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.
Highlights
A closed quantum system is prepared in some initial state and evolves unitarily over time
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 |ψin of the walker is orthogonal to the detected state |ψd
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
Summary
A closed quantum system is prepared in some initial state and evolves unitarily over time. This line of research was considered by Krovi and Brun [10,11,12], who showed that even for small systems the hitting time can be infinite These authors discussed the specific example of a quantum walk on a hypercube in detail and demonstrated that there is either a speed-up or a slow down in comparison with the classical random walk, depending on the conditions of the initial and the final state as well as on the evolution operator. We will show how the mean FDT time is related to the stationary points of a set of classical charges positioned on the unit circle in the complex plane with locations exp(iE jτ ), where E j is an energy level of the underlying Hamiltonian This charge picture was previously promoted in the context of the return problem [13], where the systems starts from an initial state and returns to this state after some time.
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