Analytic Benchmarks for Coherence-to-Entanglement Conversion under Post-Gate Noise in CNOT-Based Protocols

Coherence-to-entanglement conversion transforms single-qubit superposition into a practical two-qubit resource, but noise limits this process in near-term quantum hardware. We derive closed-form benchmarks for a minimal CNOT primitive in which a coherent qubit and an incoherent ancilla generate entanglement before undergoing phase damping, global depolarizing, amplitude damping, or independent local depolarizing noise. Using the $\ell_1$-norm of coherence and negativity, we prove the noiseless law $\mathcal{N}_0=C_{\ell_1}/2$, valid for arbitrary mixed inputs, and obtain exact negativities, survival fractions, and entanglement-sudden-death thresholds. For all $X$-state-preserving channels, a master relation shows that entanglement loss results from the competition between coherence suppression and partial-transpose spectral shifts. Phase damping yields $\eta=1-p$ without finite-noise sudden death; global depolarization gives coherence-dependent sudden death; amplitude damping adds an excited-population penalty and sudden death only for $\theta>\pi/4$; while local depolarization is most destructive at equal depolarizing strength. The initial survival slopes, $-1$, $-3/2$, $-2$, and $-3$, act as compact noise fingerprints. Since concurrence satisfies $C=2\mathcal{N}$ for the generated states, all robustness rankings remain unchanged. Mapping channel parameters to $T_1$, $T_\varphi$, and average gate fidelity connects the theory to hardware-level performance.