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Majorization between symplectic spectra of positive semidefinite matrices

Temjensangba, Hemant K. Mishra, Niloy Paul

TL;DR

This work develops symplectic analogues of classical majorization–eigenvalue relations for real PSD matrices with symplectic kernels. It proves that $d(A) \prec d(B)$ implies $A$ lies in the convex hull of the symplectic orbit of $B$, and that a weak converse holds: membership in the hull implies $d(A) \prec^w d(B)$. The results are obtained by weaving together majorization theory, doubly stochastic/superstochastic connections, and symplectic structure via Williamson diagonalization, symplectic direct sums, and pinching, yielding several concrete corollaries for diagonal and block-structured cases. The findings extend Schur–Horn-type insights to the symplectic setting and have potential implications for bosonic Gaussian quantum information and Hamiltonian dynamics by clarifying how symplectic spectra constrain operator orbits.

Abstract

Given $2n \times 2n$ real symmetric positive semidefinite matrix $A$ with symplectic kernel, there exists a real $2n \times 2n$ \emph{symplectic matrix} $M$ such that $M^TAM= D \oplus D$, where $D$ is an $n \times n$ non-negative diagonal matrix which is unique up to permutation of its diagonal entries. The diagonal entries of $D$ are called the \emph{symplectic eigenvalues} or symplectic spectrum of $A$. In this work, we investigate some majorization and weak supermajorization relations between the symplectic spectra of two positive semidefinite matrices. More explicitly, suppose $A$ and $B$ are $2n \times 2n$ real symmetric positive semidefinite matrices with symplectic kernels. We show that if the symplectic spectrum of $A$ is majorized by the symplectic spectrum of $B$, then $A$ lies in the convex hull of the symplectic orbit of $B$. We also establish that only a weak converse of this statement holds; i.e., if $A$ lies in the convex hull of the symplectic orbit of $B$ then the symplectic spectrum of $A$ is \emph{weakly supermajorized} by the symplectic spectrum of $B$. Several consequences of our results are also presented. Our methods make use of well-known connections between the theory of majorization, doubly stochastic, doubly superstochastic, and symplectic matrices.

Majorization between symplectic spectra of positive semidefinite matrices

TL;DR

This work develops symplectic analogues of classical majorization–eigenvalue relations for real PSD matrices with symplectic kernels. It proves that implies lies in the convex hull of the symplectic orbit of , and that a weak converse holds: membership in the hull implies . The results are obtained by weaving together majorization theory, doubly stochastic/superstochastic connections, and symplectic structure via Williamson diagonalization, symplectic direct sums, and pinching, yielding several concrete corollaries for diagonal and block-structured cases. The findings extend Schur–Horn-type insights to the symplectic setting and have potential implications for bosonic Gaussian quantum information and Hamiltonian dynamics by clarifying how symplectic spectra constrain operator orbits.

Abstract

Given real symmetric positive semidefinite matrix with symplectic kernel, there exists a real \emph{symplectic matrix} such that , where is an non-negative diagonal matrix which is unique up to permutation of its diagonal entries. The diagonal entries of are called the \emph{symplectic eigenvalues} or symplectic spectrum of . In this work, we investigate some majorization and weak supermajorization relations between the symplectic spectra of two positive semidefinite matrices. More explicitly, suppose and are real symmetric positive semidefinite matrices with symplectic kernels. We show that if the symplectic spectrum of is majorized by the symplectic spectrum of , then lies in the convex hull of the symplectic orbit of . We also establish that only a weak converse of this statement holds; i.e., if lies in the convex hull of the symplectic orbit of then the symplectic spectrum of is \emph{weakly supermajorized} by the symplectic spectrum of . Several consequences of our results are also presented. Our methods make use of well-known connections between the theory of majorization, doubly stochastic, doubly superstochastic, and symplectic matrices.
Paper Structure (13 sections, 6 theorems, 38 equations)

This paper contains 13 sections, 6 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Relevant concepts from majorization theory
  4. Certain vectors associated with a positive definite matrix
  5. Symplectic direct sum and symplectic pinching
  6. Results
  7. Proofs
  8. Proof of Theorem \ref{['thm:maj_implies_convex_symplectic']}.
  9. Proof of Theorem \ref{['thm:modifiedversion']}.
  10. Proof of Theorem \ref{['thm:theorem2']}.
  11. Proof of Corollary \ref{['thm:firstcorollary']}.
  12. Proof of Corollary \ref{['thm:secondcorollary']}.
  13. Proof of Corollary \ref{['thm:thirdcorollary']}.

Key Result

Theorem 3.1

Let $A, B \in \operatorname{Psd}(2n)$ with symplectic kernels. If $d(A) \prec d(B)$, then there exist a family of symplectic matrices $M_\pi$ for $\pi \in \operatorname{S}_n$ and a probability vector $( p(\pi) )_{\pi \in S_n}$ such that

Theorems & Definitions (7)

  • Theorem 3.1
  • Theorem 3.2
  • Example 3.3
  • Theorem 3.4
  • Corollary 3.5
  • Corollary 3.6
  • Corollary 3.7