Timelike Entanglement Entropy

• Timelike entanglement entropy is a measure defined for timelike-separated regions in quantum field theory, capturing correlations via analytic continuation or operator-algebraic approaches.
• It presents as a complex (pseudo)entropy in analytic continuation frameworks and as a real-valued quantity in operator-algebraic formulations, reflecting distinct underlying physics.
• Holographic methods extend this concept by using complex or mixed-signature extremal surfaces, linking timelike entanglement to replica methods, black hole interiors, and thermal effects.

Timelike entanglement entropy denotes an entanglement-like quantity associated with timelike-separated regions or intervals in quantum field theory, and in holography it is represented by Lorentzian or complexified extremal geometry rather than by the standard Ryu–Takayanagi surface on a fixed time slice. In much of the recent literature, it is obtained by analytically continuing spatial entanglement entropy to timelike kinematics, so the result is generally complex and is interpreted as a pseudoentropy; in a distinct operator-algebraic formulation, it is instead defined from the algebra of observables on a timelike interval and is real-valued because the timelike tube theorem identifies that algebra with the algebra of its timelike envelope (Heller et al., 2024, Jiang et al., 25 Mar 2025).

1. Definitions and conceptual scope

For an ordinary spatial subregion $A$ on a fixed time slice, entanglement entropy is

$S(A)=-\mathrm{Tr}(\rho_A\log\rho_A),$

and in holography is computed by the area of a codimension-two extremal surface $\Gamma_A$ with $\partial\Gamma_A=\partial A$. Timelike entanglement entropy replaces the spatial interval by a timelike interval or timelike strip. The central obstruction is that timelike regions are not subsystems in the usual tensor-factor sense, so the quantity need not be a standard von Neumann entropy (Heller et al., 2024, Doi et al., 2023).

A compact way to organize the literature is the following.

Framework	Defining object	Characteristic output
Analytic continuation / pseudoentropy	Wick-rotated twist correlator or reduced transition matrix	Generally complex
Operator-algebraic	Split property plus timelike tube theorem	Real-valued
Holographic geometric	Complex extremal surface or mixed spacelike–timelike extremal surface	Complex in general

In the pseudoentropy-based approach, timelike entanglement entropy is identified with the entropy of a reduced transition matrix

$$\tau_A=\mathrm{Tr}_{A^c}\!\left[\frac{|\psi\rangle\langle\phi|}{\langle\phi|\psi\rangle}\right], \qquad S_{\rm ps}(A)=-\mathrm{Tr}(\tau_A\log\tau_A),$$

so complex values are expected when $\tau_A$ is non-Hermitian (Doi et al., 2023). In the operator-algebraic approach, a timelike interval

$$\mathcal T=\{x=(t,\vec x);\, |t|<T/2,\ \vec x=\vec x_0\}$$

generates an algebra $\mathcal A(\mathcal T)$ by smearing local fields in time, and the timelike tube theorem gives $$\mathcal A(\mathcal T)=\mathcal A(\mathcal E_{\mathcal T})$$, where $\mathcal E_{\mathcal T}$ is the timelike envelope. In favorable cases $S(A)=-\mathrm{Tr}(\rho_A\log\rho_A),$0 is the causal diamond of a spatial ball $S(A)=-\mathrm{Tr}(\rho_A\log\rho_A),$1, so

$S(A)=-\mathrm{Tr}(\rho_A\log\rho_A),$2

which is real-valued by construction (Jiang et al., 25 Mar 2025).

This split in definitions is not merely terminological. It produces two distinct research programs: one centered on analytic continuation, pseudoentropy, and complex holographic saddles, and another centered on local algebras, the split property, and equality with spatial-ball entropy. A plausible implication is that “timelike entanglement entropy” currently names a family of related but inequivalent constructions rather than a single universally accepted observable.

2. Analytic continuation, replica methods, and the universal imaginary term

The basic analytic-continuation construction starts from a known spatial interval or strip formula and continues the spatial width to a timelike separation. In a $S(A)=-\mathrm{Tr}(\rho_A\log\rho_A),$3D CFT at zero temperature,

$S(A)=-\mathrm{Tr}(\rho_A\log\rho_A),$4

becomes

$S(A)=-\mathrm{Tr}(\rho_A\log\rho_A),$5

for the principal branch used in the timelike literature (Doi et al., 2023). At finite temperature $S(A)=-\mathrm{Tr}(\rho_A\log\rho_A),$6,

$S(A)=-\mathrm{Tr}(\rho_A\log\rho_A),$7

and on a circle of circumference $S(A)=-\mathrm{Tr}(\rho_A\log\rho_A),$8,

$S(A)=-\mathrm{Tr}(\rho_A\log\rho_A),$9

(Doi et al., 2023, Jiang et al., 2023).

