Uncertainty relation between detection probability and energy fluctuations

TL;DR

This paper establishes an uncertainty relation linking the deviation of quantum detection probability from classical expectations to energy fluctuations, highlighting quantum interference effects in search processes on graphs.

Contribution

It introduces a novel uncertainty relation connecting detection probability deviations with energy fluctuations in quantum walks, considering interference effects.

Findings

Detection probability deviations are bounded by energy fluctuations.

Quantum interference can cause non-ergodic search behavior.

The relation generalizes classical search probabilities to quantum scenarios.

Abstract

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 is fully exploring 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.