Curvature of Entanglement

• Curvature of entanglement is a family of constructions that use geometric features—derived from quantum states, surfaces, or spacetime—to diagnose and reinterpret quantum entanglement.
• Different frameworks, such as Weyl-geometric, modular-Berry, and Hilbert-space trajectory approaches, apply curvature to distinguish separable and entangled states and optimize metrological protocols.
• Holographic and gravitational models leverage curvature invariants and geometric flows to connect entanglement entropy with bulk spacetime dynamics, offering insights into quantum gravity and information.

“Curvature of entanglement” does not denote a single standardized object across current research. In the surveyed literature, the expression is used for several non-equivalent constructions: a Weyl-geometric scalar curvature induced by the wavefunction amplitude and proposed as a numerical diagnostic of separable versus entangled states (Liang et al., 2023); the Berry curvature of families of modular Hamiltonians, whose holographic dual probes bulk Riemann curvature (Czech et al., 2019); the second derivative of concurrence with respect to a coupling parameter, related at special points to quantum Fisher information (Saleem et al., 18 Apr 2025); curvature-dependent generation, redistribution, or harvesting of entanglement in de Sitter and related settings (Wang et al., 2019, Brahma et al., 2023, K et al., 2023); and curvature- or geometry-dependent corrections to entanglement entropy, area laws, and negativity in holography, condensed matter, and loop quantum gravity (Paul, 2022, Taylor et al., 2020, Cepollaro et al., 2023). The common theme is not a universal invariant of entanglement itself, but the use of geometric curvature, geometric response, or geometric flow to diagnose, generate, constrain, or reinterpret entanglement.

1. Terminological scope and recurrent distinctions

A first distinction concerns whether curvature is treated as a quantity derived from the quantum state or as a background structure acting on the state. In the Weyl-geometric construction, the geometry is induced by the wavefunction amplitude through $f=-\lambda\ln\Omega$, with $\Omega=\sqrt{\rho}=|\psi|$, so the Weyl scalar curvature becomes a functional of $|\psi|$ and its derivatives (Liang et al., 2023). In the modular-Berry construction, the relevant curvature is the field strength of a connection on the space of subregions, associated with modular zero-mode ambiguity, and in the holographic regime it includes the bulk spin-connection curvature and probes the bulk Riemann tensor (Czech et al., 2019). In metrological work, by contrast, “curvature of entanglement” means $\partial_g^2 C(g,t)$, the second derivative of concurrence with respect to a coupling parameter $g$ (Saleem et al., 18 Apr 2025).

A second distinction concerns whether curvature is ontological, diagnostic, or operational. Some papers use curvature as a diagnostic signal: the Weyl scalar curvature changes from a negative dip peak for a separate state to a positive peak for the entangled state near the origin in the two-oscillator example (Liang et al., 2023). Others use curvature operationally: maximally path-entangled N00N states amplify the curvature-sensitive gravitational phase in a Mach–Zehnder interferometer, with $$\mathbf R_{0303}=\frac{c^5}{n\omega l}\frac{\partial^2}{\partial h^2}\Delta\phi$$ (Mieling et al., 2022). Still others argue only for suggestive links. The Kramers–Kronig paper explicitly states that it does not prove an entanglement–curvature relation and instead offers indirect clues based on a shared signature, namely KK-violation (Tasgin, 2019).

A third distinction is between curvature of spacetime, curvature of a state manifold, curvature of an embedded surface, and momentum-space quantum geometry. These are not interchangeable. The de Sitter entanglement-harvesting and gravity-mediated-entanglement papers concern spacetime curvature (K et al., 2023, Brahma et al., 2023). The projective-Hilbert-space analysis of entanglement-generating Hamiltonians defines curvature through the bending of quantum trajectories, via the curvature coefficient $\kappa_{\mathrm{AC}}^2$ (Cafaro et al., 15 Jan 2026). The holographic surface/state work identifies geodesic curvature of a convex bulk curve as a local measure of factorization (Prudenziati, 2019). The multiband-fermion work uses “quantum geometry” to mean the quantum metric and Berry curvature of Bloch states and studies their contribution to area-law entanglement (Paul, 2022).

