Holographic Timelike Entanglement Entropy

• Holographic timelike entanglement entropy is a framework that extends the standard RT prescription using complex-valued weak extremal surfaces combining spacelike and timelike segments.
• It employs analytic continuation to derive a complex entropy with real and imaginary parts, revealing insights into emergent time and causal structure.
• The approach consistently reproduces known results in AdS/CFT and BCFT settings, offering a robust probe of non-unitary dynamics and interior spacetime geometry.

Holographic timelike entanglement entropy extends the notion of entanglement entropy to subregions of a quantum field theory that are separated along a timelike interval, rather than a purely spatial one. In the holographic context, this entails generalizing the standard Ryu–Takayanagi (RT) prescription by introducing complex-valued extremal surfaces composed of spacelike and timelike segments, with the associated geometric area functional yielding a complex entropy—often termed "pseudo entropy." This framework provides a natural probe of causal structure, the interplay between space and time in quantum entanglement, and the dynamics of renormalization group flows, particularly in systems lacking Lorentz invariance.

1. Generalization of the RT Prescription: Complex Weak Extremal Surfaces

The standard RT prescription computes the entanglement entropy $S_A$ for a spacelike region $A$ in a $d$-dimensional CFT from the area of a smooth, spacelike codimension-2 extremal surface $\gamma$,

$S_A = \frac{\mathrm{Area}(\gamma)}{4 G_N^{d+1}}\,,$

where $\gamma$ is homologous to $A$. For timelike subregions, smooth spacelike extremal surfaces connecting the region’s endpoints do not generally exist. The proposal (Li et al., 2022) is to generalize to "complex-valued weak extremal surfaces" (CWES), which are piecewise smooth surfaces comprising both spacelike and timelike extremal segments. The area now becomes a complex-valued functional

$\mathcal{A} = \int \sqrt{h} \, d^{d-1}x$

with $h$ the determinant of the induced metric, without enforcing the absolute value, so that time-like contributions yield purely imaginary terms.

CWES are required to satisfy:

• Each segment must independently extremize the area functional.
• Among multiple options compatible with the local boundary data, the one with minimal real area is selected.
• Junction (“joint”) positions between segments are to be determined by extremizing the total complex area with respect to their location, requiring $\delta \mathcal{A}/\delta E_{ij} = 0$.

Because the area is not generally real, the extremality prescription requires an ordering of complex numbers: $z_1 \succ z_2$ if $\operatorname{Im}(z_1) > \operatorname{Im}(z_2)$, or for equal imaginary parts, $\operatorname{Re}(z_1) > \operatorname{Re}(z_2)$. The entropy is the minimum CWES area in this ordering.

2. Complex Entropy: Structure and Analytic Continuation

Timelike entanglement entropy (TEE) is naturally obtained by analytic continuation from its spacelike counterpart, for example, by the substitution $L \to iT$ in the CFT replica method. This yields, for a pure timelike interval,

$$S_A^T = \frac{c}{3} \log\left(\frac{T}{\epsilon}\right) + i \frac{\pi c}{6}$$

where $\epsilon$ is a UV cutoff and $c$ is the central charge (Doi et al., 2023). The imaginary part is universal and arises from the analytic properties of the two-point function of twist operators when continued to Lorentzian signature. It reflects the emergence of a new notion of "entanglement in time," and connects to the concept of pseudo entropy, the von Neumann entropy of a non-Hermitian reduced transition matrix.

In holography, extremal surfaces corresponding to timelike subregions are complex codimension-2 surfaces constructed as unions of spacelike and timelike geodesic segments, glued together according to joint extremality and minimal area ordering constraints (Li et al., 2022, Doi et al., 2023).

