Symplectic analogs of polar decomposition and their applications to bosonic Gaussian channels

TL;DR

This paper develops symplectic analogs of polar decomposition for matrices and applies these results to analyze bosonic Gaussian channels using symplectic transformations.

Contribution

It introduces new symplectic matrix decompositions and demonstrates their application to the study of bosonic Gaussian channels.

Findings

Decomposition of matrices into Hamiltonian and anti-symplectic parts

Application to bosonic Gaussian channels analysis

Extension of polar decomposition concepts to symplectic matrices

Abstract

We obtain several analogs of real polar decomposition for even dimensional matrices. In particular, we decompose a non-degenerate matrix as a product of a Hamiltonian and an anti-symplectic matrix and under additional requirements we decompose a matrix as a skew-Hamiltonian and a symplectic matrix. We apply our results to study bosonic Gaussian channels up to inhomogeneous symplectic transforms.