2. Geometric diagnostics built directly from states, surfaces, and modular structure

In the Weyl-geometric approach, the starting point is a Weyl geometry built on a flat Euclidean background, with non-metricity determined by an exact Weyl one-form $\phi_a=\partial_a f$. The Weyl scalar field is identified with the logarithm of the wavefunction amplitude, $f=-\lambda\ln\Omega$, so the Weyl scalar curvature becomes amplitude-dependent. In $n$ dimensions,

$\Omega=\sqrt{\rho}=|\psi|$0

For $\Omega=\sqrt{\rho}=|\psi|$1, choosing $\Omega=\sqrt{\rho}=|\psi|$2 yields $\Omega=\sqrt{\rho}=|\psi|$3, and with $\Omega=\sqrt{\rho}=|\psi|$4 one obtains $\Omega=\sqrt{\rho}=|\psi|$5. The same paper introduces a “quantum entropy” $\Omega=\sqrt{\rho}=|\psi|$6, for which $\Omega=\sqrt{\rho}=|\psi|$7, hence $\Omega=\sqrt{\rho}=|\psi|$8. The direct curvature–quantum-potential correspondence is claimed to work only in three dimensions; the further suggestion that this may explain why physical space is 3D is presented as speculative (Liang et al., 2023).

That Weyl program does not define a general entanglement monotone from curvature. Instead, it studies whether curvature profiles distinguish separable from entangled states. In the bipartite two-oscillator example, entanglement is quantified by the von Neumann entropy of the reduced density matrix, $\Omega=\sqrt{\rho}=|\psi|$9, while the Weyl scalar curvature is computed from $|\psi|$0. The reported numerical result is that the Weyl scalar curvature shows a negative dip peak for the separate state and a positive peak for the entangled state near the origin, which is proposed as a geometric signal of entanglement (Liang et al., 2023).

A different state-derived construction appears in the modular-Berry framework. For a family of modular Hamiltonians $|\psi|$1, the modular Berry connection is

$|\psi|$2

with $|\psi|$3 the projector onto modular zero modes. Its curvature is the usual field strength $|\psi|$4. In holography, via JLMS, this becomes a bulk connection sewing together the edge-mode frames of neighboring HRRT surfaces; in the smooth-surface semiclassical regime, the boost part of this modular curvature is the normal-bundle spin-connection curvature and hence contains projected components of the bulk Riemann tensor (Czech et al., 2019).

A third geometric diagnostic is the geodesic curvature of a convex bulk curve in the surface/state correspondence. For a convex curve $|\psi|$5 parameterized by proper length, the proposal is

$|\psi|$6

with $|\psi|$7 the geodesic curvature. This is interpreted as a local measure of factorization failure of the dual CFT state. Its integral,

$|\psi|$8

is interpreted as the total bipartite entanglement on the chosen domain. Using Gauss–Bonnet, differences of integrated curvature relative to a geodesic reference become bulk scalar-curvature integrals, $|\psi|$9 (Prudenziati, 2019).

3. Differential-response notions: concurrence curvature, modular reflection, and trajectory bending

In two-qubit metrology, the curvature of entanglement is defined directly as the second derivative of concurrence with respect to the estimated coupling: $\partial_g^2 C(g,t)$0 For the Jaynes–Cummings-type interaction $\partial_g^2 C(g,t)$1, the paper studies both initially separable and initially entangled probes under symmetric amplitude damping. The central relation is

$\partial_g^2 C(g,t)$2

and for the main models the concurrence maxima occur at $\partial_g^2 C(g,t)$3. At those same points, the eigenstates of the symmetric logarithmic derivative become unentangled, so simple product measurements suffice to saturate the quantum Cramér–Rao bound; away from them, entangled measurements are generally required. The paper is explicit that this behavior is not universal, since counterexamples arise for other interactions or noise models (Saleem et al., 18 Apr 2025).

