Shell game with a quantum pea

We examine the role of quantum measurement and coherence, considering the shell game with a quantum particle. The game consists of three sites arranged to form a ring geometry and a particle visiting the sites through a unitary evolution. A player selects one of the sites as a target site and wins if the target site contains the particle at a preassigned time. The player has two options for determining win or loss: local and global measurements, which perform projective measurements on the occupancy of the target site and the particle's position, respectively. The game can be played over multiple rounds, and each round begins with the state reached at the end of the previous round. We calculate the probability of winning at each round and the probability of consecutive losses and find that those probabilities have characteristic asymptotes. We obtain the analytic expressions of the asymptote and scrutinize their essential dependence on the measurement options and coherence in the initial ensemble.