No-cloning theorem for two pure states
= No-cloning theorem for two pure states
No <unitary operator> $U$ can satisfy
$$
U|c_j\rangle|0\rangle=|c_j\rangle|c_j\rangle,
\qquad j=0,1,
$$
for two distinct nonorthogonal pure states. Taking the inner product of these two equations would require
$$
\langle c_0|c_1\rangle
=\langle c_0|c_1\rangle^2,
$$
which is impossible when $0<|\langle c_0|c_1\rangle|<1$.