= Solution
\b[No.] <Perfect discrimination of pure states requires orthogonality>, whereas
$$
\langle\phi_0|\phi_1\rangle=\frac14-\frac14+\frac12=\frac12\ne0.
$$
To see the obstruction directly for a <basic measurement>, let $\{|b_j\rangle\}$ be its <orthonormal basis>. Certain identification would require disjoint sets of possible outcome labels for two different preparations. For every $j$, at least one of $\langle b_j|\phi_0\rangle$ and $\langle b_j|\phi_1\rangle$ would therefore vanish. Completeness would then give
$$
\langle\phi_0|\phi_1\rangle=\sum_j\langle\phi_0|b_j\rangle\langle b_j|\phi_1\rangle=0,
$$
contradicting the computed overlap. Failure to distinguish just this pair already rules out certain identification of the three-state family.
Back to article page
