Transition in Splitting Probabilities of Quantum Walks

We investigate the splitting probability of a monitored continuous-time quantum walk with two targets and show that, in stark contrast to a classical random walk, it exhibits a nonanalytic, phase-transition-like behavior controlled by the sampling time at the targets. For large systems and sampling times smaller than a critical value τ_c = 2π/ΔE, where ΔE is the energy bandwidth, the splitting probability is universal and equal to 1/2, independent of the initial condition and the sampling time. Above the critical sampling, a nonuniversal regime emerges in which the splitting probability deviates from 1/2 and develops a fluctuating pattern of pronounced peaks and dips dependent on both the sampling time and the initial condition. These results follow from a nontrivial mapping of the splitting problem onto a pair of single-target detection problems enabled by the superposition principle.