Quantum Markov chains on the line: matrix orthogonal polynomials, spectral measures and their statistics

TL;DR

This paper extends classical spectral analysis techniques to quantum Markov chains using matrix orthogonal polynomials, providing tools for analyzing quantum walks and their statistical properties.

Contribution

It introduces a framework for spectral analysis of quantum Markov chains with matrix-valued orthogonal polynomials and explores their statistical measures, extending classical methods to quantum settings.

Findings

Development of spectral analysis methods for quantum walks

Calculation of recurrence and passage probabilities in quantum chains

Extension of classical results to non-symmetric weight matrices

Abstract

Inspired by the classical spectral analysis of birth-death chains using orthogonal polynomials, we study an analogous set of constructions in the context of open quantum dynamics and related walks. In such setting, block tridiagonal matrices and matrix-valued orthogonal polynomials are the natural objects to be considered. We recall the problems of the existence of a matrix of measures or weight matrix together with concrete calculations of basic statistics of the walk, such as site recurrence and first passage time probabilities, with these notions being defined in terms of a quantum trajectories formalism. The discussion concentrates on the models of quantum Markov chains, due to S. Gudder, and on the particular class of open quantum walks, due to S. Attal et al. The folding trick for birth-death chains on the integers is revisited in this setting together with applications of the matrix-valued Stieltjes transform associated with the measures, thus extending recent results on the subject. Finally, we consider the case of non-symmetric weight matrices and explore some examples.