The imaginary part is the defining signature of this continuation. It arises from branch choices in complex logarithms and from the Lorentzian $\Gamma_A$0 prescription. In the simplest formulation, crossing the negative real axis gives $\Gamma_A$1, while in thermal formulas the continuation of $\Gamma_A$2 or the associated boost parameter supplies the same constant $\Gamma_A$3 for a pure timelike interval (Heller et al., 2024, Chu et al., 28 Apr 2025).

The replica formulation makes this more precise. Twist fields $\Gamma_A$4 in the $\Gamma_A$5-copy theory produce the Rényi entropies, and Wick rotation of their correlators yields the timelike quantity. A later development shows that, for a broad range of states in $\Gamma_A$6D CFT, timelike entanglement entropy can be written as a linear combination of spacelike entanglement entropy on a Cauchy slice and its first-order temporal derivatives. In that relation, the imaginary part originates from the non-commutativity between the twist operator and its first-order temporal derivative (Guo et al., 2024). A further refinement shows that this first-derivative commutator is universal in many cases, while more general states can require higher odd temporal derivatives; for strip geometry in higher dimensions, however, the imaginary part again reduces to commutators of the twist operator and its first-order temporal derivative (Guo et al., 2024).

These results sharpen a common misconception. The imaginary part is not treated merely as an arbitrary branch artifact; rather, within the pseudoentropy framework it is tied to operator ordering, Lorentzian continuation, and local commutators.

3. Holographic formulations

The standard RT/HRT prescription for a spatial region $\Gamma_A$7 is

$\Gamma_A$8

with $\Gamma_A$9 a codimension-two extremal surface satisfying the homology constraint. A direct holographic extension to timelike subregions was formulated by defining, for a timelike boundary strip $\partial\Gamma_A=\partial A$0,

$\partial\Gamma_A=\partial A$1

where $\partial\Gamma_A=\partial A$2 is a boundary-anchored codimension-two extremal surface in a complexified bulk geometry, subject to $\partial\Gamma_A=\partial A$3 and an appropriate homology condition in complex geometry (Heller et al., 2024). In this construction, bulk coordinates, including the radial coordinate, are treated as complex, the asymptotic AdS boundary remains real, and both the real and imaginary parts of the entropy arise from a single complex area.

An alternative Lorentzian prescription builds the holographic object from a stationary union of spacelike and timelike extremal segments. In AdS$\partial\Gamma_A=\partial A$4, the total complex area of this mixed surface $\partial\Gamma_A=\partial A$5 defines

$\partial\Gamma_A=\partial A$6

with the timelike segment contributing $\partial\Gamma_A=\partial A$7 times a real length and the spacelike segments contributing the real part (Doi et al., 2023). A closely related proposal introduces “Complex-Valued Weak Extremal Surfaces” and a selection rule that orders candidate complex areas first by imaginary part and then by real part, thereby choosing a unique holographic value from a non-unique family of mixed-signature piecewise extremal surfaces (Li et al., 2022).

A third AdS$\partial\Gamma_A=\partial A$8 approach uses the Rindler method. There, timelike entanglement entropy is identified with the thermal entropy of the Rindlerized CFT plus a universal constant,

$\partial\Gamma_A=\partial A$9

with the imaginary term attributed to the covariant timelike cutoff and the analytic continuation of the logarithm. In this AdS$$\tau_A=\mathrm{Tr}_{A^c}\!\left[\frac{|\psi\rangle\langle\phi|}{\langle\phi|\psi\rangle}\right], \qquad S_{\rm ps}(A)=-\mathrm{Tr}(\tau_A\log\tau_A),$$0 construction, the real part is reproduced by the AdS-Rindler horizon area, while the constant imaginary term is a boundary phase fixed by the Brown–Henneaux relation rather than by a real bulk area (He et al., 2023).

