Time-Like Entanglement Entropy: QFT & Holography

• Time-like entanglement entropy is an extension of conventional entanglement to timelike intervals, defined through analytic continuation, pseudo-entropy, and operator-algebraic methods.
• It features a universal imaginary term in 2D CFTs and is applied in holographic duals, black-hole physics, and time-dependent QFT to probe causal and modular structures.
• Distinct frameworks—ranging from reduced transition matrices to causal diamond approaches and complex extremal surfaces—highlight theoretical diversity and ongoing debates over its physical interpretation.

Searching arXiv for papers on timelike entanglement entropy and closely related holographic/QFT constructions. {"query":"all:\"timelike entanglement entropy\" OR ti:\"timelike entanglement entropy\" OR abs:\"timelike entanglement entropy\"","max_results":10,"sort_by":"submittedDate","sort_order":"descending"} I found recent arXiv entries directly relevant to the topic, including operator-algebraic, holographic, black-hole, non-relativistic, and time-dependent formulations: (Jiang et al., 25 Mar 2025, Nunez et al., 26 May 2025, Nunez et al., 18 Aug 2025, Li et al., 21 Nov 2025, Afrasiar et al., 24 Dec 2025, Li et al., 26 Jan 2026, Ladghami et al., 6 Feb 2026), as well as foundational 2022–2024 papers such as (Doi et al., 2023, Li et al., 2022, He et al., 2023, Jiang et al., 2023, Jiang et al., 2023, Chu et al., 2023, Guo et al., 2024, Afrasiar et al., 2024, Heller et al., 2024, Jena et al., 2024, Afrasiar et al., 2024), and (Grieninger et al., 2023). Time-like entanglement entropy denotes an entanglement-like quantity assigned to timelike intervals or timelike strips rather than to spacelike subregions. In much of the recent $1+1$-dimensional CFT and holography literature, it is defined by analytic continuation of ordinary entanglement entropy, by a reduced transition matrix or pseudo-entropy, or by a spacetime density matrix, and it is generally complex; in the simplest relativistic $2$D examples one repeatedly finds

$$S_T=\frac{c}{3}\ln\frac{\Delta t}{\epsilon}+\frac{i\pi c}{6}$$

for a purely timelike interval of duration $\Delta t$ and UV cutoff $\epsilon$ (Doi et al., 2023). A distinct operator-algebraic formulation instead defines the entropy of a timelike interval $\mathcal T$ as the entropy of its causal diamond $\mathcal E_{\mathcal T}$ using the timelike tube theorem, and in that framework the result is real-valued (Jiang et al., 25 Mar 2025). The subject therefore comprises a family of related, but not identical, constructions linking causality, modular structure, replica methods, and bulk extremal geometry.

1. Foundational definitions and competing frameworks

A recurring starting point is the statement that in a Lorentz-invariant QFT the entanglement entropy of a region $A$ should depend on its domain of dependence $\mathcal D(A)$ and on the regulator, rather than on a particular spacelike slice. In the timelike case, this viewpoint leads to a corresponding dependence on $\mathcal D(A_t)$ and on an appropriate timelike regulator $2$0, heuristically

$2$1

with the spacelike short-distance cutoff replaced by a short time-scale cutoff (He et al., 2023).

One influential line of work identifies time-like entanglement entropy with pseudo-entropy. In that language, one introduces a non-Hermitian reduced transition matrix

$2$2

and defines

$2$3

Ordinary entanglement entropy is recovered when $2$4 (Doi et al., 2023). Closely related constructions use a spacetime density matrix $2$5 associated to two Cauchy slices and define Rényi and von Neumann entropies after tracing out complements on both slices; in that framework, timelike and spacelike intervals are treated within a unified replica formalism (Guo et al., 23 Dec 2025).

A materially different framework is operator-algebraic. There the observable algebra $2$6 associated with a timelike segment is equated, by the Borchers-Araki timelike tube theorem, with the algebra of its timelike envelope or causal diamond: $2$7 The entropy of $2$8 is then defined as the entanglement entropy of $2$9, using the split property and type-I approximants. In this formulation, the timelike entropy is real and, in $$S_T=\frac{c}{3}\ln\frac{\Delta t}{\epsilon}+\frac{i\pi c}{6}$$0D CFT at zero temperature, takes the form

$$S_T=\frac{c}{3}\ln\frac{\Delta t}{\epsilon}+\frac{i\pi c}{6}$$1

without an imaginary contribution (Jiang et al., 25 Mar 2025).

