All operations on quantum states are unitary operations. We don't make the rules, this is just how nature seems to work. So if you want to define an operation that copies a qbit, it has to be a unitary operation. That unitary operation would look like this:

$U|\psi\rangle_A|0\rangle_B=|\psi\rangle_A|\psi\rangle_B$

So you have the qbit you want to copy, $|\psi\rangle_A$, and the qbit into which you want to copy it, $|0\rangle_B$. This is the most general way to write the copy operation, although any other way of writing it reaches the same conclusion: it cannot be done.

The reason for this is as follows. Consider your starting state:

$|\psi\rangle|0\rangle = \begin{bmatrix} \alpha \\ \beta \end{bmatrix} ⊗ \begin{bmatrix} 1 \\ 0 \end{bmatrix} = \begin{bmatrix} \alpha \\ 0 \\ \beta \\ 0 \end{bmatrix}$

And now consider your desired ending state:

$|\psi\rangle|\psi\rangle = \begin{bmatrix} \alpha \\ \beta \end{bmatrix} ⊗ \begin{bmatrix} \alpha \\ \beta \end{bmatrix} = \begin{bmatrix} \alpha^2 \\ \alpha\beta \\ \beta\alpha \\ \beta^2 \end{bmatrix}$

So you want to go from here to here:

$\begin{bmatrix} \alpha \\ 0 \\ \beta \\ 0 \end{bmatrix} \rightarrow \begin{bmatrix} \alpha^2 \\ \alpha\beta \\ \beta\alpha \\ \beta^2 \end{bmatrix}$

But see those exponents? They mean this is not a *linear* operation! And since we can only perform linear operations on quantum states, no operation exists which can take us from the first state to the second (other than an operation which itself uses the values of $\alpha$ and $\beta$). Thus, copying (cloning) is impossible when you don't know $\alpha$ or $\beta$.

As for why we don't just use a different unitary transformation for each copy, that would require us knowing the exact quantum state we want to copy. If we know the exact quantum state, we can just take a blank qbit and reconstruct the same quantum state on the qbit. Which is fine, but pretty useless considering the *reason* we want to be able to copy a quantum state is so we can find the value of the quantum state in the first place.

Classical bits can always be copied, as you discovered. Of course, we copy classical bits all the time in the real world (you're reading copied classical bits right now!).

“No quantum copier rule” is misleading. It should be “no linear quantum copier rule”. Nature is not limited by a vanilla non-relativistic quantum physics. It could be that the linear structure of the postulates is just an approximation. There are suggestions to add non-linear terms (https://www.sciencedirect.com/science/article/abs/pii/0003491689902765).

– facetus – 2020-05-31T21:45:03.040Thank you for explaining it from basics, I have some query (i)

`We don't make the rules, this is just how nature seems to work`

- how behavior of nature affecting it? (ii)`we can just take a blank qbit`

- what does it mean by blank qubit, $|0\rangle$? – tarit goswami – 2018-09-13T18:09:28.5372Nature is quantum. We are modeling quantum mechanics mathematically. In all experiments, quantum states change according to unitary operations. Thus, we model quantum state changes as unitary operators. A blank qbit is $|0\rangle$, yes. – ahelwer – 2018-09-13T18:47:39.347

Thank you, you mean "Nature is quantized" ? – tarit goswami – 2018-09-13T18:50:56.033

2Nature appears to work according to the rules of quantum mechanics as we understand them. – ahelwer – 2018-09-13T18:56:23.797