Symplectic decomposition from submatrix determinants

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.

10 pages, supplementary files available at https://github.com/softquanta/symplectic_decomposition

References in corpus (7)

• Quantum Illumination with Gaussian States
• The Quantum Chernoff Bound
• Computable bounds for the discrimination of Gaussian states
• Correlation Matrices of Two-Mode Bosonic Systems
• Quantum Illumination with a generic Gaussian source
• Optimal measurements for quantum fidelity between Gaussian states and its relevance to quantum metrology
• Device-independent quantum reading and noise-assisted quantum transmitters