Matrix decompositions in Quantum Optics: Takagi/Autonne, Bloch-Messiah/Euler, Iwasawa, and Williamson

TL;DR

This paper reviews four key matrix decompositions used in quantum optics, providing explicit constructions and implementation details for each, facilitating their practical use in quantum information processing.

Contribution

It offers explicit algorithms and implementation guidance for Takagi/Autonne, Bloch-Messiah/Euler, Iwasawa, and Williamson decompositions in quantum optics.

Findings

Provides explicit construction methods for each decomposition.

Demonstrates implementation using standard linear algebra packages.

Facilitates practical applications in quantum optics and quantum information.

Abstract

In this note we summarize four important matrix decompositions commonly used in quantum optics, namely the Takagi/Autonne, Bloch-Messiah/Euler, Iwasawa, and Williamson decompositions. The first two of these decompositions are specialized versions of the singular-value decomposition when applied to symmetric or symplectic matrices. The third factors any symplectic matrix in a unique way in terms of matrices that belong to different subgroups of the symplectic group. The last one instead gives the symplectic diagonalization of real, positive definite matrices of even size. While proofs of the existence of these decompositions exist in the literature, we focus on providing explicit constructions to implement these decompositions using standard linear algebra packages and functionalities such as singular-value, polar, Schur and QR decompositions, and matrix square roots and inverses.