Okay, hello.

We were just discussing some applications of entanglement and so the next application

I want to discuss is quantum teleportation.

So what is the setting?

The setting is you are given some unknown quantum state and you want to transfer it

from A to B but without physically transporting it.

You could always send a spin from A to B but you don't want to do this.

And so the question is what are your options?

So let's imagine I have some unknown quantum state of a single qubit and what would be

psi equals some arbitrary superposition of up and down and I want to transfer this without

actually physically transporting the state.

And so if this were a classical bit one option would be just to produce a copy of this bit

and then send along the copy so you can keep your state.

But this is not possible in quantum mechanics because in quantum mechanics it's actually

not allowed to do a direct copy of a quantum state and this is known as the no cloning

theorem which tells you that you can't just take psi and somehow turn it into psi product

state psi again.

Now of course this wouldn't work anyway because the Hilbert space here is smaller than the

Hilbert space there but you could imagine that for example here you take a product of an

arbitrary spin state with the state you want to copy and still it's not possible and the

easiest way to see this is that if you look into it and look at these coefficients on

the right hand side if you expand the coefficients would appear like alpha squared and beta squared

and so on.

On the left hand side they only appear linearly and since quantum mechanics is linear it cannot

possibly transfer a state where alpha and beta appear only linearly to something where

they appear quadratically.

And if you don't like that kind of argument you could also then just take the special

cases where say you take a copy of up where alpha equals one and beta equals zero and

you also take a copy of down where alpha equals zero and beta equals one and then because

of the linearity of quantum mechanics you automatically know what will happen if you

have such a superposition state and you won't get this kind of product of the state of this

copy.

Okay so cloning is not allowed.

What would be the other option?

Well classically it would always be possible just to take a look at the state of the bit

and then have some communication line to the other place and tell them how to recreate

that state.

But quantum mechanically again this won't work because the state is completely unknown

and if you want to do a measurement you only have one chance and you have to pick your

measurement axis and for example if you take the z axis you might get up but then what

do you know?

Was the state really purely up or was it some superposition of up and down like shown here

so you only have one chance so this doesn't work either.

So then what is the trick?

The trick goes under the name quantum teleportation so I'll tell you the ingredients of this so-called

teleportation quantum quantum.

So suppose you want to transfer your unknown quantum state from A to B without actually

physically transporting it so here we have your particle one that carries the unknown

quantum state psi and then what you need is first of all you need to have created a pair

of particles that is entangled and which is shared by A and B so let me just draw a line