A modular generalization replaces concurrence by a functional $\partial_g^2 C(g,t)$4 built from the state and its modular reflection. The modular curvature of entanglement is

$\partial_g^2 C(g,t)$5

For two qubits, the paper realizes $\partial_g^2 C(g,t)$6 by the spin-flip antiunitary $\partial_g^2 C(g,t)$7, so that $\partial_g^2 C(g,t)$8 is exactly the Wootters spin-flipped state entering concurrence. At modular self-duality points, $\partial_g^2 C(g,t)$9, the paper finds $g$0 in the explicit dissipative flip-flop model. Away from self-duality, $g$1 and $g$2 generally diverge, so the claimed coincidence is explicitly local and model-dependent (Chatterjee, 17 Aug 2025).

The 2026 trajectory-geometry analysis uses a different notion again. There the relevant curvature is not $g$3 but the curvature coefficient $g$4 of the projective-Hilbert-space path traced by the evolving state. For stationary Hamiltonians this reduces to

$g$5

Time-optimal evolutions from separable to maximally entangled two-qubit states have $g$6, no energy-resource wastage, and $g$7, while suboptimal evolutions have nonzero curvature. The same paper emphasizes that optimal geodesic evolutions can have lower average path entanglement than suboptimal evolutions; curvature therefore diagnoses how direct the entanglement-generation route is, not how much entanglement is present at intermediate times (Cafaro et al., 15 Jan 2026).

These three uses should not be conflated. In one case curvature is $g$8 (Saleem et al., 18 Apr 2025); in another it is $g$9 (Chatterjee, 17 Aug 2025); in the third it is the covariant bending of a state-space trajectory (Cafaro et al., 15 Jan 2026). A plausible implication is that “curvature of entanglement” has become a family resemblance term rather than a uniquely defined observable.

4. Spacetime curvature as a generator, redistributor, or probe of entanglement

In de Sitter quantum field theory, curvature can generate or redistribute multipartite entanglement through Bogoliubov mixing. For a free scalar field in the Bunch–Davies vacuum, the mismatch between global modes and open-chart modes makes the vacuum a two-mode squeezed state of the causally disconnected $$\mathbf R_{0303}=\frac{c^5}{n\omega l}\frac{\partial^2}{\partial h^2}\Delta\phi$$0 and $$\mathbf R_{0303}=\frac{c^5}{n\omega l}\frac{\partial^2}{\partial h^2}\Delta\phi$$1 modes: $$\mathbf R_{0303}=\frac{c^5}{n\omega l}\frac{\partial^2}{\partial h^2}\Delta\phi$$2 With an initially entangled Alice–Bob Gaussian state, de Sitter curvature acts as an effective two-mode squeezing operation on Bob and anti-Bob, generating genuine tripartite entanglement for any nonzero initial squeezing and any nonzero $$\mathbf R_{0303}=\frac{c^5}{n\omega l}\frac{\partial^2}{\partial h^2}\Delta\phi$$3. Stronger curvature corresponds to smaller $$\mathbf R_{0303}=\frac{c^5}{n\omega l}\frac{\partial^2}{\partial h^2}\Delta\phi$$4, hence larger $$\mathbf R_{0303}=\frac{c^5}{n\omega l}\frac{\partial^2}{\partial h^2}\Delta\phi$$5, and redistributes the original $$\mathbf R_{0303}=\frac{c^5}{n\omega l}\frac{\partial^2}{\partial h^2}\Delta\phi$$6-$$\mathbf R_{0303}=\frac{c^5}{n\omega l}\frac{\partial^2}{\partial h^2}\Delta\phi$$7 resource into $$\mathbf R_{0303}=\frac{c^5}{n\omega l}\frac{\partial^2}{\partial h^2}\Delta\phi$$8, $$\mathbf R_{0303}=\frac{c^5}{n\omega l}\frac{\partial^2}{\partial h^2}\Delta\phi$$9, and $\kappa_{\mathrm{AC}}^2$0, including horizon-separated entanglement between $\kappa_{\mathrm{AC}}^2$1 and $\kappa_{\mathrm{AC}}^2$2 when curvature is strong enough (Wang et al., 2019).

