Equality in some symplectic eigenvalue inequalities

Abstract

In the last decade, numerous works have investigated several properties of symplectic eigenvalues. Remarkably, the results on symplectic eigenvalues have been found to be analogous to those of eigenvalues of Hermitian matrices with appropriate interpretations. In particular, symplectic analogs of famous eigenvalue inequalities are known today such as Weyl's inequalities, Lidskii's inequalities, and Schur--Horn majorization inequalities. In this paper, we provide necessary and sufficient conditions for equality in the symplectic analogs of the aforementioned inequalities. The equality conditions for the symplectic Weyl's and Lidskii's inequalities turn out to be analogous to the known equality conditions for eigenvalues.

arXiv comment19 pages, 41 references, some new results are added/updated

References in corpus (13)

• Extremal entanglement and mixedness in continuous variable systems
• The Real Symplectic Groups in Quantum Mechanics and Optics
• Congruences and Canonical Forms for a Positive Matrix: Application to the Schweinler-Wigner Extremum Principle
• On symplectic eigenvalues of positive definite matrices
• Gaussian relative entropy of entanglement
• Symplectic eigenvalue problem via trace minimization and Riemannian optimization
• Derivatives of symplectic eigenvalues and a Lidskii type theorem
• Entanglement dynamics of coupled quantum oscillators in independent nonMarkovian baths
• Real Normal Operators and Williamson's Normal Form
• First order sensitivity analysis of symplectic eigenvalues
• Block perturbation of symplectic matrices in Williamson's theorem
• Majorization in some symplectic weak supermajorizations
• An Order Relation between Eigenvalues and Symplectic Eigenvalues of a Class of Infinite-Dimensional Operators