Quantum Entanglement Geometry on Severi-Brauer Schemes: Subsystem Reductions of Azumaya Algebras

Abstract

Quantum entanglement is a basic resource of quantum theory, but its usual definition assumes a fixed decomposition into subsystems. We develop an algebro--geometric framework to address quantum entanglement. For the Severi--Brauer scheme $SB(\mathcal{A})\to X$ of an Azumaya algebra and a factorization type $\mathbf{d}$, we prove that a global locus of product states exists exactly when the associated torsor under $\mathrm{PGL}_n$ reduces to the stabilizer $G_{\mathbf d}$ of the Segre variety. Thus entanglement in families is measured by a geometric obstruction to global subsystem structure. Beyond this reduction criterion, we identify the cohomology that carries computable entangling obstructions. After determinant and incidence data $L\subset R\subset K$ are chosen, the relevant classes lie in the mapping cone relative cohomology $E^\bullet(BL\to BR)$. Their absolute images on $X$, modulo classes induced from $BK$, give determinant entangling obstructions beyond the Brauer class. We construct the moduli of subsystem structures as the quotient $P/G_{\mathbf d}$ and realize it as a locally closed locus in the relative Hilbert scheme, with a compactification by degenerations of product state loci. In the bipartite case, a chosen subsystem structure globalizes the Schmidt rank stratification to a flat filtration with base change compatible resolutions and fiberwise constant numerical invariants. Examples show that reducibility can depend on the underlying torsor, not only on the Brauer class, with an interpretation by entangling monodromy.