Simple factorization of unitary transformations

TL;DR

This paper presents a recursive method to factorize any SU(n) matrix into simpler components using SU(2) transformations, applicable across various physical systems like optics, microwaves, and acoustics.

Contribution

It introduces a novel, efficient recursive factorization technique for SU(n) matrices based solely on linear relationships, broadening applications in different wave-based systems.

Findings

Efficient recursive factorization of SU(n) matrices.

Applicable to multiple physical systems including optics and microwaves.

Simplifies the implementation of complex unitary transformations.

Abstract

We demonstrate a method for general linear optical networks that allows one to factorize any SU($n$) matrix in terms of two SU($n-1)$ blocks coupled by an SU(2) entangling beam splitter. The process can be recursively continued in an efficient way, ending in a tidy arrangement of SU(2) transformations. The method hinges only on a linear relationship between input and output states, and can thus be applied to a variety of scenarios, such as microwaves, acoustics, and quantum fields.