Symplectic eigenvalue problem via trace minimization and Riemannian optimization

Abstract

We address the problem of computing the smallest symplectic eigenvalues and the corresponding eigenvectors of symmetric positive-definite matrices in the sense of Williamson's theorem. It is formulated as minimizing a trace cost function over the symplectic Stiefel manifold. We first investigate various theoretical aspects of this optimization problem such as characterizing the sets of critical points, saddle points, and global minimizers as well as proving that non-global local minimizers do not exist. Based on our recent results on constructing Riemannian structures on the symplectic Stiefel manifold and the associated optimization algorithms, we then propose solving the symplectic eigenvalue problem in the framework of Riemannian optimization. Moreover, a connection of the sought solution with the eigenvalues of a special class of Hamiltonian matrices is discussed. Numerical examples are presented.

arXiv comment24 pages, 2 figures

References in corpus (8)

• Congruences and Canonical Forms for a Positive Matrix: Application to the Schweinler-Wigner Extremum Principle
• On symplectic eigenvalues of positive definite matrices
• Gaussian quantum marginal problem
• Additivity and multiplicativity properties of some Gaussian channels for Gaussian inputs
• Riemannian Optimization on the Symplectic Stiefel Manifold
• High Performance Solution of Skew-symmetric Eigenvalue Problems with Applications in Solving the Bethe-Salpeter Eigenvalue Problem
• Optimization on the symplectic group
• Symplectic spaces and pairs of symmetric and nonsingular skew-symmetric matrices under congruence

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• Triplet pairing, orbital selectivity and correlations in Iron-based superconductors
• A Result About the Classification of Quantum Covariance Matrices Based on Their Eigenspectra
• Simultaneous symplectic spectral decomposition of positive semidefinite matrices
• A contour for the entanglement negativity of bosonic Gaussian states