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

TL;DR

This paper introduces a geometric framework using Severi-Brauer schemes and Azumaya algebras to analyze quantum entanglement in families of pure states, revealing obstructions to subsystem decomposition due to geometric and Brauer-theoretic factors.

Contribution

It develops a novel geometric approach to study entanglement in twisted families of quantum states, connecting subsystem structures with algebraic geometry and Brauer theory.

Findings

Global product-state loci correspond to reductions of projective linear torsors.

Constructed the moduli space of subsystem structures as a torsor quotient.

Demonstrated that reducibility depends on torsor properties, not just Brauer class.

Abstract

Quantum entanglement is a defining signature and resource of quantum theory, but its standard definition presupposes a globally fixed decomposition into subsystems. We develop a geometric framework that detects when such a decomposition cannot be globalized for twisted families of pure-state spaces. Using Severi--Brauer schemes associated to Azumaya algebras over a base scheme, we study pure-state entanglement in families of projective state spaces that are locally trivial but globally twisted. For a given factorization type, we show that the existence of a global locus of product states is equivalent to a reduction of the underlying projective linear torsor to the stabilizer of the corresponding Segre variety, so entanglement in families becomes a geometric obstruction to globalizing subsystem structure. We construct the moduli space of subsystem structures, identify it with a natural torsor quotient, and realize it as a locally closed locus in the relative Hilbert scheme, yielding a canonical compactification by degenerations of product-state loci. In the bipartite case, once a subsystem structure is chosen, the Schmidt-rank stratification globalizes to a flat filtration with base-change compatible incidence resolutions and fiberwise constant numerical invariants. We complement this with Brauer-theoretic constraints and explicit examples showing that reducibility can depend on the underlying torsor rather than only on the Brauer class, with an interpretation via entangling monodromy.