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Equality in some symplectic eigenvalue inequalities

Hemant K. Mishra

TL;DR

The paper investigates when equality is attained in the symplectic analogs of Weyl's, Lidskii's, and Schur–Horn inequalities for symplectic eigenvalues. It develops necessary and sufficient conditions using Williamson's theorem, symplectic eigenvector pairs, and invariant symplectic subspaces, establishing a clear parallel with classical Hermitian eigenvalue theory. For Weyl, equality requires a common normalized symplectic eigenvector of $A$, $B$, and $A+B$. For Lidskii, it provides several equivalent formulations, including a linearity in $t$ for $d_i(A+tB)$ and the existence of structured symplectic subspaces and $M(t)$. For Schur–Horn, equality in the weak majorization is characterized by an orthosymplectic conjugation $A=N(D\oplus D)N^T$. These results refine our understanding of when symplectic eigenvalue inequalities saturate and may impact areas such as continuous-variable quantum information and symplectic geometry.

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.

Equality in some symplectic eigenvalue inequalities

TL;DR

The paper investigates when equality is attained in the symplectic analogs of Weyl's, Lidskii's, and Schur–Horn inequalities for symplectic eigenvalues. It develops necessary and sufficient conditions using Williamson's theorem, symplectic eigenvector pairs, and invariant symplectic subspaces, establishing a clear parallel with classical Hermitian eigenvalue theory. For Weyl, equality requires a common normalized symplectic eigenvector of , , and . For Lidskii, it provides several equivalent formulations, including a linearity in for and the existence of structured symplectic subspaces and . For Schur–Horn, equality in the weak majorization is characterized by an orthosymplectic conjugation . These results refine our understanding of when symplectic eigenvalue inequalities saturate and may impact areas such as continuous-variable quantum information and symplectic geometry.

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.
Paper Structure (6 sections, 8 theorems, 49 equations)

This paper contains 6 sections, 8 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Equality in symplectic Weyl's inequalities
  4. Equality in symplectic Lidskii's inequalities
  5. Majorization in symplectic Schur--Horn weak supermajorization
  6. Summary

Key Result

Proposition 1

Let $A \in \mathbb{P}_{2n}(\mathbb{R})$ and let $d>0$ be a symplectic eigenvalue of $A$. Suppose $(u_1,v_1),\ldots, (u_k, v_k)$ are normalized symplectic eigenvector pairs of $A$ corresponding to the common symplectic eigenvalue $d$. Any non-zero pair of vectors $(u,v)$ of the form $u = \sum_{\ell=1

Theorems & Definitions (19)

  • Proposition 1
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Proposition 2
  • proof
  • Definition 1
  • Proposition 3
  • ...and 9 more