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Converse symplectic interlacing via low-rank singular value perturbations
Priyanka Grover, Hemant K. Mishra, Sarishti Singh, Temjensangba
TL;DR
The paper asks whether symplectic interlacing inequalities suffice to realize prescribed global and local spectra. It constructs a positive definite matrix with those spectra, showing the inequalities give a complete characterization of the corresponding Gaussian marginal spectra.
Problem
Whether symplectic interlacing inequalities suffice to realize prescribed positive global and local spectra is the question addressed.
Source: §1 · p. 2Method
Results
Takeaways & Limitations
The interlacing inequalities completely characterize possible relations between global and subsystem thermal-mode mean photon numbers in the stated Gaussian setting.
Source: §1 · pp. 2–3The stated construction assumes positive global and local symplectic spectra.
Source: §4 · p. 7
Abstract
from arXiv · showhide
The symplectic interlacing theorem states that the symplectic spectrum of a $2n \times 2n$ real symmetric positive definite matrix satisfies a $2k$-step interlacing relation with the symplectic spectrum of every $2(n-k) \times 2(n-k)$ $s$-principal submatrix. We prove the converse in this paper. More precisely, given a positive global symplectic spectrum of size $n$ and a positive local symplectic spectrum of size $n-k$ satisfying the $2k$-step interlacing inequalities, we construct a $2n \times 2n$ real symmetric positive definite matrix having the prescribed global symplectic spectrum and a $2(n-k) \times 2(n-k)$ $s$-principal submatrix of it having the prescribed local symplectic spectrum. Our main tool is a reduction of $s$-principal compression to a rank-$k$ singular value perturbation. In the context of the Gaussian quantum marginal problem, our result implies that the symplectic interlacing inequalities are not merely necessary conditions on Gaussian marginals, but are also the complete spectral compatibility conditions for the global-versus-local-mode marginal problem.
1 Introduction
The paper proves that every positive global and local symplectic spectrum satisfying the interlacing inequalities is realizable, completing the converse to the symplectic interlacing theorem. It reduces the construction to a restricted-rank singular-value reassignment problem and applies the result to Gaussian quantum marginals.
1 Introduction
- Symplectic eigenvalues are positive ordered values in Williamson’s diagonal form, with analogues of several familiar Hermitian-eigenvalue results.The cited analogues include variational principles, Weyl- and Lidskii-type inequalities, and Horn-type descriptions.Source: §1 · p. 1
- Krbek, Tyc, and Vlach established necessary symplectic interlacing inequalities and a stronger relation for passive Gaussian transformations, but left the general converse open.They reported numerical evidence for the converse when k = 1.Source: §1 · p. 2
- Every positive ordered global and local spectrum satisfying the interlacing inequalities is realized by a positive definite matrix and one of its s-principal submatrices.The realizing matrix can be chosen on a point-symplectic orbit.Source: §1 · p. 2
- The retained submatrix’s symplectic eigenvalues are exactly the nonzero singular values of an n × n defect matrix formed by a rank-k perturbation.This recasts the inverse problem as restricted-rank singular-value reassignment.Source: §1 · p. 2
- Every subsystem collection of mean photon numbers satisfying the symplectic interlacing inequalities can be realized with prescribed global mean photon numbers.The paper interprets symplectic eigenvalues as thermal-mode mean photon numbers, up to normalization.Source: §1 · pp. 2–3
2 Symplectic eigenvalues and s-principal submatrices
The section defines symplectic eigenvalues through Williamson’s theorem and specifies the paired-index principal submatrices used for symplectic interlacing. It also introduces a restricted symplectic orbit whose members retain the same symplectic spectrum and suffice to realize every allowed s-principal spectrum.
