Exact pure-state overlap readout would allow superluminal signalling ($|⟨0|\psi ⟩{|}^2\in \{0,1\}\text{or}\{1/2\}$)

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.