Conditional limit measure of one-dimensional quantum walk with absorbing sink

TL;DR

This paper analyzes a one-dimensional quantum walk with an absorbing sink at the origin, deriving a limit theorem for the conditioned probability distribution and exploring recurrence properties.

Contribution

It introduces a limit theorem for the conditioned distribution of a quantum walk with an absorbing sink, revealing a modified Konno density function that vanishes at the origin.

Findings

The conditioned probability distribution converges to a modified Konno density function.

The distribution vanishes at the origin due to the absorbing sink.

The Polya number and recurrence properties are characterized.

Abstract

We consider a two-state quantum walk on a line where after the first step an absorbing sink is placed at the origin. The probability of finding the walker at position $j$, conditioned on that it has not returned to the origin, is investigated in the asymptotic limit. We prove a limit theorem for the conditional probability distribution and show that it is given by the Konno's density function modified by a pre-factor ensuring that the distribution vanishes at the origin. In addition, we discuss the relation to the problem of recurrence of a quantum walk and determine the Polya number. Our approach is based on path counting and stationary phase approximation.