Phase information beyond entanglement sudden death in coherence-to-entanglement conversion under post-gate noise
An ideal CNOT maps the phase of a coherent qubit onto the coherence between $\ket{00}$ and $\ket{11}$ of a two-qubit state, producing an output that carries both entanglement and estimable phase information. We ask how post-gate noise degrades these two quantities, and find that they are not lost together. For the phase-encoded X states generated by the protocol, the negativity is a thresholded difference of the surviving coherence $z=fκ$ and a population penalty $g$, vanishing once $fκ\le g$, while the phase quantum Fisher information (QFI) is the smooth ratio $F_ϕ=4z^2/(a+b)$, which stays positive for any nonzero coherence. As a result there is an exact region of state space in which the output is separable but still phase-sensitive. We characterize this region, give the residual QFI $F_ϕ^\star=4g_\star^2/(1-2g_\star)$ at entanglement death, and show that channels reaching death at the same coordinate share this residual, with global and independent local depolarization forming one such class and $F_ϕ^\star=1/6$ at maximal input coherence. Four standard channels appear as trajectories through this common geometry, and asymmetric population transfer adds a third coordinate that changes the entanglement but leaves the QFI unchanged, which marks where the two-coordinate description applies. We identify a measurement that attains the bound and compare with a direct single-qubit probe, which is more precise under matched exposure; the results are therefore reference benchmarks for phase-information retention, not a claim of metrological advantage.