It is written in some places that the unitarity of time evolution is what prevents quantum cloning. However, consider the typical definition of a cloning operator $A$. For all $\left|\psi\right>$ and a standard state $\left|0\right>$, $$ A[\left|\psi\right>\otimes \left|0\right>] = \left|\psi\right>\otimes \left|\psi\right> $$ Without using the unitarity of $A$, I can follow the proof in these notes to demonstrate no-cloning. With a superposition state $\left|\chi\right> = a\left|\psi\right>+b\left|\phi\right>$, $A$ can be appled to find, $$ A[\left|\chi\right>\otimes \left|0\right>] = a(\left|\psi\right>\otimes \left|\psi\right>)+b(\left|\phi\right>\otimes\left|\phi\right>) $$ $A$ could also be applied to find, $$ A[\left|\chi\right>\otimes \left|0\right>] =\left|\chi\right>\otimes\left|\chi\right> = (a\left|\psi\right>+b\left|\phi\right>)\otimes(a\left|\psi\right>+b\left|\phi\right>) $$
These expressions are not equal, so there is a contradiction. This appears to be a proof that no-cloning is impossible for any linear time evolution, even in an alternate universe where quantum time evolution does not have to be unitary. Is this reasoning correct?