= Solution
A unitary cloner with blank state $|0\rangle$ would satisfy
$$
U(|\psi\rangle|0\rangle)=|\psi\rangle|\psi\rangle
$$
for every $|\psi\rangle\in S$. For $|\psi\rangle,|\phi\rangle\in S$, preservation of inner products gives
$$
\langle\psi|\phi\rangle
=\langle\psi|\phi\rangle^2.
$$
Hence every pair is either orthogonal or represents the same state. Conversely, an orthonormal set can be cloned by defining the map on its basis states and extending it to a unitary. Therefore the <Quantum cloning>[no-cloning theorem] says that a set of pure states is unitarily clonable exactly when its distinct members are mutually orthogonal.
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