Entanglement-Assisted Timing Optimization for Discriminating Amplitude-Damping Dynamics

We analyze minimum-error discrimination of two qubit dynamical processes generated by phase-covariant amplitude-damping Lindbladians in a single-use scenario. The optimization involves both the input probe, possibly entangled with an isolated ancilla, and the interrogation time. For unassisted probes we obtain a closed expression for the optimal trace-norm distinguishability at fixed time, with distinct interior and boundary branches. For entanglement-assisted probes, the common phase covariance of the two channels reduces the diamond-norm optimization to a one-parameter Schmidt family. The resulting formula gives transparent sufficient conditions for fixed-time entanglement advantage and separates this local advantage from advantage after global optimization over time. We exhibit examples in which the best unassisted strategy is approached only asymptotically, whereas an entangled probe achieves a strictly smaller error probability at finite time.