Quantum walks: the first detected transition time

TL;DR

This paper derives a general formula for the mean first detection time in quantum walks on graphs, revealing critical behavior and divergence near phase transition points, with implications for quantum measurement theory.

Contribution

It introduces a new analytical approach to calculate mean detection times in quantum walks, especially near critical parameters where divergence occurs.

Findings

Derived a formula for mean first detected transition time.

Identified critical parameters causing divergence of detection time.

Established a relation between mean transition time and fluctuations near criticality.

Abstract

We consider the quantum first detection problem for a particle evolving on a graph under repeated projective measurements with fixed rate $1/\tau$. A general formula for the mean first detected transition time is obtained for a quantum walk in a finite-dimensional Hilbert space where the initial state $|\psi_{\rm in}\rangle$ of the walker is orthogonal to the detected state $|\psi_{\rm d}\rangle$. We focus on diverging mean transition times, where the total detection probability exhibits a discontinuous drop of its value, by mapping the problem onto a theory of fields of classical charges located on the unit disk. Close to the critical parameter of the model, which exhibits a blow-up of the mean transition time, we get simple expressions for the mean transition time. Using previous results on the fluctuations of the return time, corresponding to $|\psi_{\rm in}\rangle = |\psi_{\rm d}\rangle$, we find close to these critical parameters that the mean transition time is proportional to the fluctuations of the return time, an expression reminiscent of the Einstein relation.