Remote preparation in conjugate bases ($({}_A⟨a|\otimes I_B)|{\Phi }_{+}⟩=|a^{\ast }{⟩}_B/\sqrt{2}$)

A projective measurement on one half of a Bell state prepares the complex-conjugate basis on the other half, conditional on the outcome. Measuring Alice in the columns of $U^{\ast }$ prepares Bob in the columns of $U$. Ignoring the equally likely outcomes leaves Bob's reduced density matrix equal to $I/2$ in either basis.

Suppose two orthonormal qubit bases have disjoint physical rays. Alice chooses which conjugate basis to measure on her half of a Bell state. A device that perfectly labels Bob's individual member of either basis would reveal Alice's choice without her outcome being communicated. Spacelike operation would therefore violate relativistic causality, even though ordinary projective measurements see the same reduced density matrix $I/2$.

With a fixed reference $|0⟩$, exact overlap readout on Bob's remotely prepared state distinguishes the computational basis from the diagonal basis in one use, by the displayed disjoint squared-overlap sets. This would reveal Alice's spacelike basis choice. The argument rules out even deterministic exact overlap magnitude, avoiding the additional unphysical global-phase dependence of a raw complex inner product.