Geometric Entanglement Entropy

Updated 3 December 2025

Geometric entanglement entropy is a quantum measure that quantifies correlations based on the geometry of subsystems, applicable to lattice models and QFT.
It exhibits an area law with subleading corrections from topological, curvature, and boundary effects, revealing universal scaling behaviors.
It serves as a diagnostic tool for characterizing topological order and multipartite entanglement in complex quantum systems including Floquet and gauge theories.

Geometric entanglement entropy is a multifaceted quantum information quantity that encodes how quantum correlations depend on the geometrical structure of subsystems and the state space. It appears in diverse contexts spanning lattice models, quantum field theories, topological phases, and projective Hilbert space geometry. Its rigorous definitions, scaling laws, and universal corrections reveal deep relationships between quantum many-body physics, quantum geometry, and topology.

1. Definitions and Core Frameworks

The geometric entanglement entropy (GEE) can reference distinct but related notions across quantum information theory and statistical mechanics:

Geometric Entanglement (GE) of many-body pure states: For a normalized pure state $|\Psi\rangle$ partitioned into $m$ parties (spins, blocks, etc.), the geometric entanglement is

$E_G(\Psi) = -\log_2 \left( \max_{|\Phi\rangle} |\langle\Phi|\Psi\rangle|^2 \right)$

where $|\Phi\rangle$ ranges over fully factorized states (product states over blocks) (Orus et al., 2013, Stéphan et al., 2010).

Geometric entropy as vacuum entanglement entropy in QFT: In continuum systems, it is the von Neumann entropy associated with the reduced density matrix of a region, sensitive to how field degrees of freedom are geometrically partitioned. The operational definition via subalgebras of observables provides an alternative to spatial tensor products, preserving finite resolution and local support (Bianchi et al., 2019).
Geometric entanglement entropy on projective Hilbert space: For the manifold of pure states endowed with the Fubini-Study metric, the geometric entanglement entropy $S_\text{geo}(e)$ at entanglement value $e$ is the logarithm of the volume of the hypersurface of states with constant entanglement, reflecting the macroscopic degeneracy of that value in state space (Cairano, 26 Nov 2025).

In gauge theory, GEE is only well-defined if the physical Hilbert space admits a geometric tensor-product decomposition; this fails for pure non-Abelian gauge theories due to nonlocal constraints, and is restored with the inclusion of matter fields (Hategan-Marandiuc, 8 Apr 2024).

2. Area Laws, Scaling, and Geometric Corrections

The leading behavior of geometric entanglement entropy in spatially extended systems is typically governed by an area (boundary) law:

$S_A \sim b_\alpha L_\partial + o(L_\partial)$

where $L_\partial$ is the boundary length/area, $b_\alpha$ a non-universal coefficient dependent on Rényi index or model parameters (Ye et al., 2022, Bianchi et al., 2019, 0801.4564).

Subleading corrections—universal and geometric—encode rich physical information:

Topological contributions: In topologically ordered states (e.g., toric code, quantum double models), GEE decomposes into bulk (area-law) and universal topological terms:

$E_G = E_\text{bulk}(L) \times n_b - E_\gamma + o(1)$

where $E_\gamma$ is constant, independent of block size, and characterizes the global entanglement pattern (Orus et al., 2013).

Corner and edge corrections: Non-smooth boundaries introduce logarithmic terms with universal angle-dependent coefficients:

$S_\text{corner}(\alpha;\theta, L_\partial) = -a_\alpha(\theta)\ln(L_\partial/\delta) + \dots$

with $a_\alpha(\theta)$ universal, determined by the central charge and operator content of the underlying CFT (Ye et al., 2022, Rozon et al., 2019).

