= Solution
The <no-cloning theorem for two pure states> states that no single <unitary operator> can clone two distinct nonorthogonal states. Suppose, to the contrary, that
$$
U|\alpha_j\rangle|0\rangle
=|\alpha_j\rangle|\alpha_j\rangle,
\qquad j=0,1.
$$
Preservation of the <inner product> by $U$ gives
$$
\langle\alpha_0|\alpha_1\rangle
=\langle\alpha_0|\alpha_1\rangle^2.
$$
Writing $z=\langle\alpha_0|\alpha_1\rangle$, this requires $z=0$ or $z=1$. The first case is orthogonality; the second means that the normalized states represent the same ray. Both contradict the assumptions. Therefore \b[distinct nonorthogonal quantum states cannot be cloned by a unitary process].
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