Introduction to Quantum Entanglement Geometry

TL;DR

This paper explores the geometric and topological aspects of quantum entanglement in finite-dimensional systems, linking entanglement properties to global geometric structures and obstructions in the underlying fiber bundle framework.

Contribution

It introduces a geometric perspective on entanglement using Azumaya algebras and Severi-Brauer schemes, highlighting conditions for global subsystem decomposition and entanglement obstructions.

Findings

Entanglement relates to the Brauer class obstruction.

Holonomy can produce entangling quantum gates.

Entanglement reflects background geometric quantities.

Abstract

This article is an expository account aimed at viewing entanglement in finite-dimensional quantum many-body systems as a phenomenon of global geometry. While the mathematics of general quantum states has been studied extensively, this article focuses specifically on their entanglement. When a quantum system varies over a classical parameter space, each fiber may look like the same Hilbert space, yet there may be no global identification because of twisting in the gluing data. Describing this situation by an Azumaya algebra, one always obtains the family of pure-state spaces as a Severi-Brauer scheme. The main focus is to characterize the condition under which the subsystem decomposition required to define entanglement exists globally and compatibly, by a reduction to the stabilizer subgroup of the Segre variety, and to explain that the obstruction appears in the Brauer class. As a consequence, quantum states yield a natural filtration dictated by entanglement on the Severi-Brauer scheme. Using a spin system on a torus as an example, we show concretely that the holonomy of the gluing can produce an entangling quantum gate, and can appear as an obstruction class distinct from the usual Berry numbers or Chern numbers. For instance, even for quantum systems that have traditionally been regarded as having no topological band structure, the entanglement of their eigenstates can be related to global geometric universal quantities, reflecting the background geometry.