Quantum Renewal Equation for the first detection time of a quantum walk

Abstract

We investigate the statistics of the first detected passage time of a quantum walk. The postulates of quantum theory, in particular the collapse of the wave function upon measurement, reveal an intimate connection between the wave function of a process free of measurements, i.e. the solution of the Schrödinger equation, and the statistics of first detection events on a site. For stroboscopic measurements a quantum renewal equation yields basic properties of quantum walks. For example, for a tight binding model on a ring we discover critical sampling times, diverging quantities such as the mean time for first detection, and an optimal detection rate. For a quantum walk on an infinite line the probability of first detection decays like $(\mbox{time})^{-3}$ with a superimposed oscillation, critical behavior for a specific choice of sampling time, and vanishing amplitude when the sampling time approaches zero due to the quantum Zeno effect.

References in corpus (6)

• Realization of quantum walks with negligible decoherence in waveguide lattices
• Experimental quantum-walk revival with a time-dependent coin
• Quantum walks with infinite hitting times
• Detection of a quantum particle on a lattice under repeated projective measurements
• Quantum walks on quotient graphs
• Recurrence properties of unbiased coined quantum walks on infinite $d$-dimensional lattices

Cited by in corpus (29)

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• Free Fermions with a Localized Source
• Dark states of quantum search cause imperfect detection
• Return to origin problem for particle on a one-dimensional lattice with quasi-Zeno dynamics
• Quantum walks: the first detected transition time
• Quantum unitary evolution interspersed with repeated non-unitary interactions at random times: The method of stochastic Liouville equation, and two examples of interactions in the context of a tight-binding chain
• Measurement induced quantum walks on an IBM Quantum Computer
• The first detection time of a quantum state under random probing
• The spectral dimension controls the decay of the quantum first detection probability
• Quantum random walk and tight-binding model subject to projective measurements at random times
• Tight-binding model subject to conditional resets at random times
• Non-Hermitian and Zeno limit of quantum systems under rapid measurements
• Randomly repeated measurements on quantum systems: Correlations and topological invariants of the quantum evolution
• Measurement-induced resetting in open quantum systems
• Causality, localization, and universality of monitored quantum walks with long-range hopping
• Thermodynamic phases in first detected return times of quantum many-body systems
• Interaction-induced transition in quantum many-body detection probability
• Quantized dynamics in closed quantum systems
• First-detection-time statistics in many-body quantum transport
• Entanglement of bosonic systems under monitored evolution
• Topological Defects from Quantum Reset Dynamics
• Mean First Passage Time of the Symmetric Noisy Voter Model with Arbitrary Initial and Boundary Conditions
• Repeated measurements and random scattering in quantum walks
• Designing non-equilibrium states of quantum matter through stochastic resetting
• Optimal detection of quantum states via projective measurements