More recently, a top-down Lorentzian formulation was developed directly in $$\tau_A=\mathrm{Tr}_{A^c}\!\left[\frac{|\psi\rangle\langle\phi|}{\langle\phi|\psi\rangle}\right], \qquad S_{\rm ps}(A)=-\mathrm{Tr}(\tau_A\log\tau_A),$$1D or $$\tau_A=\mathrm{Tr}_{A^c}\!\left[\frac{|\psi\rangle\langle\phi|}{\langle\phi|\psi\rangle}\right], \qquad S_{\rm ps}(A)=-\mathrm{Tr}(\tau_A\log\tau_A),$$2D supergravity. Its key device is a signature parameter $$\tau_A=\mathrm{Tr}_{A^c}\!\left[\frac{|\psi\rangle\langle\phi|}{\langle\phi|\psi\rangle}\right], \qquad S_{\rm ps}(A)=-\mathrm{Tr}(\tau_A\log\tau_A),$$3 multiplying $$\tau_A=\mathrm{Tr}_{A^c}\!\left[\frac{|\psi\rangle\langle\phi|}{\langle\phi|\psi\rangle}\right], \qquad S_{\rm ps}(A)=-\mathrm{Tr}(\tau_A\log\tau_A),$$4 throughout the calculation: $$\tau_A=\mathrm{Tr}_{A^c}\!\left[\frac{|\psi\rangle\langle\phi|}{\langle\phi|\psi\rangle}\right], \qquad S_{\rm ps}(A)=-\mathrm{Tr}(\tau_A\log\tau_A),$$5 reproduces Euclidean RT entropy and $$\tau_A=\mathrm{Tr}_{A^c}\!\left[\frac{|\psi\rangle\langle\phi|}{\langle\phi|\psi\rangle}\right], \qquad S_{\rm ps}(A)=-\mathrm{Tr}(\tau_A\log\tau_A),$$6 gives Lorentzian timelike entanglement entropy without a post hoc $$\tau_A=\mathrm{Tr}_{A^c}\!\left[\frac{|\psi\rangle\langle\phi|}{\langle\phi|\psi\rangle}\right], \qquad S_{\rm ps}(A)=-\mathrm{Tr}(\tau_A\log\tau_A),$$7 rotation. This yields a stability criterion for bulk embeddings, analytic approximations in slab geometries, central-charge extraction, and confining-model phase transitions (Nunez et al., 26 May 2025).

Across these proposals, the main structural issue is not the existence of candidate bulk objects but the choice of physical saddle. Complexification, mixed signature, and top-down Lorentzian extremization all solve the kinematic problem of anchoring to timelike boundary data; they differ in how they impose homology, contour selection, and reality conditions.

4. Exact examples, higher-dimensional strips, and multi-saddle structure

The complex-extremal-surface proposal admits explicit solutions in AdS$$\tau_A=\mathrm{Tr}_{A^c}\!\left[\frac{|\psi\rangle\langle\phi|}{\langle\phi|\psi\rangle}\right], \qquad S_{\rm ps}(A)=-\mathrm{Tr}(\tau_A\log\tau_A),$$8. For the vacuum, with boundary conditions $$\tau_A=\mathrm{Tr}_{A^c}\!\left[\frac{|\psi\rangle\langle\phi|}{\langle\phi|\psi\rangle}\right], \qquad S_{\rm ps}(A)=-\mathrm{Tr}(\tau_A\log\tau_A),$$9 and $\tau_A$0, the solution is

$\tau_A$1

with

$\tau_A$2

Its length yields

$\tau_A$3

which matches the $\tau_A$4D CFT continuation (Heller et al., 2024).

For the AdS$\tau_A$5 black brane, with $\tau_A$6 and $\tau_A$7, one finds

$\tau_A$8

and the timelike entropy becomes

$\tau_A$9

again reproducing the analytically continued boundary result (Heller et al., 2024).

In higher-dimensional vacuum AdS$$\mathcal T=\{x=(t,\vec x);\, |t|<T/2,\ \vec x=\vec x_0\}$$0, the same framework gives a strip formula

$$\mathcal T=\{x=(t,\vec x);\, |t|<T/2,\ \vec x=\vec x_0\}$$1

with

$$\mathcal T=\{x=(t,\vec x);\, |t|<T/2,\ \vec x=\vec x_0\}$$2

This is the higher-dimensional timelike analogue of the spatial strip law and makes explicit that the phase depends on dimension through $$\mathcal T=\{x=(t,\vec x);\, |t|<T/2,\ \vec x=\vec x_0\}$$3 (Heller et al., 2024).

