Uncertainty Relation between Detection Probability and Energy Fluctuations

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.

Automated summary

A classical random walker on a finite graph will always reach any other node, while quantum walks may exhibit non-ergodic features due to destructive interference. The final detection of the system is not guaranteed under repeated projective local measurements, as the Hilbert space is split into bright and dark subspaces. An uncertainty relation for detection probability deviations from the classical counterpart is found in terms of energy fluctuations.