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

• URL: http://arxiv.org/abs/2403.04596v2
• Date: Wed, 13 Mar 2024 15:55:37 GMT
• Title: Matrix decompositions in Quantum Optics: Takagi/Autonne, Bloch-Messiah/Euler, Iwasawa, and Williamson
• Authors: Martin Houde, Will McCutcheon, Nicol\'as Quesada
• Abstract summary: We present four important matrix decompositions commonly used in quantum optics. The first two of these decompositions are specialized versions of the singular-value decomposition. The third factors any symplectic matrix in a unique way in terms of matrices that belong to different subgroups of the symplectic group.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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