Quantum entanglement is a basic resource of quantum theory, but its usual definition assumes a fixed decomposition into subsystems. We develop an algebro--geometric framework to address quantum entanglement. For the Severi--Brauer scheme
SB(A)→X of an Azumaya algebra and a factorization type
d, we prove that a global locus of product states exists exactly when the associated torsor under
PGLn reduces to the stabilizer
Gd of the Segre variety. Thus entanglement in families is measured by a geometric obstruction to global subsystem structure.
Beyond this reduction criterion, we identify the cohomology that carries computable entangling obstructions. After determinant and incidence data
L⊂R⊂K are chosen, the relevant classes lie in the mapping cone relative cohomology
E∙(BL→BR). Their absolute images on
X, modulo classes induced from
BK, give determinant entangling obstructions beyond the Brauer class. We construct the moduli of subsystem structures as the quotient
P/Gd and realize it as a locally closed locus in the relative Hilbert scheme, with a compactification by degenerations of product state loci. In the bipartite case, a chosen subsystem structure globalizes the Schmidt rank stratification to a flat filtration with base change compatible resolutions and fiberwise constant numerical invariants. Examples show that reducibility can depend on the underlying torsor, not only on the Brauer class, with an interpretation by entangling monodromy.
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.
In a previous paper with Adam Brandenburger, we used sheaf theory to analyze the structure of non-locality and contextuality. Moreover, on the basis of this formulation, we showed that the phenomena of non-locality and contextuality can be characterized precisely in terms of obstructions to the existence of global sections. Our aim in the present work is to build on these results, and to use the powerful tools of sheaf cohomology to study the structure of non-locality and contextuality. We use the Cech cohomology on an abelian presheaf derived from the support of a probabilistic model, viewed as a compatible family of distributions, in order to define a cohomological obstruction for the family as a certain cohomology class. This class vanishes if the family has a global section. Thus the non-vanishing of the obstruction provides a sufficient (but not necessary) condition for the model to be contextual. We show that for a number of salient examples, including PR boxes, GHZ states, the Peres-Mermin magic square, and the 18-vector configuration due to Cabello et al. giving a proof of the Kochen-Specker theorem in four dimensions, the obstruction does not vanish, thus yielding cohomological witnesses for contextuality.
Entanglement cohomology assigns a graded cohomology ring to a multipartite pure state, providing homological invariants that are stable under local unitaries and characterize inequivalent patterns of entanglement. In this work we derive exact expressions for the dimensions of these cohomology groups in two canonical entanglement classes, generalized GHZ and W states on an arbitrary number of parties and local Hilbert space dimensions, thus proving conjectures of arXiv:1901.02011. Using the additional structure of the Hodge star and wedge product operations, we propose two new classes of local unitary invariants: the spectrum of the natural Laplacian acting on entanglement
k-forms, and the intersection numbers obtained from wedge products of representatives for cohomology classes. We present numerical experiments which investigate these invariants in particular states, suggesting that they may provide useful quantities for describing multipartite entanglement.
Quantum entanglement is a fundamental resource in quantum information theory, yet its general characterization and quantification remain challenging, especially in multipartite systems. In this work we investigate entanglement from a geometric perspective, focusing on the Riemannian structure induced by the Fubini--Study metric on the projective Hilbert space of multi-qubit quantum states. By exploiting the local-unitary invariance of this metric, we derive the entanglement distance (ED), a geometric measure that quantifies entanglement as an obstruction to locally minimizing the sum of squared Fubini--Study distances generated by local operations.
We analyze the properties of ED for pure multi-qubit states and discuss its behavior under local operations and classical communication. In particular, we show that ED reproduces established entanglement measures in well-defined and restricted settings. For pure states of two qubits, ED reduces to an exact monotone function of the concurrence and, independently, to an explicit monotone function of the entropy of entanglement. These results provide a clear geometric interpretation of standard bipartite entanglement measures within the present framework, while highlighting the limitations of such correspondences beyond the two-qubit case.
This paper is an introduction to work motivated by the question "can multipartite entanglement be detected by homological algebra?" We introduce cochain complexes associated to multipartite density states whose cohomology detects factorizability. The
kth cohomology components of such cochain complexes produce tuples of
(k+1)-body operators that are non-locally correlated due to the non-factorizability of the state. Associated Poincare polynomials are invariant under local invertible linear transformations (automorphisms that decompose as tensor products). These complexes can be considered as a step toward realizing mutual information as an Euler characteristic. We motivate the definition of the "state index" associated to a multipartite state: a three-parameter function which is: invariant under local invertible transformations, well-behaved under tensor products of states, and interpolates between multipartite mutual information and the integer-valued Euler characteristics of our complexes. We compute cohomologies and state indices of multipartite W and GHZ states. The approach in this paper is directed toward practitioners of finite-dimensional quantum mechanics, although the machinery developed generalizes far beyond. Some results are applicable in infinite dimensions and should be generalizable to the context of quantum field theory. In order to compensate for the long length, a detailed summary is provided in the introduction section.