I've been studying quantum mechanics recently, which is basically just a lot of linear algebra. There's a standard theorem called the no-cloning theorem (and related no-deleting theorem) that says you cannot clone/copy an arbitrary quantum state. I.e., there's no map from $\Psi \otimes e \rightarrow \Psi\otimes\Psi$ where $\Psi$ is some unknown quantum state (which is just a vector in a Hilbert space) and $e$ is a "substrate" that we want to copy $\Psi$ onto.
This no-cloning theorem is supposedly derived from the linearity of linear transformations in Hilbert space. Does this mean that linear transformations have something to do with information conservation? And hence non-linear functions are functions that create/destroy information? I realize I haven't defined "information" and what information means in a vector space...
*(edit): Rahul answered why in the quantum case information is preserved due to operators being unitary and hence norm-preserving, however, the motivation for this question runs deeper.
The reason I initially posed this question was because ordinary non-linear functions e.g. a polynomial like $f(x)=x^2=x*x$ appear to violate a no-cloning-like rule as well, namely that the function accepts one copy of the variable $x$ yet it gets duplicated in the function expression. Whereas a linear function only uses its input argument exactly once. Is the connection to quantum mechanics' no-cloning theorem more than skin deep?