2 Symplectic eigenvalues and s-principal submatrices
- Williamson’s theorem represents a real symmetric positive definite matrix by a symplectic congruence to Γ ⊕ Γ, where Γ’s diagonal entries are its ordered symplectic eigenvalues.Equivalently, the eigenvalues of JA are ±iµ_j.Source: §2 · p. 3
- An s-principal submatrix retains the same index set in both coordinate blocks, so deleting a mode removes the corresponding index from each block.The index set is defined using standard basis vectors.Source: §2 · p. 3
- The point-symplectic orbit of Λ ⊕ Λ consists of matrices A_X indexed by invertible X, and every member has symplectic spectrum equal to the diagonal of Λ.These transformations do not mix the two n-dimensional coordinate sectors; the orbit suffices to realize every s-principal spectrum allowed by the interlacing inequalities.Source: §2 · p. 3
3 Symplectic spectrum of s-principal compression and singular values of low-rank perturbation
The section reduces the symplectic spectrum of an s-principal compression to the non-zero singular values of a low-rank perturbation. It also establishes a normalization that lets the perturbation data be realized by a change of coordinates.
Symplectic spectrum of s-principal compression and singular values of low-rank perturbation
- The compression spectrum can therefore be studied through singular values of a rank-k perturbation of the matrix associated with the given matrix.The section uses classical singular-value interlacing results to support this reduction.Source: §3 · pp. 4–5
- For a block-diagonal positive definite matrix P ⊕ Q, its symplectic eigenvalues are the singular values of P^1/2Q^1/2.They are also the positive square roots of the eigenvalues of PQ.Source: §3 · p. 4
- The non-zero singular values of the compression-associated matrix equal the symplectic eigenvalues of the retained s-principal submatrix, and its nullity is k.This correspondence turns the compression problem into a singular-value problem for a rank-k perturbation.Source: §3 · pp. 4–5
- The normalization is automatic for a low-rank update that creates exactly k zero singular values.The construction of X realizes the perturbation factors relative to Λ, enabling reconstruction of a point transformation.Source: §3 · pp. 5–6
4 The converse of the symplectic interlacing theorem
The paper proves the converse of symplectic interlacing: every positive global and local spectrum satisfying the interlacing inequalities is realized by a positive definite matrix and one of its s-principal submatrices. The construction can be chosen on a point-symplectic orbit.
The converse of the symplectic interlacing theorem
- The construction recasts the reversed spectra as a singular-value reassignment problem solvable by a rank-at-most-k update.The update has nullity k, so its rank is k and it can be factored into full-column-rank matrices for the realization step.Source: §4 · pp. 6–7
- Every positive global spectrum μ and positive local spectrum d satisfying the interlacing inequalities are realized by a positive definite matrix and an s-principal submatrix, respectively.The realizing matrix can always be chosen on a point-symplectic orbit.Source: §4 · p. 7
- The resulting perturbation’s non-zero singular values match the prescribed local symplectic spectrum, while the global symplectic spectrum remains μ.The construction uses X and the compression correspondence to realize the local spectrum.Source: §4 · p. 7
5 Application to the Gaussian quantum marginal problem
The converse theorem, combined with symplectic interlacing, gives an if-and-only-if criterion for global and subsystem symplectic spectra of Gaussian states. The criterion also describes compatible thermal-mode mean photon numbers.
Application to the Gaussian quantum marginal problem
- A Gaussian state with global spectrum μ and retained subsystem spectrum d exists if and only if the interlacing inequalities hold.For sufficiency, the constructed positive definite matrix is a physical covariance matrix when its global symplectic eigenvalues satisfy μ_j ≥ 1.Source: §5 · p. 8
- The criterion completely characterizes possible relations between global and subsystem thermal-mode mean photon numbers.The paper relates these numbers to symplectic eigenvalues by μ_j = 2N_j + 1 and d_j = 2N′_j + 1.Source: §5 · p. 8
- For k = 1, the paper gives an explicit way to prepare a global Gaussian state with prescribed global and local symplectic spectra.It applies the Gaussian unitary corresponding to X ⊕ X^-T and traces out the selected mode.Source: §5 · p. 8
AI declaration
The authors report that ChatGPT helped develop technical parts of the paper after they had identified the problem and made initial progress. They state that they independently verified the mathematical details and take responsibility for the manuscript.
AI declaration
- ChatGPT helped develop technical parts of the paper, including the idea of linking compression spectra to singular values of a rank-k perturbed matrix; the authors say they independently verified the details.The authors take full responsibility for the manuscript’s correctness, accuracy, and content.Source: AI declaration · p. 8