Effect of quantum mechanical global phase factor on error versus sensitivity limitation in quantum routing

In this paper, we explore the effect of the purely quantum mechanical global phase factor on the problem of controlling a ring-shaped quantum router to transfer its excitation from an initial spin to a specified target spin. "Quantum routing" on coherent spin networks is achieved by shaping the energy landscape with static bias control fields, which already results in the nonclassical feature of purely oscillatory closed-loop poles. However, more to the point, it is shown that the global phase factor requires a projective reinterpretation of the traditional tracking error where the wave function state is considered modulo its global phase factor. This results in a time-domain relaxation of the conflict between small tracking error and small sensitivity of the tracking error to structured uncertainties. While fundamentally quantum routing is achieved at a specific final time and hence calls for time-domain techniques, we also develop a projective s-domain limitation.