Timelike Entanglement Entropy

• Timelike entanglement entropy is a quantum measure that extends conventional entanglement entropy to timelike intervals, yielding a complex value with a universal imaginary part.
• It is derived using analytic continuation of the replica method and holographic CWES, linking field theory computations with piecewise extremal surfaces in gravity.
• The measure reveals insights into phase transitions, gravitational anomalies, and RG behavior, thereby probing emergent time and the causal structure of quantum systems.

Timelike entanglement entropy (tEE) is a complex-valued quantum information measure that extends conventional (spacelike) entanglement entropy to subsystems oriented along timelike directions. Unlike the standard von Neumann entropy associated with reduced density matrices for spacelike subregions, tEE is typically computed for non-Hermitian, transition-type matrices and generically acquires a universal imaginary part. This quantity has been developed within field theory, holography, and operator-algebraic frameworks, providing new insights into information-theoretic aspects of causal structure, pseudoentropy, and the emergence of spacetime.

1. Field Theory Definition, Pseudoentropy, and Analytic Continuation

In quantum field theory, traditional entanglement entropy (EE) is well-defined for spacelike subsystems by tracing out spatial degrees of freedom, yielding

$S_A = -\operatorname{Tr}(\rho_A \log \rho_A)$

where $\rho_A$ is a reduced density matrix. For a spacelike interval in 2D CFT, one finds $S_A = (c/3) \log(\ell/\epsilon)$ with $\ell$ the spatial extent.

Timelike entanglement entropy arises when this construction is applied to subsystems extended along the time direction rather than space. One defines a "transition matrix" (or more generally, a non-Hermitian operator) via a Wick rotation in the replica method or path integral, such that the subsystem supports are timelike related. This gives a pseudoentropy,

$$S_A^{\text{pseudo}} = -\operatorname{Tr}(\tau_A \log \tau_A)$$

with

$$\tau_A = \operatorname{Tr}_B \frac{|\Psi_1\rangle\langle\Psi_2|}{\langle\Psi_2|\Psi_1\rangle}$$

for two possibly distinct states $|\Psi_1\rangle$, $|\Psi_2\rangle$. Analytic continuation of twist operator correlators from spacelike to timelike intervals yields

$$S_A^\text{timelike} = \frac{c}{3} \log (T_0/\epsilon) + i \frac{\pi c}{6}$$

for a pure timelike interval of duration $T_0$ in a 2D CFT, where the imaginary part is independent of the cutoff and universal (Doi et al., 2023).

2. Holographic Construction and the Complex-Valued Weak Extremal Surface (CWES) Proposal

The canonical holographic prescription for entanglement entropy is the Ryu-Takayanagi (RT) formula, evaluating the minimal area of a bulk codimension-2 extremal surface homologous to the boundary subregion: $$S_A = \frac{\operatorname{Area}(\gamma)}{4 G_N^{d+1}}$$ For timelike intervals, no smooth spacelike extremal surface exists connecting the endpoints, necessitating piecewise smooth surfaces composed of both spacelike and timelike segments. In this regime, the area functional is redefined without the modulus: $$\mathcal{A}(\Gamma) = \int_\Gamma \sqrt{h} \, dx^{d-1}$$ where $\mathcal{A}$ becomes generically complex: spacelike segments contributing real parts, timelike segments pure imaginary (Li et al., 2022).

To identify a unique bulk surface, the CWES is defined by the following conditions:

• Each segment is extremal.
• For multiple candidate extremal segments, the minimal area is selected using an ordering: $z_1\succ z_2$ if $\operatorname{Im} z_1 > \operatorname{Im} z_2$, or if equal, $\operatorname{Re} z_1 > \operatorname{Re} z_2$.
• The total complex area is stationary under deformations at the joints:

$\delta \mathcal{A}(\Gamma) / \delta E_{ij} = 0$

The extended holographic tEE formula is thus

$$S_A = \min \Bigg\{\operatorname{Ext}_{\partial\Gamma = \partial A} \left[\frac{\mathcal{A}(\Gamma)}{4G_N^{d+1}}\right]\Bigg\}$$

The real part always reproduces the characteristic UV divergence of spacelike EE, while the imaginary part is universal.

3. Examples, Universal Imaginary Part, and Phase Structure

Several explicit calculations illustrate the structure:

• AdS$_3$/CFT$_2$: For a time interval $T_0$,

$S_A = (c/3) \log(T_0/\epsilon) + i (\pi c/6)$

This matches field theory results from analytic continuation (Li et al., 2022, Doi et al., 2023).

• BTZ black holes: Timelike entropy at finite temperature is

$$S_A = \frac{c}{6} \left[ \log\left(\frac{\beta}{\pi\epsilon} \sinh\frac{\pi T_0}{\beta} \right) + i\pi \right]$$

(Li et al., 2022)

• Boundary Conformal Field Theory (BCFT): The tEE in BCFT$_2$ manifests three phases:
  • Bulk phase: complex tEE,
  • Regge phase: when an endpoint approaches the light cone of its mirror, tEE can be real or pick up a jump in the imaginary part,
  • Boundary phase: tEE is purely real and incorporates boundary entropy (Chu et al., 2023).