Thermal higher-dimensional geometries introduce genuinely new structure. In the AdS$$\mathcal T=\{x=(t,\vec x);\, |t|<T/2,\ \vec x=\vec x_0\}$$4 black brane, the equations admit multiple complex extremal surfaces anchored on the same timelike strip. Two classes appear: “vacuum-connected,” for which the tip approaches the boundary as $$\mathcal T=\{x=(t,\vec x);\, |t|<T/2,\ \vec x=\vec x_0\}$$5, and “vacuum-disconnected,” for which the tip moves toward the singularity and $$\mathcal T=\{x=(t,\vec x);\, |t|<T/2,\ \vec x=\vec x_0\}$$6 in the same limit. Critical points $$\mathcal T=\{x=(t,\vec x);\, |t|<T/2,\ \vec x=\vec x_0\}$$7 governing the late-time behavior satisfy

$$\mathcal T=\{x=(t,\vec x);\, |t|<T/2,\ \vec x=\vec x_0\}$$8

and for $$\mathcal T=\{x=(t,\vec x);\, |t|<T/2,\ \vec x=\vec x_0\}$$9 give three roots $\mathcal A(\mathcal T)$0. At large $\mathcal A(\mathcal T)$1, both classes have linear area growth, while at small $\mathcal A(\mathcal T)$2 the vacuum-connected branch reproduces the vacuum result and the vacuum-disconnected branch approaches a singularity-probing constant (Heller et al., 2024).

This produces the central saddle-selection controversy. A minimal-$\mathcal A(\mathcal T)$3 rule would choose the vacuum-disconnected branch at small $\mathcal A(\mathcal T)$4, but that violates the requirement that the timelike entropy reduce continuously to the vacuum limit and would imply a UV/UV map from boundary short distance to bulk scales near the singularity. The alternative principle advocated in this setting is to view timelike entropy as an analytic continuation of spacelike holographic entanglement entropy and enforce the vacuum limit, which selects the vacuum-connected saddles. Picard–Lefschetz analysis, replica guidance, homology in complex geometry, and branch-structure constraints are proposed as more rigorous criteria, but a general resolution remains open (Heller et al., 2024).

5. Anomalies, boundaries, deformations, and non-relativistic or non-conformal extensions

In anomalous $\mathcal A(\mathcal T)$5D CFTs with $\mathcal A(\mathcal T)$6, timelike entanglement entropy acquires a chiral imaginary part. For a pure timelike interval,

$\mathcal A(\mathcal T)$7

The real part depends symmetrically on $\mathcal A(\mathcal T)$8, but the imaginary part depends only on one chirality in the conventions used. This asymmetric dependence provides a diagnostic of gravitational anomalies. Holographically, the result is reproduced in AdS$\mathcal A(\mathcal T)$9 topologically massive gravity, where

$$\mathcal A(\mathcal T)=\mathcal A(\mathcal E_{\mathcal T})$$0

and the timelike entropy is obtained from the on-shell action of a massive spinning particle, with the Chern–Simons term measuring the twist or rotation of the normal frame (Chu et al., 28 Apr 2025).

Boundary conformal field theory introduces further kinematic structure. For a pure timelike interval in BCFT$$\mathcal A(\mathcal T)=\mathcal A(\mathcal E_{\mathcal T})$$1, three phases appear. In the bulk phase,

$$\mathcal A(\mathcal T)=\mathcal A(\mathcal E_{\mathcal T})$$2

In the boundary phase,

$$\mathcal A(\mathcal T)=\mathcal A(\mathcal E_{\mathcal T})$$3

which is purely real and includes the boundary entropy. In the Regge phase near $$\mathcal A(\mathcal T)=\mathcal A(\mathcal E_{\mathcal T})$$4, the entropy has no boundary entropy term and can be real or complex depending on the side from which the Regge limit is approached. The AdS/BCFT dual reproduces these three phases through different geodesic configurations (Chu et al., 2023).

Irrelevant deformations provide another selective probe. In $$\mathcal A(\mathcal T)=\mathcal A(\mathcal E_{\mathcal T})$$5-deformed CFT$$\mathcal A(\mathcal T)=\mathcal A(\mathcal E_{\mathcal T})$$6, finite-size and finite-temperature systems behave differently: in the finite-size system only timelike entanglement entropy receives a correction, while in the finite-temperature system only the usual spacelike entanglement entropy is corrected at leading order (Jiang et al., 2023). This complements the cylinder-exchange symmetry between space and time and supports the claim that a “general entanglement entropy” on the cylinder requires both spacelike and timelike sectors.

Lorentz-symmetry breaking sharpens the sensitivity of timelike entropy. In non-relativistic holographic theories with hyperscaling violation and Lifshitz-like anisotropy, the properties of the extremal surfaces and of the entropy depend heavily on the symmetry-breaking parameters. In the hyperscaling-violating case, the scaling of the finite part is modified to

$$\mathcal A(\mathcal T)=\mathcal A(\mathcal E_{\mathcal T})$$7

and for $$\mathcal A(\mathcal T)=\mathcal A(\mathcal E_{\mathcal T})$$8 one finds a Fermi-surface signal: $$\mathcal A(\mathcal T)=\mathcal A(\mathcal E_{\mathcal T})$$9 Thus the logarithmic real part and the constant imaginary part $\mathcal E_{\mathcal T}$0 both identify Fermi surfaces, and the imaginary part carries explicit dependence on the Lifshitz exponent (Afrasiar et al., 2024).

