# Open quantum random walks on the half-line: the Karlin-McGregor formula, path counting and Foster's Theorem

## TL;DR

This paper develops a matrix-based approach to analyze open quantum random walks on the half-line, deriving transition probabilities, path counting methods, and an open quantum analogue of Foster's Theorem, with applications to boundary behaviors.

## Contribution

It introduces a matrix Karlin-McGregor formula and non-commutative gambler's ruin analysis for open quantum walks, expanding classical stochastic tools into the quantum domain.

## Key findings

- Derived transition probability expressions using orthogonal matrix polynomials.
- Established a non-commutative gambler's ruin model with generating functions.
- Proposed an open quantum Foster's Theorem for expected return times.

## Abstract

In this work we consider open quantum random walks on the non-negative integers. By considering orthogonal matrix polynomials we are able to describe transition probability expressions for classes of walks via a matrix version of the Karlin-McGregor formula. We focus on absorbing boundary conditions and, for simpler classes of examples, we consider path counting and the corresponding combinatorial tools. A non-commutative version of the gambler's ruin is studied by obtaining the probability of reaching a certain fortune and the mean time to reach a fortune or ruin in terms of generating functions. In the case of the Hadamard coin, a counting technique for boundary restricted paths in a lattice is also presented. We discuss an open quantum version of Foster's Theorem for the expected return time together with applications.