In all cases, the tEE is complex-valued, and the imaginary part typically encodes a universal term interpreted as emergent time (Doi et al., 2023). The appearance or disappearance of the imaginary component may indicate phase transitions in confining backgrounds (Afrasiar et al., 1 Apr 2024).

4. Relations to Spacelike Entanglement, Operator Content, and Imaginary Component

For a wide class of states and geometries, the tEE is not an independent quantity but can be reconstructed from data on a single Cauchy surface. A general formula relates tEE to the spacelike EE and its first-order temporal derivatives: $$S(t, x; t', x') = \text{linear combination of}~ S(0, \cdot; 0, \cdot)~ \text{and}~ \partial_t S(0, \cdot; 0, \cdot)$$ The imaginary part arises from the non-commutativity between the twist operator and its first temporal derivative. In CFT$_2$,

$$\operatorname{Im} S = \frac{1}{2}[S(t, x; t', x') - S(t', x'; t, x)] = i \frac{\pi c}{12}$$

for vacuum, thermal, and holographic states (Guo et al., 1 Feb 2024, Guo et al., 30 Oct 2024). The operator product expansion (OPE) of twist operators refines this, showing that contributions involving operators with fractional dimensions yield state-dependent corrections to the imaginary part (Guo et al., 30 Oct 2024).

5. Non-Conformal, Non-Relativistic, and Anomalous Backgrounds

In non-conformal (confining) theories, the holographic tEE involves merging timelike and spacelike extremal surfaces. The imaginary part is associated with the segment featuring a turning point in the bulk, while the real part comes from spacelike branches. There exists a critical interval size beyond which the connected extremal surface ceases to exist and the imaginary part vanishes, signaling a phase transition associated with confinement (Afrasiar et al., 1 Apr 2024, Nunez et al., 26 May 2025).

For non-relativistic and Lifshitz/hyperscaling-violating backgrounds, tEE is highly sensitive to parameters such as the dynamical exponent $z$. Its imaginary part can, for certain parameter choices, signal the presence of Fermi surfaces via either a logarithmic violation of the real part or a constant imaginary component dependent on $z$ (Afrasiar et al., 27 Nov 2024, Jena et al., 1 Oct 2024).

In CFTs with gravitational anomalies, the imaginary part of tEE is sensitive to the central charges $c_L$ and $c_R$, depending asymmetrically on $c_R$: $$S_A^{\text{T}} = \frac{c_L + c_R}{6} \log(T/\epsilon) + \frac{c_R}{6} i \pi$$ This property allows tEE to serve as a probe for gravitational anomalies (Chu et al., 28 Apr 2025).

6. Operator-Algebraic and Generalized Perspectives

A rigorous operator-algebraic definition of tEE in QFT can be formulated using the split property and the timelike tube theorem. The entropy associated to a timelike interval $\mathcal{T}$ is defined as the von Neumann entropy of the algebra associated with its timelike envelope (the causal diamond generated by $\mathcal{T}$): $S(\mathcal{T}) = S(\mathcal{D}_\mathcal{T})$ The entropy computed in this way is always real due to the equivalence $$\mathcal{A}(\mathcal{T}) = \mathcal{A}(\mathcal{D}_\mathcal{T})$$ (Jiang et al., 25 Mar 2025).

7. Temporal Entanglement Entropy, RG, and Extensions

Temporal entanglement entropy extends tEE to Euclidean time intervals, connecting it directly to coarse-graining and the renormalization group (RG) flow. Tracing over a Euclidean time interval reduces high-frequency resolution, and the corresponding entropy

$$S_{\mathrm{temp}} \sim -\operatorname{Tr}(\rho_{\tau_0} \log \rho_{\tau_0})$$

probes the scale-dependence of quantum information (Grieninger et al., 2023). In holographic cutoff geometries, temporal entanglement entropy can be used to extract information such as the Lifshitz scaling exponent $z$, which is invisible to spatial EE at zero temperature and density.

Further, in the context of causally connected (not necessarily spacelike) subregions, the transition operator $T_{AB}$ can be constructed using the Schwinger-Keldysh real-time formalism and the real-time replica method. The associated pseudoentropies generalize tEE, and timelike extensions of measures such as entanglement wedge cross section and reflected entropy become computable and holographically meaningful, exhibiting properties distinct from their spacelike analogs (Gong et al., 7 Aug 2025).

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The body of research on timelike entanglement entropy has revealed its utility as a complex-valued generalization of entanglement measures, encoding information about emergent time, phase transitions (including confinement), gravitational anomalies, and non-relativistic scaling in both field theory and holographic settings. The modern developments around CWES, top-down holographic prescriptions, commutator-based imaginary component, and operator-algebraic approaches have significantly expanded the understanding of causal entanglement structure and its role in quantum gravity, conformal field theory, and many-body systems.