Timelike Entanglement Entropy in Higher Curvature Gravity (2509.04181v1)

Abstract: This work investigates holographic timelike entanglement entropy in higher curvature gravity, with a particular focus on Lovelock theories and on the role of excited states. For strip subsystems, higher-curvature terms are found to affect the imaginary part of the entropy in a dimension-dependent manner, while excited states contribute solely to the real part. For the cases analyzed, spacelike and timelike entanglement entropies exhibit proportional relations: vacuum contributions differ by universal phase factors, while excitation contributions are linked by dimension-dependent rational coefficients. For hyperbolic subsystems, the timelike entanglement entropy computed via complex extremal surfaces is shown to agree with results obtained through analytic continuation, with imaginary contributions appearing in all dimensions. Higher-curvature corrections are explicitly calculated in five- and $(d+1)$-dimensional Gauss-Bonnet gravity, illustrating the applicability of the complex surface prescription to general Lovelock corrections. These results provide a controlled setting to examine the influence of higher-curvature interactions on holographic timelike entanglement entropy, and clarify its relation to vacuum and excited-state contributions.

• The paper derives a perturbative framework for timelike entanglement entropy by analytically continuing the RT prescription, capturing both real and imaginary components.
• It employs Lovelock and Gauss-Bonnet gravity models to compute explicit corrections, revealing universal mappings between timelike and spacelike entropies.
• Key findings include dimension-dependent analytic phase relations and the systematic impact of higher-curvature corrections on holographic quantum information.

Timelike Entanglement Entropy in Higher Curvature Gravity: A Technical Summary

Introduction and Motivation

This work systematically investigates the structure of timelike entanglement entropy (TEE) in the context of higher-curvature gravity, focusing on Lovelock theories. The paper is motivated by the recent extension of the Ryu-Takayanagi (RT) prescription to timelike subregions, where the entanglement entropy becomes complex-valued and encodes information about quantum correlations across timelike-separated regions. The analysis is particularly relevant for understanding quantum information-theoretic aspects of holography beyond Einstein gravity, as higher-curvature corrections naturally arise in string-theoretic and other UV-complete models of gravity.

Timelike Entanglement Entropy and Lovelock Gravity

TEE is defined via analytic continuation of the standard (spacelike) entanglement entropy to timelike intervals, resulting in a complex-valued quantity. In the AdS/CFT context, this is realized holographically by associating TEE with the area of a complex extremal surface anchored on the timelike subregion. The prescription generalizes the RT formula to higher-curvature gravity by employing the appropriate Wald-like entropy functionals, with Lovelock gravity providing a tractable and physically motivated setting.

Lovelock gravity extends the Einstein-Hilbert action by including higher-order curvature invariants (Euler densities), with the action in $d+1$ dimensions given by

$$I = \frac{1}{2\ell_P^{d-1}} \int d^{d+1}x \sqrt{-g} \left[ \frac{d(d-1)}{L^2} + R + \sum_{p=2}^{\lfloor\frac{d+1}{2}\rfloor} c_p L^{2p-2} \mathcal{L}_{2p}(R) \right].$$

The analysis is performed in the regime of small higher-curvature couplings, ensuring a smooth connection to Einstein gravity.

Strip Subsystems: Explicit Computations and Universal Patterns

Five-Dimensional Gauss-Bonnet Gravity

For a strip subsystem in five-dimensional Gauss-Bonnet gravity, the TEE is computed perturbatively in both the Gauss-Bonnet coupling $\lambda_5$ and the excitation parameter $m$ (characterizing deviations from pure AdS). The entropy functional includes both intrinsic and extrinsic curvature terms, and the extremal surface is complexified. The leading-order result is

$$S_A^{(T)} = \frac{1}{4G} \left[ \frac{1}{\epsilon^2} + \frac{c_4}{2} \frac{1}{(\Delta t)^2} \right] + \frac{f_\infty \lambda_5}{4G} \left( \frac{2}{\epsilon^2} - \frac{\text{finite}}{(\Delta t)^2} \right) - \frac{\Delta t^2 m}{8G} \text{(finite)} + \frac{f_\infty \lambda_5 m \Delta t^2}{8G} \text{(finite)} + \cdots$$

where $f_\infty$ encodes the effective AdS radius, and the "finite" terms involve explicit Gamma function ratios. Notably, higher-curvature corrections affect both the divergent and finite parts, and the imaginary part of the entropy is modified in a dimension-dependent manner. Excited states contribute only to the real part.

Generalization to $(d+1)$-Dimensional Gauss-Bonnet and Lovelock Gravity

The analysis is extended to arbitrary dimensions and to cubic Lovelock gravity. The structure of the corrections is preserved: higher-curvature terms universally rescale the vacuum (divergent and finite) contributions, while excitation-induced corrections are modulated by dimension-dependent rational coefficients. The general result for TEE in $(d+1)$-dimensional Lovelock gravity (truncated at cubic order) is

$$S_A^{(T)} = \text{vacuum terms} + \text{(Gauss-Bonnet and cubic Lovelock corrections)} + \text{excitation corrections},$$

with explicit expressions for each sector involving Gamma functions and analytic continuation phases.

Analytic Continuation and Relation to Spacelike Entanglement Entropy

A central result is the precise mapping between timelike and spacelike entanglement entropies. For the vacuum sector, the mapping is governed by analytic continuation $a \to i\Delta t$, introducing universal phase factors (e.g., $i^{2-d}$). For excitation-induced terms, the coefficients in TEE and spacelike EE are related by simple rational functions of the spacetime dimension $d$. This mapping holds for all orders in the perturbative expansion in higher-curvature couplings and excitation parameters.

The explicit relations are:

• Vacuum higher-curvature corrections: proportional by $-(i)^{-d}$.
• Excitation corrections: proportional by $-(d-3)/(d-1)$ (Einstein), $-(d+1)/(d-1)$ (Gauss-Bonnet), and $-(3d-1)/(3d-3)$ (cubic Lovelock).

This demonstrates that TEE is not an independent quantity but is systematically determined by the analytic structure of the spacelike EE.

Hyperbolic Subsystems and Geometric Dependence

For hyperbolic subsystems, the extremal surface is parametrized in hyperbolic coordinates and embedded in complexified AdS. The TEE is computed as a single complex integral, unifying the real and imaginary parts. In contrast to the strip case, higher-curvature corrections to the imaginary part of TEE appear in all dimensions for hyperbolic subsystems, not just in odd dimensions. This highlights the sensitivity of the analytic structure of TEE to the geometry of the entangling region.

Implications, Limitations, and Future Directions

The results provide a controlled setting for analyzing the impact of higher-curvature corrections on holographic TEE, clarifying the interplay between vacuum and excitation contributions. The analytic continuation framework offers a systematic map between timelike and spacelike entanglement measures, with universal features determined by the spacetime dimension and the geometry of the entangling region.

However, the analysis is perturbative in higher-curvature couplings and excitation parameters. Nonperturbative effects, as well as the selection of physically relevant complex extremal surfaces in the presence of multiple saddles, remain open problems. The extension to more general higher-derivative gravities and the development of a general principle for saddle selection are important directions for future research.

Conclusion

This work establishes a comprehensive framework for computing and interpreting timelike entanglement entropy in higher-curvature gravity, with explicit results for Lovelock theories. The findings elucidate the universal structure of higher-curvature corrections, the analytic continuation relations to spacelike entanglement entropy, and the dependence on subsystem geometry. These results have significant implications for the paper of quantum information in holography and for probing the structure of quantum gravity beyond Einstein theory.