Gravity-mediated entanglement in de Sitter provides a distinct mechanism. For two identical massive oscillators, integrating out the graviton field in linearized de Sitter space yields an effective potential

$\kappa_{\mathrm{AC}}^2$3

Expanding at small oscillator fluctuations gives an entangling coupling

$\kappa_{\mathrm{AC}}^2$4

and the von Neumann entanglement entropy of the resulting two-component state is written explicitly in terms of $\kappa_{\mathrm{AC}}^2$5 and $\kappa_{\mathrm{AC}}^2$6. The main result is a non-monotonic entanglement profile: flat-space-like decrease at short proper distance, followed by a turnover near the Hubble scale as the de Sitter correction becomes dominant (Brahma et al., 2023).

A related essay reaches a more cautious conclusion. For two masses in superposition, de Sitter graviton mode functions modify the effective potential to

$\kappa_{\mathrm{AC}}^2$7

The paper argues that curvature modifies the interaction potential through the background-dependent graviton vacuum and propagator, but in de Sitter this special correction does not alter the leading nonrelativistic entanglement profile, whereas a generic FLRW background is expected to do so. That expectation is presented as a thought-experimental extension rather than a full derivation (Brahma et al., 2023).

Entanglement harvesting makes the curvature dependence fully covariant. For two geodesic Unruh–DeWitt detectors, the paper parameterizes the problem by $\kappa_{\mathrm{AC}}^2$8, where $\kappa_{\mathrm{AC}}^2$9 and $\phi_a=\partial_a f$0 are defined in Fermi normal coordinates along detector 1’s worldline. The reduced detector state is controlled by local terms $\phi_a=\partial_a f$1 and a nonlocal term $\phi_a=\partial_a f$2, constructed from the Wightman and Feynman propagators. The geodesic interval between detector events is expanded as

$\phi_a=\partial_a f$3

so curvature enters the competition between local noise and nonlocal vacuum correlations through the covariant detector separation. The paper’s stated conclusion is that curvature can induce entanglement features in certain regions of the parameter space and thereby facilitate using entanglement as a probe of spacetime curvature (K et al., 2023).

Not all proposed curvature–entanglement links are equally direct. The KK-relation paper does not derive spacetime curvature from entanglement. It shows that, in one optomechanical model, the onset of single-mode nonclassicality and KK-violation coincide, while in curved-spacetime QED vacuum polarization also violates KK relations without violating causality. The proposed entanglement–curvature connection is explicitly heuristic (Tasgin, 2019).

5. Entropy scaling, quantum geometry, analog curvature, and loop-quantum-gravity tags

In multiband free-fermion systems, quantum geometry contributes a distinct boundary term to real-space entanglement entropy. The entropy scales as

$\phi_a=\partial_a f$4

and the quantum-geometric entanglement entropy is defined by

$\phi_a=\partial_a f$5

Here the relevant geometry is the full Bloch-overlap structure, whose infinitesimal content is the quantum metric and Berry curvature. The paper shows that $\phi_a=\partial_a f$6 is sensitive to nontrivial $\phi_a=\partial_a f$7-dependence of Bloch states, but also stresses that, in general, it is not reducible to a simple Brillouin-zone integral of either the metric or the Berry curvature alone (Paul, 2022).

A curved-spacetime field-theory counterpart appears for a massive scalar nonminimally coupled to curvature through $\phi_a=\partial_a f$8. After spherical reduction and lattice discretization, the Hamiltonian contains the effective local term $\phi_a=\partial_a f$9. For $f=-\lambda\ln\Omega$0, the ground-state entropy preserves the usual area law in the static spherically symmetric backgrounds studied. For $f=-\lambda\ln\Omega$1, especially large positive $f=-\lambda\ln\Omega$2, the entropy ceases to be linear in the boundary area in regions where $f=-\lambda\ln\Omega$3 varies appreciably. Pure de Sitter, with constant $f=-\lambda\ln\Omega$4, behaves instead like a mass shift and continues to obey area scaling. The paper’s central lesson is therefore that curvature alone is not enough; nonconstant curvature combined with nonminimal coupling is the mechanism behind the observed area-law violation (Belfiglio et al., 2023).

