Introduction to Quantum Entanglement Geometry
Kazuki Ikeda
TL;DR
The article reframes quantum entanglement in finite-dimensional systems as a problem of global geometry over a parameter space, introducing twisted families of pure states that form Severi–Brauer schemes when a global vector bundle is absent. It develops an algebraic framework for entanglement by identifying product states with Segre varieties and expressing entanglement through tensor- and border-rank conditions via determinantal and secant varieties, while highlighting obstructions captured by Brauer classes to globally defining subsystem decompositions. The first half analyzes pure-state geometry, partitions, and GME criteria, and connects these to reductions of structure groups and Brauer obstructions; the second half lifts these notions to Severi–Brauer schemes, showing how global entanglement structure hinges on a reducible gauge group to the stabilizer G_d, with Hilbert schemes modularizing subsystem data. A concrete spin-torus example demonstrates holonomy that both entangles states and represents a Brauer-theoretic obstruction, illustrating the deep link between global geometric background and quantum information properties. Overall, the work offers a principled geometric and cohomological account of when and how entanglement structures can be defined and maintained across parameterized quantum systems, tying topological invariants to observable quantum correlations and state decompositions.
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.