Boundary conditions: In critical chains (e.g., XXZ, Ising) and systems with conformal interfaces, the subleading constant is controlled by Affleck-Ludwig boundary entropy, tied to specific boundary conditions (Neumann, ferromagnetic, etc.) (Stéphan et al., 2010, Gutperle et al., 2015).
Curvature or geometry-dependent terms: For smooth surfaces, subleading corrections involve integrals over intrinsic/extrinsic curvature, e.g., in 4d CFTs Rényi entropy (Lee et al., 2014):

$S_n[\Sigma]_\text{univ} \sim f_a(n)\int_\Sigma E_2 + f_b(n)\int_\Sigma(\text{Tr}\,K^2 - \tfrac12 K^2) - f_c(n)\int_\Sigma C^{ab}_{ab}$

capturing topological (Euler), extrinsic curvature, and Weyl contributions.

3. Topological and Multipartite Entanglement

GEE serves as a robust signature of topological order, producing a universal "topological entanglement entropy" $E_\gamma$ that is stable under RG flows and local perturbations (Orus et al., 2013):

Toric code, double semion: $E_\gamma=1$
Color code: $E_\gamma=2$ (double toric code)
Quantum doubles D(G): For non-Abelian finite groups, $E_\gamma = \log_2 |G|$ with detailed bounding formalism (Orus et al., 2013).

These topological terms are invariant under entanglement RG, insensitive to non-relevant perturbations, and vanish only when the topological phase is destroyed.

Geometric entanglement is also connected to multipartite measures, such as the maximal overlap with product states over blocks, and in permutation-symmetric many-body states (spin- $s$ chains) is governed by the Hausdorff dimension (possibly fractal) of the support in projective space (Castro-Alvaredo et al., 2012):

$S(m) \sim \frac{d}{2} \ln m + S_\text{geo} + o(1)$

where $S_\text{geo}$ is the Shannon entropy of the density on support, a direct geometric entropy in quantum mechanics.

4. Entanglement Contour, Susceptibilities, and Shape Dependence

Beyond global entropy, one can resolve entanglement into spatially or geometrically defined "contours" or "susceptibilities":

Entanglement contour $s_A(x)$ : The local density of entanglement assigned to points in region $A$ , constructed to satisfy normalization, additivity, and positivity, and matching area laws in higher dimensions (Han et al., 2019, Roy et al., 2021).
Entanglement susceptibilities $\chi^{(1)}, \chi^{(2)}$ : The response of EE to infinitesimal shape deformations of the entangling surface, leading to quadratic nonlocal corrections and universal log-divergent contributions at corners or cones (Witczak-Krempa, 2018, Rosenhaus et al., 2014).
Corner functions and scaling: In d=2, the universal corner function $a_\alpha(\theta)$ is given analytically for CFTs, e.g., $a_1(\theta) = \frac{c}{24}[1 + (\pi-\theta)\cot\theta]$ (Ye et al., 2022). In quantum Hall systems, measured corner coefficients match CFT predictions to high precision (Ye et al., 2022, Rozon et al., 2019).

5. Quantum Geometry, Projective Space, and Microcanonical Entropy

Entanglement structure is intricately related to the geometry of the state manifold:

Projective Hilbert space/Fubini-Study metric: The natural geometric setting for pure state quantum mechanics, with entanglement entropy $E([ψ])$ as a functional on $P(\mathcal H)$ (Cairano, 26 Nov 2025).
Geometric entanglement entropy $S_\text{geo}(e)$ : Defined as the log-volume of the level set $\Sigma_e = \{ [ψ] : E([ψ]) = e \}$ , weighted by the Fubini-Study gradient:

$S_\text{geo}(e) = \log \omega(e),\quad \omega(e) = \int_{\Sigma_e} \frac{d\sigma_{\rm FS}}{\|\nabla^{\rm FS} E\|}$

Large $S_\text{geo}$ signals typical states, small $S_\text{geo}$ rare ones, encoding a microcanonical density of entanglement values (Cairano, 26 Nov 2025).

Multipartite measures and convex geometry: Geometric (wedge-product, determinant) measures of entanglement are monotonic with the usual entropy for bipartite pure states; higher-dimensional analogs (volumes, areas) capture the multipartite structure (Sarkar et al., 24 Feb 2024).