Non-conformal confining backgrounds exhibit an additional geometric transition. In holographic confining theories, the timelike entropy is built from spacelike surfaces plus a finite timelike bulk surface with mirror symmetry, merged at the infrared tip of the geometry. There exists a critical length within which a connected non-trivial surface can exist and the imaginary part is non-zero. Beyond that critical length, the timelike part collapses to the infrared tip, the imaginary contribution vanishes, and the connected timelike branch ceases to exist (Afrasiar et al., 2024).

6. Time dependence, black holes, and unresolved questions

Real-time dynamics shows that timelike entanglement is not restricted to stationary states. In a $\mathcal E_{\mathcal T}$1-dimensional CFT with a global quench prepared by a boundary state, the spacetime-density-matrix formalism gives a timelike entropy

$\mathcal E_{\mathcal T}$2

in the large-time regime. Its notable features are time-independence with respect to the absolute times, dependence only on the temporal separation $\mathcal E_{\mathcal T}$3, and the persistence of the constant imaginary term. In local quenches, the excess Rényi entropy is piecewise and determined by the operator quantum dimension $\mathcal E_{\mathcal T}$4, while the universal imaginary part remains supplied by the vacuum twist sector (Guo et al., 23 Dec 2025).

Black-hole backgrounds turn timelike entropy into a probe of the interior. In higher-dimensional AdS–Schwarzschild geometries, timelike extremal surfaces exhibit a dimension-dependent critical turning point at which the boundary subsystem length diverges. For large subsystem length, the finite part of the holographic timelike entropy takes a “volume-plus-area” form,

$\mathcal E_{\mathcal T}$5

with a real volume term and a complex coefficient for the area term. The same analysis introduces a timelike entanglement density and shows that, in the large-$\mathcal E_{\mathcal T}$6 regime, its monotonicity is violated rather than supporting a general timelike area theorem. Near the horizon, both the spacelike and timelike branches show exponential growth of the form

$\mathcal E_{\mathcal T}$7

saturating the MSS chaos bound (Afrasiar et al., 24 Dec 2025).

A more explicitly interior-focused proposal studies a timelike strip in Schwarzschild–AdS and in charged scalar-hairy black holes through a Complex-valued Weak Extremal Surface prescription. In Schwarzschild–AdS, the real part grows linearly at large temporal width and the imaginary part is

$\mathcal E_{\mathcal T}$8

so the imaginary contribution is not merely a branch constant but carries horizon data. In the hairy geometries, a critical temporal width $\mathcal E_{\mathcal T}$9 separates Type-I and Type-II interiors; for $S(A)=-\mathrm{Tr}(\rho_A\log\rho_A),$00 the system is in a distinct timelike entanglement phase dominated purely by timelike contributions up to regulator effects, while for $S(A)=-\mathrm{Tr}(\rho_A\log\rho_A),$01 spacelike entanglement re-emerges. The proposal further suggests that a Cauchy horizon drives $S(A)=-\mathrm{Tr}(\rho_A\log\rho_A),$02 to infinity, producing pure timelike entanglement (Li et al., 26 Jan 2026).

These developments leave several basic questions open. The geometric proposal based on complex extremal surfaces explicitly lists the unresolved issues: uniqueness and saddle selection, a rigorous Picard–Lefschetz prescription, precise homology in complex geometry, a Lorentzian replica-trick derivation, first-law and modular-Hamiltonian interpretations for timelike subregions, strong-subadditivity analogues for complex areas, higher-curvature corrections, and extension beyond strips and black branes (Heller et al., 2024). The operator-algebraic program adds a different challenge: it establishes a real-valued timelike entropy equal to spatial-ball entropy and thereby sharpens, rather than removes, the conceptual divide between algebraic and pseudoentropic notions (Jiang et al., 25 Mar 2025).

Timelike entanglement entropy is therefore best viewed as a technically rich and still unsettled domain. In one branch of the subject it is a complex pseudoentropy governed by Lorentzian branch structure, twist-operator commutators, and complex or mixed-signature holographic saddles; in another it is a real algebraic entropy tied to the timelike tube theorem. The convergence of these viewpoints, if it exists, remains one of the central problems in the theory of entanglement across time.