Quantum Entanglement Geometry on Severi-Brauer Schemes: Subsystem Reductions of Azumaya Algebras
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866912848242802688 |
|---|---|
| author | Ikeda, Kazuki |
| author_facet | Ikeda, Kazuki |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_13764 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Quantum Entanglement Geometry on Severi-Brauer Schemes: Subsystem Reductions of Azumaya Algebras Ikeda, Kazuki Algebraic Geometry High Energy Physics - Theory Mathematical Physics Rings and Algebras Quantum Physics 14F22, 16K50, 14C05, 14M12, 14M15, 14L30, 81P40 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. |
| title | Quantum Entanglement Geometry on Severi-Brauer Schemes: Subsystem Reductions of Azumaya Algebras |
| topic | Algebraic Geometry High Energy Physics - Theory Mathematical Physics Rings and Algebras Quantum Physics 14F22, 16K50, 14C05, 14M12, 14M15, 14L30, 81P40 |
| url | https://arxiv.org/abs/2601.13764 |