Localized states in monitored quantum walks

In this paper we study localized states in a monitored evolution on a finite graph and how they are distinguished from the delocalized states in terms of the transition probabilities and the mean transition times. Monitoring is performed by repeated projective measurements with respect to a single quantum state. Our constructive approach is based on a mapping from a set of energy levels and an eigenvector basis onto the monitored evolution matrix. The eigenvalues of the latter are distributed over the complex unit disk and the corresponding transition probabilities decay quickly in the quantum Zeno regime at frequent measurements. A localized basis favors the return to the initial state, while a delocalized basis favors transitions between different states. This provides a practical criterion to identify localized states by measuring the mean transition time.