6. Geometric Entanglement in Dynamical and Topological Matter

GEE is diagnostic for Floquet topological phases, critical points, and band geometry:

Floquet systems: The quantum metric tensor of Floquet-Bloch states encodes geometric information, with GEE exhibiting area-law scaling away from phase transitions and universal logarithmic scaling at critical points (Zhou, 10 Aug 2024).
Critical chains: In XXZ/Ising chains, the subleading GEE correction coincides with the Affleck-Ludwig boundary entropy and encodes boundary condition universality (Stéphan et al., 2010).
Gauge theories: Geometric scaling of the physical Hilbert space is a necessary condition for defining GEE; non-Abelian gauge theories without matter lack local factorizability and thus a well-defined geometric entropy (Hategan-Marandiuc, 8 Apr 2024).

7. Holography and Geometric Entropy

Holographic approaches relate GEE to extremal surface areas and encode interface/boundary effects via minimal surface prescriptions and warp factors (Gutperle et al., 2015, Jiang et al., 2017). Interface terms and generalized boundary entropies emerge in the holographic calculation of entanglement entropy in nontrivial geometries.

Table: Universal Geometric Corrections by Model

Model/Phase	Leading Law	Universal Correction	Reference
Toric code (Z₂), Double semion	$E_G = \frac{L}{4}n_b - 1$	Topological $E_\gamma=1$	(Orus et al., 2013)
Color code	$E_G = (\cdots) - 2$	Topological $E_\gamma=2$	(Orus et al., 2013)
XXZ/Ising critical chain	$E_G(L) = \alpha L + s$	$s = \ln g$ (Affleck-Ludwig entropy)	(Stéphan et al., 2010)
IQH (edge + corners)	$S = b L + S_\text{edge} - \sum b(\theta)$	$S_\text{edge} = (c/3)\log \ell$ ; $b(\theta)$ universal	(Ye et al., 2022, Rozon et al., 2019)
Permutation symmetric states	$S(m) = \frac{d}{2}\log m + S_\text{geo}$	$S_\text{geo}$ = Shannon entropy on $CP^{2s}$	(Castro-Alvaredo et al., 2012)
Floquet topological phases	Area or log law depending on gap	GEE logarithmic at transitions	(Zhou, 10 Aug 2024)

References

(Orus et al., 2013) Geometric Entanglement in Topologically Ordered States
(Stéphan et al., 2010) Geometric entanglement of critical XXZ and Ising chains and Affleck-Ludwig boundary entropies
(Bianchi et al., 2019) Entropy of a subalgebra of observables and the geometric entanglement entropy
(Ye et al., 2022) The entanglement entropy of the quantum Hall edge and its geometric contribution
(Lee et al., 2014) Renyi Entropy and Geometry
(Rozon et al., 2019) Geometric entanglement in integer quantum Hall states with boundaries
(Witczak-Krempa, 2018) Entanglement susceptibilities and universal geometric entanglement entropy
(Castro-Alvaredo et al., 2012) Entanglement in permutation symmetric states, fractal dimensions, and geometric quantum mechanics
(Sarkar et al., 24 Feb 2024) A Geometry of entanglement and entropy
(Cairano, 26 Nov 2025) Geometric Entanglement Entropy on Projective Hilbert Space
(Zhou, 10 Aug 2024) Quantum geometry and geometric entanglement entropy of one-dimensional Floquet topological matter
(Hategan-Marandiuc, 8 Apr 2024) Entanglement entropy in lattices with non-abelian gauge groups
(Gutperle et al., 2015) Entanglement entropy at holographic interfaces
(Jiang et al., 2017) Entanglement Entropy in Flat Holography
(Han et al., 2019) Entanglement entropy from entanglement contour: higher dimensions
(0801.4564) Entanglement Entropy and Spatial Geometry
(Rosenhaus et al., 2014) Entanglement Entropy for Relevant and Geometric Perturbations
(Roy et al., 2021) Emergent geometry from entanglement structure
(Belfiglio et al., 2022) Geometric corrections to cosmological entanglement

Geometric entanglement entropy, in its multiple rigorous forms, unifies the description of quantum correlations, topological order, universal corrections, and the geometry of state space, thus serving as a fundamental diagnostic and analytical tool in modern quantum many-body theory, quantum information, and quantum geometry.