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math.FA — Functional Analysis

Weak positivity of Schur forms under decomposable curvature

Contributed by Suvrit Sra

Griffiths's 1969 conjecture predicts positivity of Schur forms for Griffiths-positive vector bundles. Finski reduced its weak form to a sign question for the double mixed discriminant $\Phi_n(H)$ of a positive linear map. Du's rank-nine counterexample shows that $\Phi_n(H)\ge0$ fails in general. We prove $\Phi_n(H)\ge0$ for every decomposable map $H$ , a sum of a completely positive and a completely copositive map, using a quaternionic Gaussian determinant representation. Combined with Finski's incidence formula, this inequality gives weak nonnegativity of every Schur form under decomposable curvature, without splitting assumptions. Beyond the decomposable cone, nonnegativity of $\Phi_k(H_{V,U})$ on every square compression is equivalent to $\Phi_n(H+S)\ge0$ for every entanglement-breaking map $S$ . This class contains explicit indecomposable maps, but is nonconvex in every size $n\ge9$ . We prove the nonconvexity by expressing Du's map as an average of four members, and the obstruction persists when all compression inequalities are strict.

A mix of human-written and AI-generated textHuman understanding: some partsBreuer-Hall mapChern formGriffiths positivitySchur formdecomposable mapentanglement-breaking mapmixed discriminantpositive linear mapquantum permanent

Nov 02, 2026PDF ↗View work →

hexagon:2610.00719v1

math.OC — Optimization and Control

Lower bounds for the quantum permanent

Contributed by Suvrit Sra

Griffiths's 1969 conjecture on the positivity of Schur forms reduces to the positivity of the quantum permanent $\Phi_n$ on positive bistochastic maps. This conjecture was refuted by Du with a map for which $\Phi_9<0$ . Therefore, we ask: how negative can $\Phi_n$ be? We prove a lower bound $\Phi_n\ge-3n/(3n^2+20n+103)$ (for $n\ge 8$ ), and an asymptotically stronger bound $\Phi_n\ge-1/n+(3e^2-7-o(1))\log n/n^2$ . The key to our bounds is the orbital polynomial of the map, which turns $\Phi_n$ into a linear function of a distribution on partitions and positivity into linear constraints. We also construct explicit maps for $n=8, 9$ that improve on Du's example. Finally, if $d_*\in\{4,\ldots,8\}$ is the first dimension for which $\Phi_n<0$ , the minimum $\mu_n$ of $\Phi_n$ over positive bistochastic maps satisfies $-\mu_n=\Omega(n^{-d_*})$ ; we conjecture the matching order $-\mu_n=O(n^{-d_*})$ .

A mix of human-written and AI-generated textHuman understanding: some partsGriffiths positivitybistochasticlower boundspositive mapsquantum permanentsemidefinite programming

Nov 02, 2026PDF ↗View work →

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