No Cloning

The No cloning principle states that we cannot copy or clone an unknown qubit. This claim is proved by contradiction. Let U be the transformation that clones, i.e. $U(\vert a0\rangle) = \vert aa\rangle$. Let $\vert a\rangle$ and $\vert b\rangle$ be two orthonormal states. Consider $\vert c\rangle=\frac{1}{\sqrt{2}}(\vert a\rangle+\vert b\rangle)$. We have by linearity,

$$U(\vert c0\rangle) = U(\frac{1}{\sqrt{2}}(\vert a0\rangle+\vert b0\rangle))$$

$$= \frac{1}{\sqrt{2}}(\vert aa\rangle + \vert bb\rangle)$$

$$= \vert cc\rangle$$

But,

$$(\vert cc\rangle) = \vert c\rangle \otimes \vert c\rangle$$

$$= \frac{1}{2}(\vert aa\rangle+\vert ab\rangle+\vert ba\rangle+\vert bb\rangle)$$

which is not equal to what was obtained above.