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Okay so, so far, we have in this lecture
we have, we have considered how the state

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of a quantum system evolves and they said
the way it evolves is, is through a

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rotation of the, of the Hilbert Space. A
unitary rotation which is of course, a

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linear transformation so now we are going
to study a very interesting consequence of

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this linearity which is called the
no-cloning theorem. What it says is that

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if I give you an unknown quantum state so
if I give you a qubit which is in an

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unknown state then you cannot make a copy
of it, you cannot create two qubits in

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exactly that same state. Okay, so before
we study this let's, let's first consider

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what happens if you have a two qubit
system? So, if you have a two qubit

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system, its state is well it's, it's a
unit vector in a four dimensional complex

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vector space. And so, the unitary
evolution axiom tells us that the way the

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state evolves is by a rotation of this
four dimensional complex vector space. So,

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it's given by some unitary transformation.
Which is now, u is now the four by four

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unitary matrix so it's a, it's a complex
matrix which, which is, which has, which

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is four by four so it has sixteen complex
numbers in it. And the fact that it's

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unitary is given by the fact that u, u
dagger is the same as u dagger u is the

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identity. And of course u is, Is, is a
linear transformation which means that if

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u of psi one = to phi one and u of psi two
= to phi two then u of psi one + psi two.

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Okay, so let's, let's write it like this
so suppose psi = to psi one plus psi two

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then u psi = to. So if you apply, if the
state psi is the sum of two, psi one and

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psi two then when you apply u to psi, you
get phi one plus phi two, okay? Added as

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vectors so that's, that's just linearity.
So now, let's, let's try to understand

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what it would mean to copy or clone a
quantum bit. So imagine now that you're,

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you're given an unknown quantum bit so
it's in some state psi which you don't

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happen to know. So let's say psi was a0 +
b1 where ab are complex numbers and you

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don't know what a and b are. Well, what
you want to do is, you want to somehow

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create a copy of this bit, so you start
with a clean qubit. It's in the state zero

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and now you want to do some transformation
to these two qubits so that you end up

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with both qubits in the state psi so
what's your output state? Well the output

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state should be well, (a(0) + b(1)) that's
the state of the first qubit and then

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(a(0) + b(1)) that's the state of the
second qubit. So output should be a^2(00)

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+ ab(01) + ba which is ab(10) + b^2(11).
So this is what our output should look

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like. Now let's look at a few cases. So
what happens if the input, what happens if

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psi = zero? What should the output look
like? Well, of course in this case, the

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output should be of course 00 because we
want each of the two qubits to be, to look

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like zero. On the other hand, if psi = one
then the output should be eleven. Right?

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Because we should copy this one over into
the other qubit. So now by linearity, when

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psi = to a(0) + b(1). What should the
output be? Well, the output should be

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a(00) + b(11) because it should be a
whatever it was in this case + B whatever

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it was in this case. And that's this so
now if we do have such a copying circuit

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then this must equal to this but how could
these two be equal? The only way they can

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be equal is if a b = zero but if a times b
is zero then either a = zero or b = zero.

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Meaning that our initial qubit must have
been either zero qubit or one qubit. Okay,

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so that says that if you achieve a copying
circuit. If you, if you can clone a

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quantum bit and if you can clone it both
in the case it happens to be in the zero

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state and in the one state then there is
no other state that you can possibly

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clone. Okay, the fact that you managed to
copy zero and one prevents you from

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copying any other state. Okay, so this is
you know, the no-cloning theorem which

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says that it's impossible to clone an
unknown quantum state. And in fact even

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says something much stronger what it says
is that if you, if you're unknown quantum

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state, if you don't know whether, you know
if there are only two possibilities per

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psi so you know that p si is one of these
two states. It's either in the state zero

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or it's in the state which is not
orthogonal to zero. It's some, it's some

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a(0) + b(1) where a and b are not equal to
zero. So if you know that psi is one of

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these two particular states where you know
a and b then you cannot even, you cannot

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clone this, this quantum state. So you
cannot make copies of, of psi. There's no

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way to transform psi and zero. There,
there's no way to transform the first

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qubit in state psi, second qubit in state
zero. There's no way to transform this

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into first qubit in state psi, second
qubit in state psi. That's a no-cloning