Hitting time expressions for quantum channels: beyond the irreducible case and applications to unitary walks

TL;DR

This paper extends the concept of hitting times in quantum channels to reducible cases and arbitrary subspaces, using generalized inverses, with applications to unitary quantum walks and connections to matrix algebra.

Contribution

It introduces a framework for calculating mean hitting times for reducible quantum channels and arbitrary subspaces, broadening previous irreducible-focused approaches.

Findings

Generalized inverses enable hitting time calculations for reducible channels.

Applicable to unitary quantum walks and other positive trace-preserving maps.

Connects group inverse with quantum matrix algebra constructions.

Abstract

In this work we make use of generalized inverses associated with quantum channels acting on finite-dimensional Hilbert spaces, so that one may calculate the mean hitting time for a particle to reach a chosen goal subspace. The questions studied in this work are motivated by recent results on quantum dynamics on graphs, most particularly quantum Markov chains. We focus on describing how generalized inverses and hitting times can be obtained, with the main novelties of this work with respect to previous ones being that a) we are able to weaken the notion of irreducibility, so that reducible examples can be considered as well, and b) one may consider arbitrary arrival subspaces for general positive, trace preserving maps. Natural examples of reducible maps are given by unitary quantum walks. We also take the opportunity to explain how a more specific inverse, namely the group inverse, appears in our context, in connection with matrix algebraic constructions which may be of independent interest.