Okay so, so far, we have in this lecture we have, we have considered how the state
of a quantum system evolves and they said the way it evolves is, is through a
rotation of the, of the Hilbert Space. A unitary rotation which is of course, a
linear transformation so now we are going to study a very interesting consequence of
this linearity which is called the no-cloning theorem. What it says is that
if I give you an unknown quantum state so if I give you a qubit which is in an
unknown state then you cannot make a copy of it, you cannot create two qubits in
exactly that same state. Okay, so before we study this let's, let's first consider
what happens if you have a two qubit system? So, if you have a two qubit
system, its state is well it's, it's a unit vector in a four dimensional complex
vector space. And so, the unitary evolution axiom tells us that the way the
state evolves is by a rotation of this four dimensional complex vector space. So,
it's given by some unitary transformation. Which is now, u is now the four by four
unitary matrix so it's a, it's a complex matrix which, which is, which has, which
is four by four so it has sixteen complex numbers in it. And the fact that it's
unitary is given by the fact that u, u dagger is the same as u dagger u is the
identity. And of course u is, Is, is a linear transformation which means that if
u of psi one = to phi one and u of psi two = to phi two then u of psi one + psi two.
Okay, so let's, let's write it like this so suppose psi = to psi one plus psi two
then u psi = to. So if you apply, if the state psi is the sum of two, psi one and
psi two then when you apply u to psi, you get phi one plus phi two, okay? Added as
vectors so that's, that's just linearity. So now, let's, let's try to understand
what it would mean to copy or clone a quantum bit. So imagine now that you're,
you're given an unknown quantum bit so it's in some state psi which you don't
happen to know. So let's say psi was a0 + b1 where ab are complex numbers and you
don't know what a and b are. Well, what you want to do is, you want to somehow
create a copy of this bit, so you start with a clean qubit. It's in the state zero
and now you want to do some transformation to these two qubits so that you end up
with both qubits in the state psi so what's your output state? Well the output
state should be well, (a(0) + b(1)) that's the state of the first qubit and then
(a(0) + b(1)) that's the state of the second qubit. So output should be a^2(00)
+ ab(01) + ba which is ab(10) + b^2(11). So this is what our output should look
like. Now let's look at a few cases. So what happens if the input, what happens if
psi = zero? What should the output look like? Well, of course in this case, the
output should be of course 00 because we want each of the two qubits to be, to look
like zero. On the other hand, if psi = one then the output should be eleven. Right?
Because we should copy this one over into the other qubit. So now by linearity, when
psi = to a(0) + b(1). What should the output be? Well, the output should be
a(00) + b(11) because it should be a whatever it was in this case + B whatever
it was in this case. And that's this so now if we do have such a copying circuit
then this must equal to this but how could these two be equal? The only way they can
be equal is if a b = zero but if a times b is zero then either a = zero or b = zero.
Meaning that our initial qubit must have been either zero qubit or one qubit. Okay,
so that says that if you achieve a copying circuit. If you, if you can clone a
quantum bit and if you can clone it both in the case it happens to be in the zero
state and in the one state then there is no other state that you can possibly
clone. Okay, the fact that you managed to copy zero and one prevents you from
copying any other state. Okay, so this is you know, the no-cloning theorem which
says that it's impossible to clone an unknown quantum state. And in fact even
says something much stronger what it says is that if you, if you're unknown quantum
state, if you don't know whether, you know if there are only two possibilities per
psi so you know that p si is one of these two states. It's either in the state zero
or it's in the state which is not orthogonal to zero. It's some, it's some
a(0) + b(1) where a and b are not equal to zero. So if you know that psi is one of
these two particular states where you know a and b then you cannot even, you cannot
clone this, this quantum state. So you cannot make copies of, of psi. There's no
way to transform psi and zero. There, there's no way to transform the first
qubit in state psi, second qubit in state zero. There's no way to transform this
into first qubit in state psi, second qubit in state psi. That's a no-cloning