Quantum Entanglement Geometry on Severi-Brauer Schemes: Subsystem Reductions of Azumaya Algebras
Kazuki Ikeda
TL;DR
The paper develops a geometric framework for entanglement in families of pure states modeled by Severi–Brauer schemes SB($\mathcal{A}$) over a base $X$, with Azumaya algebra $\mathcal{A}$ of degree $n$. It shows that a global product-state (Segre) locus of type $\mathbf d$ exists if and only if the associated $PGL_n$-torsor reduces to the stabilizer $G_{\mathbf d}$, introducing the notion of a $\mathbf d$-subsystem structure and the Brauer-admissible subset $\mathrm{Br}_{\mathbf d}(X)$. The authors construct a moduli Hilbert-scheme locus $\mathrm{Hilb}^{\Sigma_{\mathbf d}}(SB(\mathcal{A})/X) \cong P/G_{\mathbf d}$, develop a Schmidt-rank filtration in the twisted setting with relative singularities and resolutions, and provide explicit obstructions and monodromy interpretations illustrating how entanglement can arise from global twisting rather than local data alone. Examples include Kummer symbol obstructions, bipartite $(p,p)$-reductions, and a spin-chain toy model linking algebraic obstructions to physical entangling gates. The results yield computable, base-change-stable invariants and a canonical projective framework for entanglement in twisted quantum-state families.
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 $Σ_{\mathbf d}\times X$.