3. Example Computations: AdS/CFT and Phases in AdS/BCFT

Explicit computations demonstrate the CWES framework recovers correct results in known examples:

• AdS$_3$/CFT$_2$ (vacuum): For a timelike interval of length $T_0$, the minimal surface combines two spacelike geodesics to infinity and a joining timelike segment. The total area is $$\mathcal{A} = 2R_{\mathrm{AdS}} \log(T_0/\epsilon) + i\pi R_{\mathrm{AdS}}$$, so that $S_A^T = (c/3) \log(T_0/\epsilon) + i(\pi c/6)$.
• BTZ Black Hole: The minimal area for a timelike boundary interval yields $$S_A^T = (c/3) \log\left[\frac{\beta}{\pi\epsilon} \sinh\left(\frac{\pi T_0}{\beta}\right)\right] + i(\pi c/6)$$, again matching the analytic continuation of the spacelike result (Li et al., 2022).
• Boundary CFT (AdS/BCFT): For time-like intervals in BCFTs, three distinct analytical phases are observed (Chu et al., 2023):
  • "Bulk phase" (complex TEE, far from boundary)
  • "Boundary phase" (purely real TEE plus boundary entropy, near boundary)
  • "Regge phase" (crossover with discontinuous imaginary part as an endpoint crosses the mirror light cone)
  • The holographic RT surfaces switch topology across these phases, always reproducing the field theory calculation.

4. Holographic Prescription and Selection Principles

The uniqueness and well-posedness of TEE rely on the CWES prescription (Li et al., 2022), which applies not only to CFTs but also to black holes and BCFT configurations. The prescription is: $$S_A = \min_{\rm CWES} \left( \frac{\mathcal{A}[\Gamma]}{4 G_N^{d+1}} \right)$$ with minimization conducted in the complex ordering (imaginary first, real second). If the area functional depends on "joint" parameters where segments meet, the stationarity condition $\delta\mathcal{A}/\delta E_{ij} = 0$ fixes the gluing uniquely. In simple settings (e.g., AdS$_3$, BTZ), the CWES construction recovers the correct phase and branch structure to match analytic continuation. The methodology generalizes to higher-dimensional settings, though some cases (e.g., strips in higher dimensions) require careful analysis as direct analytic continuation may not coincide with geometric extremization.

5. Physical and Conceptual Implications

Holographic timelike entanglement entropy achieves several key objectives:

• Unified Framework: It extends the RT prescription to arbitrary (spacelike or timelike) subregions using complex extremal surfaces, allowing the paper of “entanglement in time” alongside traditional spatial entanglement (Li et al., 2022, Doi et al., 2023).
• Emergent Time Interpretation: The imaginary part of the entropy—originating solely from the timelike segments of the extremal surface—is interpreted as a quantifier of emergent time from entanglement, analogous to how spatial entanglement encodes emergent geometry (Doi et al., 2023).
• Ambiguity Resolution: The CWES prescription resolves ambiguities arising from multiple possible ways to join spacelike and timelike segments by enforcing joint extremality and area minimization in the prescribed complex ordering.
• Consistency Across Examples: In all explicit cases tested (vacuum, thermal BTZ, global AdS, BCFT), the CWES proposal reproduces the analytic continuation of known entanglement results.

Because some CWES surfaces can penetrate black hole interiors, the formalism enables new probes of interior dynamics and causal structure through the lens of entanglement (Li et al., 2022). In some multiboundary setups, the TEE may be real, suggesting pseudo entropy can reflect physical "real" measures of quantum information in special circumstances.

6. Connections to Pseudo Entropy and Extensions

TEE is closely related to the pseudo entropy, the von Neumann entropy of a reduced transition matrix, relevant in settings with non-unitary evolution or quantum quenches (Doi et al., 2023). The universal imaginary part observed in holographic computations matches the constant imaginary pseudo entropy found via replica or boundary CFT methods.

Higher-dimensional generalizations—such as hemispherical and strip-like regions—have been considered via analytic continuation and show characteristic complex phases tied to the dimensionality (e.g., overall $i^{2-d}$ factors). The method extends to BCFTs by enforcing holographic boundaries via end-of-the-world branes and reproducing the phase structure, including the emergence of “boundary” and “Regge” phases seen in BCFT correlators (Chu et al., 2023).

7. Summary

The theory of holographic timelike entanglement entropy generalizes the RT prescription to arbitrary-causal subregions, introducing a robust geometric prescription—complex-valued weak extremal surfaces (CWES)—that provides a unified, well-posed framework for both spacelike and timelike entanglement entropy. This prescription accounts for the analytic structure, phase, and universal constant terms arising from the analytic continuation of spacelike to timelike intervals in quantum field theory. Timelike entanglement entropy provides a new tool for probing emergent time, interior spacetime structure, and non-unitary dynamics in holography and field theory, and forms a bridge to recent developments in the paper of pseudo entropy.