Entanglement and Optimal Timing in Discriminating Quantum Dynamical Processes

I address the problem of optimally discriminating between two open quantum dynamical processes. This problem is solved by identifying the optimal time at which two quantum channels, drawn from distinct sets of time-dependent channels, can be distinguished with minimal error probability, along with the optimal (potentially entangled) input state. I illustrate the richness of the solutions through an explicit study of Pauli dynamical maps and their corresponding sets of Pauli channels. Notably, I identify a scenario in which optimal discrimination without entanglement requires waiting indefinitely -- i.e., until the unknown process reaches its stationary state -- whereas the presence of entanglement allows for discrimination at a finite optimal time, along with a strict improvement in distinguishability.