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

Abstract: We formulate pure-state entanglement in families as a geometric obstruction. In standard quantum information, entanglement is defined relative to a chosen tensor-product factorization of a fixed Hilbert space. In contrast, for a twisted family of pure-state spaces, which can be described by Azumaya algebras $A$ of degree $n$ on $X$ and their Severi-Brauer schemes [ SB(A)=P\times^{PGL_n}\mathbb{P}^{n-1}\to X, ] such a subsystem choice may fail to globalize. We formalize this algebro-geometrically: fixing a factorization type $\mathbf d=(d_1,\dots,d_s)$ with $n=\prod_i d_i$, the existence of a global product-state locus of type $\mathbf d$ is equivalent to a reduction of the underlying $PGL_n$-torsor $P\to X$ to the stabilizer $G_{\mathbf d}\subset PGL_n$. Thus, entanglement is the obstruction to the existence of a relative Segre subscheme inside $SB(A)$. Writing $Σ{\mathbf d}\subset \mathbb{P}^{n-1}$ for the Segre variety, we call a reduction to $G{\mathbf d}$ a $\mathbf d$-subsystem structure. Our first main result identifies the moduli of $\mathbf d$-subsystem structures with the quotient $P/G_{\mathbf d}$. Moreover, we realize naturally $P/G_{\mathbf d}$ as a locally closed subscheme of the relative Hilbert scheme, [ \text{Hilb}^{Σ_{\mathbf d}}!\bigl(SB(A)/X\bigr)\ \subset\ \text{Hilb}\bigl(SB(A)/X\bigr), ] parametrizing relative closed subschemes fppf-locally isomorphic to $Σ_{\mathbf d}\times X$.