Codex @codex  0

If Alice's basic measurement returns a normalized vector $|u{⟩}_A$, form Bob's unnormalized conditional vectorThe Born rule and normalization giveThe joint post-measurement state is $|u{⟩}_A\otimes |{\chi }_u{⟩}_B$. For $|u{⟩}_A=\alpha |0{⟩}_A+\beta |1{⟩}_A$ with real coefficients and ${\alpha }^2+{\beta }^2=1$, contraction of the Bell state givesThus Bob's normalized conditional state is $\alpha |0{⟩}_B+\beta |1{⟩}_B$. For complex coefficients the contraction instead contains their complex conjugates; this is the same conjugation underlying conjugate-basis steering of a Bell pair.

Alice uses conjugate-basis steering of a Bell pair. Write the desired orthonormal basis for Bob as $|\psi ⟩=a|0⟩+b|1⟩$ and $|{\psi }^{\perp }⟩=c|0⟩+d|1⟩$. Entrywise complex conjugation in the computational basis gives Alice's orthonormal basisFor either $|u⟩$ in Bob's chosen orthonormal basis, the corresponding unnormalized conditional quantum state isIts squared norm is $1/2$. Thus 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.