Symplectic decomposition from submatrix determinants

TL;DR

This paper introduces a new method to find the symplectic transformation that diagonalizes Gaussian state covariance matrices using submatrix determinants, simplifying the process in Gaussian quantum information.

Contribution

It presents a novel technique for deriving the diagonalising symplectic from submatrix determinants, inspired by eigenvector extraction methods for Hermitian matrices.

Findings

Provides a new analytical approach for symplectic diagonalization

Reduces computational complexity compared to existing methods

Potentially useful for Gaussian quantum information processing

Abstract

An important theorem in Gaussian quantum information tells us that we can diagonalise the covariance matrix of any Gaussian state via a symplectic transformation. Whilst the diagonal form is easy to find, the process for finding the diagonalising symplectic can be more difficult, and a common, existing method requires taking matrix powers, which can be demanding analytically. Inspired by a recently presented technique for finding the eigenvectors of a Hermitian matrix from certain submatrix eigenvalues, we derive a similar method for finding the diagonalising symplectic from certain submatrix determinants, which could prove useful in Gaussian quantum information.