In a measurement-induced continuous-time quantum walk, we address the problem of detecting a particle in a subspace, instead of a fixed position. In this configuration, we develop an approach of bright and dark states based on the unit and vanishing detection probability respectively for a particle-detection in the subspace. Specifically, by employing the rank-nullity theorem, we determine several properties of dark and bright states in terms of energy spectrum of the Hamiltonian used for a quantum walk and the projectors applied to detect the subspace. We provide certain conditions on the position and the rank of the subspace to be detected, resulting in the unit total detection probability, which has broad implications for quantum computing. Further, we illustrate the forms of dark as well as bright states and the dependence of detection probability on the number of dark states by considering a cyclic graph with nearest-neighbor and next nearest-neighbor hopping. Moreover, we observe that the divergence in the average number of measurements for detecting a particle successfully in a subspace can be reduced by performing high rank projectors.
In a measurement-induced continuous-time quantum walk, we address the problem of detecting a particle in a subspace, instead of a fixed position. In this configuration, we develop an approach of bright and dark states based on the unit and vanishing detection probability respectively for a particle-detection in the subspace. Specifically, by employing the rank-nullity theorem, we determine several properties of dark and bright states in terms of energy spectrum of the Hamiltonian used for a quantum walk and the projectors applied to detect the subspace. We provide certain conditions on the position and the rank of the subspace to be detected, resulting in the unit total detection probability, which has broad implications for quantum computing. Further, we illustrate the forms of dark as well as bright states and the dependence of detection probability on the number of dark states by considering a cyclic graph with nearest-neighbor and next nearest-neighbor hopping. Moreover, we observe that the divergence in the average number of measurements for detecting a particle successfully in a subspace can be reduced by performing high rank projectors.
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.
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.
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.
Building upon the recently introduced particle interpretation of the double-slit experiment [Phys. Rev. Lett. 134, 133603 (2025)] which attributes interference phenomena to detector-coupled (bright) and detector-uncoupled (dark) states of light, we develop a continuous-mode extension of the bright- and dark-state framework. This extension addresses a conceptual distinction between interference and diffraction, that is, the transition from a finite set of discrete paths to a continuum of modes. Through the construction of a complete detector-oriented basis for single-slit diffraction, we demonstrate that the observed diffraction pattern arises from projection of the photon state onto a single bright mode by identifying the detectable and undetectable modes, with photons detected at intensity minima having zero probability, as they reside in modes spanning an infinite-dimensional dark subspace. Our approach thus provides a unified particle-based explanation of diffraction that connects quantum and classical wave optics, and reveals distinctive quantum signatures in higher-order correlations.
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.
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.