= Solution
Alice uses <conjugate-basis steering of a Bell pair>. Write the desired <orthonormal basis> for Bob as $|\psi\rangle=a|0\rangle+b|1\rangle$ and $|\psi^\perp\rangle=c|0\rangle+d|1\rangle$. Entrywise <complex conjugation> in the <computational basis> gives Alice's <orthonormal basis>
$$
|\overline\psi\rangle=\overline a|0\rangle+\overline b|1\rangle,
\qquad
|\overline{\psi^\perp}\rangle=\overline c|0\rangle+\overline d|1\rangle.
$$
For either $|u\rangle$ in Bob's chosen <orthonormal basis>, the corresponding unnormalized conditional <quantum state> is
$$
(\langle\overline u|_A\otimes I_B)|\phi_+\rangle_{AB}
=\frac1{\sqrt2}\sum_{j=0}^1\langle\overline u|j\rangle|j\rangle_B
=\frac{|u\rangle_B}{\sqrt2}.
$$
Its squared <norm> is $1/2$. Thus \b[Alice measures in the complex-conjugate basis], obtaining the required Bob <pure states> with <probability> $1/2$ each. Averaging over Alice's undisclosed outcome leaves Bob's <density matrix> equal to $I/2$, consistent with the <no-communication theorem>.
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