Quantum random walks on a beam splitter array

TL;DR

This paper introduces a matrix-based framework for analyzing quantum random walks in beam splitter arrays, enabling calculation of system responses and photon probability distributions.

Contribution

It provides a general matrix representation of beam splitter arrays using rotations in high-dimensional space, facilitating analysis of quantum walks.

Findings

Matrix representation allows systematic analysis of beam splitter arrays.

The approach enables calculation of photon probability distributions.

Provides a mathematical tool for quantum walk studies.

Abstract

The general matrix representation of a beam splitter array is presented. Each beam splitter has a transmission/reflection coefficient that determines the behavior of these individual devices and, in consequence, the whole system response. The general matrix representation of each beam splitter is given as rotations of a $2n-{th}$ dimensional space. With these operators, the matrix that describes the entire array and, consequently, the final probability distribution of an input photon state can be calculated.