 Solution

Apply the hypothetical quantum cloning device repeatedly to obtain $N$ copies of the unknown state. If $c=|⟨\varphi |\psi ⟩|<1$, the two possibilities arewhose overlap has modulus $c^N$. For equal prior probabilities, the Helstrom-Holevo bound gives optimal success probabilityThus, for any $\epsilon >0$, sufficiently many clones permit success probability above $1-\epsilon$. Orthogonal states already have $c=0$ and can be distinguished perfectly. This is clone-assisted asymptotic state discrimination.

Here distinct states means distinct rays. Vectors differing only by a global phase represent the same quantum state and cannot be distinguished.