Localized states in monitored quantum walks

TL;DR

This paper investigates localized versus delocalized states in monitored quantum walks on finite graphs, using eigenvalue analysis and transition times to distinguish them, with implications for quantum state control.

Contribution

It introduces a constructive method linking energy levels and eigenvectors to monitored evolution, providing a practical criterion for identifying localized states.

Findings

Eigenvalues are distributed over the complex unit disk.

Transition probabilities decay rapidly in the quantum Zeno regime.

Localized states favor return to initial state, delocalized states favor transitions.

Abstract

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.