On symplectic eigenvalues of positive definite matrices

Abstract

If $A$ is a $2n\times 2n$ real positive definite matrix, then there exists a symplectic matrix $M$ such that $M^{T}AM=\left[\begin{array}{cc}D& O\\ O& D\end{array}\right]$ where $D= \diag (d_1 (A), \ldots, d_n(A))$ is a diagonal matrix with positive diagonal entries, which are called the symplectic eigenvalues of $A.$ In this paper we derive several fundamental inequalities about these numbers. Among them are relations between the symplectic eigenvalues of $A$ and those of $A^t,$ between the symplectic eigenvalues of $m$ matrices $A_1,\dots ,A_m$ and of their Riemannian mean, a perturbation theorem, some variational principles, and some inequalities between the symplectic and ordinary eigenvalues.

References in corpus (3)

• The Real Symplectic Groups in Quantum Mechanics and Optics
• Gaussian quantum marginal problem
• Additivity and multiplicativity properties of some Gaussian channels for Gaussian inputs

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