These definitions are not equivalent by construction. The complex-valued pseudo-entropy literature and the real-valued algebraic literature assign different roles to analytic continuation, state pairs, and causal algebras. A common misconception is that there is already a single universally accepted definition; the current literature instead exhibits multiple formulations with overlapping motivations but distinct mathematical inputs.

2. $$S_T=\frac{c}{3}\ln\frac{\Delta t}{\epsilon}+\frac{i\pi c}{6}$$2D CFT constructions and the universal imaginary term

For a single interval in a $$S_T=\frac{c}{3}\ln\frac{\Delta t}{\epsilon}+\frac{i\pi c}{6}$$3D CFT on Minkowski space, written in null coordinates $$S_T=\frac{c}{3}\ln\frac{\Delta t}{\epsilon}+\frac{i\pi c}{6}$$4, $$S_T=\frac{c}{3}\ln\frac{\Delta t}{\epsilon}+\frac{i\pi c}{6}$$5, the standard spacelike result is

$$S_T=\frac{c}{3}\ln\frac{\Delta t}{\epsilon}+\frac{i\pi c}{6}$$6

Analytically continuing to a timelike interval amounts to $$S_T=\frac{c}{3}\ln\frac{\Delta t}{\epsilon}+\frac{i\pi c}{6}$$7, so that $$S_T=\frac{c}{3}\ln\frac{\Delta t}{\epsilon}+\frac{i\pi c}{6}$$8, producing

$$S_T=\frac{c}{3}\ln\frac{\Delta t}{\epsilon}+\frac{i\pi c}{6}$$9

For a purely timelike cut, this is the familiar

$\Delta t$0

(He et al., 2023).

Replica derivations in $\Delta t$1D CFT implement this continuation directly in twist-operator correlators. In the spacetime density matrix approach, $\Delta t$2 is represented by a Schwinger-Keldysh replica path integral with twist insertions at the interval endpoints, and the real-time continuation $\Delta t$3 generates a nontrivial branch structure. In global and local quench states, the resulting timelike entanglement entropy remains well defined for arbitrary spacetime intervals, and timelike separations universally exhibit a constant imaginary contribution (Guo et al., 23 Dec 2025).

The same constant appears in several other settings. In the vacuum, in BTZ, and in boundary CFT bulk phases, the imaginary part is again $\Delta t$4 (Chu et al., 2023). In the relation between timelike and spacelike entropies derived for a broad range of $\Delta t$5-dimensional states, the imaginary contribution is traced to the non-commutativity between the twist operator and its first-order temporal derivative; specifically, the relevant equal-time commutator is nonzero at order $\Delta t$6 in the replica index (Guo et al., 2024).

This repeated appearance of $\Delta t$7 has sometimes been interpreted as purely kinematical. That interpretation is accurate for many $\Delta t$8D relativistic CFT examples, but later black-hole and higher-dimensional results indicate that the imaginary sector need not always remain a state-independent constant.

3. Holographic prescriptions

The earliest holographic prescriptions generalized the Ryu-Takayanagi or HRT logic by allowing extremal surfaces with both spacelike and timelike segments. In AdS$\Delta t$9, the real part is carried by spacelike legs and the imaginary part by a timelike segment, yielding precisely the analytically continued field-theory answer for pure AdS and BTZ (Doi et al., 2023).

A technical difficulty is non-uniqueness. For a timelike boundary interval, naive piecewise assemblies of spacelike and timelike geodesics can produce infinitely many complex areas. To address this, the notion of a complex-valued weak extremal surface (CWES) was introduced. A CWES is piecewise smooth, each segment is extremal in its own causal class, and the junctions are stationary under infinitesimal corner variations. The prescription then minimizes the complex area using an ordering that first compares imaginary parts and then real parts. In Poincaré AdS$\epsilon$0, global AdS$\epsilon$1, and BTZ, this selects a unique complex area and reproduces the known analytic-continuation results (Li et al., 2022).

A conceptually different holographic derivation uses the Rindler method. For a spacelike interval, a conformal map sends the causal diamond to a thermal strip, and the bulk lift yields a horizon whose Bekenstein-Hawking entropy reproduces the ordinary entanglement entropy. Replacing $\epsilon$2 in the Rindler map gives a timelike version with the same thermal entropy as the real part,

$\epsilon$3

supplemented by the universal phase shift $\epsilon$4. In the bulk, the pulled-back horizon coincides with two spacelike geodesics, while an interior timelike segment contributes the imaginary part (He et al., 2023).

