We consider a quantum walk where a detector repeatedly probes the system with fixed rate 1/τ until the walker is detected.This is a quantum version of the first-passage problem.We focus on the total probability P det that the particle is eventually detected in some target state, for example, on a node r d on a graph, after an arbitrary number of detection attempts.Analyzing the dark and bright states for finite graphs and more generally for systems with a discrete spectrum, we provide an explicit formula for P det in terms of the energy eigenstates which is generically τ independent.We find that disorder in the underlying Hamiltonian renders perfect detection, P det = 1, and then expose the role of symmetry with respect to suboptimal detection.Specifically, we give a simple upper bound for P det that is controlled by the number of equivalent (with respect to the detection) states in the system.We also extend our results to infinite systems, for example, the detection probability of a quantum walk on a line, which is τ dependent and less than half, well below Polya's optimal detection for a classical random walk.
We consider a quantum walk where a detector repeatedly probes the system with fixed rate 1/τ until the walker is detected.This is a quantum version of the first-passage problem.We focus on the total probability P det that the particle is eventually detected in some target state, for example, on a node r d on a graph, after an arbitrary number of detection attempts.Analyzing the dark and bright states for finite graphs and more generally for systems with a discrete spectrum, we provide an explicit formula for P det in terms of the energy eigenstates which is generically τ independent.We find that disorder in the underlying Hamiltonian renders perfect detection, P det = 1, and then expose the role of symmetry with respect to suboptimal detection.Specifically, we give a simple upper bound for P det that is controlled by the number of equivalent (with respect to the detection) states in the system.We also extend our results to infinite systems, for example, the detection probability of a quantum walk on a line, which is τ dependent and less than half, well below Polya's optimal detection for a classical random walk.
We investigate a generic discrete quantum system prepared in state |ψ in under repeated detection attempts, aimed to find the particle in state |d , for example, a quantum walker on a finite graph searching for a node.For the corresponding classical random walk, the total detection probability P det is unity.Due to destructive interference, one may find initial states |ψ in with P det < 1.We first obtain an uncertainty relation which yields insight on this deviation from classical behavior, showing the relation between P det and energy fluctuations: P Var[ Ĥ ] d | d|[ Ĥ, D]|ψ in | 2 , where P = P det -| ψ in |d | 2 and D = |d d| is the measurement projector.Secondly, exploiting symmetry we show that P det 1/ν, where the integer ν is the number of states equivalent to the initial state.These bounds are compared with the exact solution for small systems, obtained from an analysis of the dark and bright subspaces, showing the usefulness of the approach.The upper bound works well even in large systems, and we show how to tighten the lower bound in this case.
A quantum walker on a graph, prepared in the state ∣ψin⟩, e.g. initially localized at node rin, is repeatedly probed, with fixed frequency 1/τ, to test its presence at some target node rd until the first successful detection. This is a quantum version of the first-passage problem. We investigate the total detection probability Pdet, i.e. the probability to eventually detect the particle after an arbitrary number of detection attempts. It is demonstrated that this total detection probability is less than unity in symmetric systems, where it is possible to find initial states which are shielded from the detector by destructive interference, so-called dark states. The identification of physically equivalent initial states yields an upper bound for Pdet in terms of the reciprocal of the number ν of physically equivalent states. The relevant subgroup of the system's symmetry operations is found to be the stabilizer of the detection state. Using this, we prove that all bright, i.e. surely detectable, states are symmetric with respect to the stabilizer. This implies that Pdet can be obtained from a diagonalization of the "symmetrized" Hamiltonian, instead of having to find all eigenstates of the Hamiltonian.
A classical random walker starting on a node of a finite graph will always reach any other node since the search is ergodic, namely it fully explores space, hence the arrival probability is unity. For quantum walks, destructive interference may induce effectively non-ergodic features in such search processes. Under repeated projective local measurements, made on a target state, the final detection of the system is not guaranteed since the Hilbert space is split into a bright subspace and an orthogonal dark one. Using this we find an uncertainty relation for the deviations of the detection probability from its classical counterpart, in terms of the energy fluctuations.
