Optimal detection of quantum states via projective measurements

Abstract We consider the quantum dynamical evolution of a single particle hopping on a graph of N nodes with a Hamiltonian H = ∑ x , y = 1 N | x ⟩ ⟨ y | that allows it to hop between any pair of sites with equal amplitudes. We try to detect the particle in a subset of sites A applying projective measurements at random times. We study analytically the distribution of the time at which the detection of the particle in A is successful for the first time. Exact analytical expressions are obtained both for the mean first detection time as well as its full distribution as functionals of the initial state | Ψ 0 ⟩ and the measurement rate r . In particular, we elucidate the role of the presence of dark components in the initial state that affect the statistics of the first detection time. Exact enumeration agrees perfectly with our analytical predictions.