An analog-model version of curvature dependence appears in stimulated parametric down-conversion seeded by two-mode sphere coherent states. There the curvature parameter is $f=-\lambda\ln\Omega$5, arising from a two-dimensional harmonic oscillator on a sphere. The output entanglement is measured by the linear entropy $f=-\lambda\ln\Omega$6, and the reported numerics show that entanglement depends on $f=-\lambda\ln\Omega$7. For the parameter choices explored, increasing curvature generally suppresses the output entanglement, especially at low gain, while larger gain $f=-\lambda\ln\Omega$8 compensates for this suppression. The paper explicitly frames this as an analog model rather than a direct gravitational effect (Akbari-Kourbolagh et al., 2024).

Loop quantum gravity supplies a multipartite version in which intrinsic curvature acts as an entangled hidden environment. Nontrivial loop holonomies are integrated into tag spins $f=-\lambda\ln\Omega$9, interpreted as closure defects or topological defects. The extended boundary Hilbert space factorizes as $n$0, where $n$1 is the set of tags, so the boundary state becomes an open quantum system after tracing out $n$2. In the large-spin random-state regime, the typical logarithmic negativity is controlled by

$n$3

The paper finds three regimes: for small curvature, $n$4; for intermediate curvature, $n$5; and for large curvature, $n$6, yielding a PPT boundary state interpreted as effective thermalization and topological disconnection of the two boundary subregions (Cepollaro et al., 2023).

6. Holographic entropy, curvature invariants, and higher-curvature gravitational dynamics

In holography, one recurring use of curvature is to rewrite renormalized entanglement entropy in explicitly geometric terms. For codimension-two static minimal surfaces in asymptotically AdS$n$7,

$n$8

so the renormalized entropy $n$9 is expressed through the Euler characteristic, the extrinsic curvature of the RT surface, and a pulled-back Weyl term. In AdS$\Omega=\sqrt{\rho}=|\psi|$00, the renormalized area is written as an Euler term plus renormalized curvature invariants $\Omega=\sqrt{\rho}=|\psi|$01, $\Omega=\sqrt{\rho}=|\psi|$02, and quadratic curvature polynomials. For spherical entangling regions in odd-dimensional CFTs, the holographic entangling surface in pure AdS is umbilic and the bulk Weyl tensor vanishes, so the renormalized entropy becomes proportional to the Euler invariant, with the proportionality coefficient identified with the renormalized $\Omega=\sqrt{\rho}=|\psi|$03 quantity (Taylor et al., 2020).

A complementary geometric bound is obtained from inverse mean curvature flow in AdS$\Omega=\sqrt{\rho}=|\psi|$04. The reduced Hawking mass functional

$\Omega=\sqrt{\rho}=|\psi|$05

obeys a monotonicity property under a suitably generalized boundary-anchored IMC flow, and this leads to

$\Omega=\sqrt{\rho}=|\psi|$06

The result is interpreted as a subregion Penrose inequality in asymptotically locally AdS spacetimes. Here entanglement is not literally a curvature scalar, but the renormalized entropy is controlled by a curvature-sensitive flow functional of the dual surface (Fischetti et al., 2016).

The higher-curvature-gravity program makes the reverse implication explicit: entanglement constraints can determine bulk curvature dynamics. For a general higher-curvature action $\Omega=\sqrt{\rho}=|\psi|$07, the holographic entanglement entropy functional is not just Wald entropy but

$\Omega=\sqrt{\rho}=|\psi|$08

Using this generalized RT functional, the paper shows that if an asymptotically AdS spacetime computes the entanglement entropies of ball-shaped CFT regions up to second order in vacuum perturbations, then the spacetime satisfies the correct higher-curvature gravitational equations of motion up to second order around AdS. In the same framework, CFT relative entropy is dual to gravitational canonical energy (Haehl et al., 2017).

These holographic results are often misread as establishing a unique “curvature of entanglement.” They do not. They show, more precisely, that renormalized entropy can be decomposed into Euler and curvature invariants (Taylor et al., 2020), that geometric flow monotonicity constrains entropy (Fischetti et al., 2016), and that consistency of entanglement and relative entropy fixes higher-curvature gravitational dynamics and the correct entropy functional, including extrinsic-curvature terms (Haehl et al., 2017). The shared conclusion is strong but specific: in semiclassical holography, entanglement is tightly constrained by intrinsic curvature, extrinsic curvature, and curvature flow of the dual geometry, yet the relevant curvature object depends on the precise construction.