The first detection of a quantum particle on a graph is shown to depend sensitively on the distance ξ between the detector and initial location of the particle, and on the sampling time τ. Here, we use the recently introduced quantum renewal equation to investigate the statistics of first detection on an infinite line, using a tight-binding lattice Hamiltonian with nearest-neighbor hops. Universal features of the first detection probability are uncovered and simple limiting cases are analyzed. These include the large ξ limit, the small τ limit, and the power law decay with the attempt number of the detection probability over which quantum oscillations are superimposed. For large ξ the first detection probability assumes a scaling form and when the sampling time is equal to the inverse of the energy band width nonanalytical behaviors arise, accompanied by a transition in the statistics. The maximum total detection probability is found to occur for τ close to this transition point. When the initial location of the particle is far from the detection node we find that the total detection probability attains a finite value that is distance independent.
Even after decades of research, the problem of first passage time statistics for quantum dynamics remains a challenging topic of fundamental and practical importance. Using a projective measurement approach, with a sampling time τ, we obtain the statistics of first detection events for quantum dynamics on a lattice, with the detector located at the origin. A quantum renewal equation for a first detection wave function, in terms of which the first detection probability can be calculated, is derived. This formula gives the relation between first detection statistics and the solution of the corresponding Schrödinger equation in the absence of measurement. We illustrate our results with tight-binding quantum walk models. We examine a closed system, i.e., a ring, and reveal the intricate influence of the sampling time τ on the statistics of detection, discussing the quantum Zeno effect, half dark states, revivals, and optimal detection. The initial condition modifies the statistics of a quantum walk on a finite ring in surprising ways. In some cases, the average detection time is independent of the sampling time while in others the average exhibits multiple divergences as the sampling time is modified. For an unbounded one-dimensional quantum walk, the probability of first detection decays like (time)^{(-3)} with superimposed oscillations, with exceptional behavior when the sampling period τ times the tunneling rate γ is a multiple of π/2. The amplitude of the power-law decay is suppressed as τ→0 due to the Zeno effect. Our work, an extended version of our previously published paper, predicts rich physical behaviors compared with classical Brownian motion, for which the first passage probability density decays monotonically like (time)^{-3/2}, as elucidated by Schrödinger in 1915.
We consider a quantum system that is initially localized at ${\mathbit{x}}_{\text{in}}$ and that is repeatedly projectively probed with a fixed period $\ensuremath{\tau}$ at position ${\mathbit{x}}_{\mathrm{d}}$. We ask for the probability Fn that the system is detected at ${\mathbit{x}}_{\mathrm{d}}$ for the very first time, where n is the number of detection attempts. We relate the asymptotic decay and oscillations of Fn with the system's energy spectrum, which is assumed to be absolutely continuous. In particular, Fn is determined by the Hamiltonian's measurement spectral density of states (MSDOS) f(E) that is closely related to the density of energy states (DOS). We find that Fn decays like a power law whose exponent is determined by the power-law exponent dS of f(E) around its singularities E∗. Our findings are analogous to the classical first passage theory of random walks. In contrast to the classical case, the decay of Fn is accompanied by oscillations with frequencies that are determined by the singularities E∗. This gives rise to critical detection periods ${\ensuremath{\tau}}_{c}$ at which the oscillations disappear. In the ordinary case dS can be identified with the spectral dimension associated with the DOS. Furthermore, the singularities E∗ are the van Hove singularities of the DOS in this case. We find that the asymptotic statistics of Fn depend crucially on the initial and detection state and can be wildly different for out-of-the-ordinary states, which is in sharp contrast to the classical theory. The properties of the first-detection probabilities can alternatively be derived from the transition amplitudes. All our results are confirmed by numerical simulations of the tight-binding model, and of a free particle in continuous space both with a normal and with an anomalous dispersion relation. We provide explicit asymptotic formulas for the first-detection probability in these models.