Entanglement and optimal timing in discriminating quantum dynamical processes

I study the problem of optimally discriminating between two open quantum dynamical processes in a single-shot scenario, with the aim of minimizing the error probability of identification. This task requires jointly optimising the input state—potentially entangled with an ancillary system isolated from the dynamics—and the time at which the resulting time-dependent quantum channels, generated by the two dynamical maps, are most distinguishable. As an illustrative case, I focus on Pauli dynamical maps and their associated families of time-dependent Pauli channels. I identify a regime in which separable strategies require waiting indefinitely until the dynamics reaches the stationary state, whereas entangled input states enable optimal discrimination at a finite time, with a strict reduction in error probability. These results highlight the crucial interplay between entanglement and timing in enhancing the distinguishability of quantum dynamical processes.