one qubit.
And another qubit, which is in a definite state.
And we'd like to turn it into two copies of the same qubit.
And I want to say we can't do that.
Well, we would need--

this goes to that.
And suppose you started this in another unknown state.
It would have to go to phi, phi.

But of course, it can't, because the no-cloning theorem
says that you can't do that.
And this says look at inner products.

So unitary transformations preserve inner products.
And I haven't proved that yet, but it's one line.
And I'll who do that after this thing.
So inner products are preserved.

So, need, well, we need to take the inner product of these two,
which was just psi, phi, 0, 0.
And we need that to be equal to psi, phi, psi, phi.

If psi, phi equals x, get--
well, this is one.
x equals x squared, which only works if x is 0 or 1.
So what does that say?
Well, if we had two distinguishable states,
we could clone them.
But that's-- now, if we have two distinguishable states,
we can measure which of these two states that they're in.
And then we can make as many copies as we want.
If we have two equal states, we can clone them.
And if we-- assuming we know what the state is,
we can make as many copies of it as we want.
But if we have two states, which we don't know,
and which are not orthogonal, then we can not clone them.

OK.
So why do unitary transformations
preserve inner products?

Recall, U is unitary if and only if U dagger equals U inverse.

So phi, psi get mapped by U to U phi U psi.

And the inner product here is just, say, phi, psi.

And the inner product here, well, we take this ket
and find the equivalent bra.
So we take the conjugate transpose.
And when we do that, it's phi U dagger.
And then this is U psi.

But remember, U dagger U is equal to U inverse U,
equals I. So this is phi times the identity,
psi, which equals phi psi.
So inner products are preserved by unitary transformations.
So this calculation was correct.