Converse symplectic interlacing via low-rank singular value perturbations
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.