Measurement-guided Zeno transfer ($P_N={\cos }^{2N}(\pi /(2N))\to 1$)

Successively measure in bases whose first vector rotates from $|0⟩$ to $|1⟩$ in $N$ equal angular steps. The probability to remain on that branch throughout is the displayed product of squared overlaps. Without postselection, the final target probability is $[1+{\cos }^N(\pi /N)]/2$. Both approach one for an initial $|0⟩$, but a general input loses its relative phase between the two measurement branches; the process does not implement a coherent Pauli X gate on unknown inputs.