More recent work proposes that the correct bulk carriers are genuinely complex extremal surfaces in a complexified spacetime rather than only piecewise real Lorentzian surfaces. In AdS$\epsilon$5 vacuum and black branes these complex geodesics reproduce the standard formulas exactly, and in AdS$\epsilon$6 black branes they lead to multiple families of complex extremal surfaces. The existence of several saddles raises a selection problem; one proposed criterion is continuity with the analytic continuation $\epsilon$7 of the ordinary strip entropy, which favors the “vacuum-connected” branch (Heller et al., 2024).

These holographic developments also support a broader gravitational role for timelike entropy. For infinitesimal perturbations around AdS, a timelike entanglement first law

$\epsilon$8

with effective temperature $\epsilon$9 has been formulated, and in asymptotically AdS spacetimes this first law is shown to be equivalent to linearized Einstein’s equations (Li et al., 21 Nov 2025).

4. Dynamics, quenches, RG flow, and deformations

In time-dependent states of $\mathcal T$0-dimensional CFTs, timelike entanglement entropy remains computable by a unified spacetime density matrix and replica formalism. For global quenches prepared by boundary states, in the regime $\mathcal T$1 with $\mathcal T$2,

$\mathcal T$3

and, with $\mathcal T$4,

$\mathcal T$5

The real part depends solely on the temporal separation and is time-independent once $\mathcal T$6 is fixed; the imaginary part remains the same constant as in the vacuum (Guo et al., 23 Dec 2025).

For local quenches generated by a primary operator $\mathcal T$7 of quantum dimension $\mathcal T$8, the excess entropy in a rational CFT obeys

$\mathcal T$9

These results are naturally organized by a generalized quasiparticle picture in which a pair contributes whenever exactly one member intersects the spacetime interval (Guo et al., 23 Dec 2025).

A distinct RG-based notion appears in dS$\mathcal E_{\mathcal T}$0/CFT$\mathcal E_{\mathcal T}$1. There the “timelike” entanglement entropy is defined by integrating the Callan-Symanzik equation from $\mathcal E_{\mathcal T}$2 to $\mathcal E_{\mathcal T}$3, yielding

$\mathcal E_{\mathcal T}$4

For a single interval in “time”, $\mathcal E_{\mathcal T}$5, and in dS$\mathcal E_{\mathcal T}$6/CFT$\mathcal E_{\mathcal T}$7, where $\mathcal E_{\mathcal T}$8, one obtains

$\mathcal E_{\mathcal T}$9

exactly matching the length of a timelike geodesic in planar dS$A$0. The same work argues that, in both AdS$A$1/CFT$A$2 and dS$A$3/CFT$A$4, there are exactly three independent entanglement entropies, sufficient to reconstruct the three-dimensional bulk geometry (Jiang et al., 2023).

Deformations sharpen the distinction between spacelike and timelike sectors. In $A$5-deformed CFT$A$6, the finite-temperature system gives a correction only to the usual spacelike entropy, while the purely timelike entropy has no $A$7-shift: $A$8 In the finite-size system the pattern reverses: the purely timelike entropy is corrected, while the purely spacelike one is not (Jiang et al., 2023).

5. Boundary conditions, non-conformal theories, and non-relativistic systems

In AdS/BCFT, timelike entanglement entropy exhibits a richer phase structure than in translationally invariant CFT. For a pure timelike interval at distance $A$9 from the boundary, three phases occur: a bulk phase, a boundary phase, and a Regge phase associated with the limit in which one endpoint approaches the light cone of the mirror image of the other. In the bulk phase,

$\mathcal D(A)$0

while in the boundary phase,

$\mathcal D(A)$1

In the Regge phase the entropy can be real or complex depending on which side of the light cone is approached, and there is no boundary entropy term (Chu et al., 2023).

For non-conformal confining theories, a Lorentzian holographic construction merges spacelike and timelike extremal pieces at the infrared tip of the geometry. In the solitonic D4 background the total boundary length $\mathcal D(A)$2 is double-valued and bounded above by a critical length $\mathcal D(A)$3. For $\mathcal D(A)$4 a connected surface exists and $\mathcal D(A)$5; for $\mathcal D(A)$6 only the trivial configuration remains and $\mathcal D(A)$7. Near $\mathcal D(A)$8, the imaginary part diverges and then jumps to zero, producing a first-order-type phase transition (Afrasiar et al., 2024). The same work emphasizes that naive analytic continuation from Euclidean entanglement entropy can fail to capture the correct bulk homology and phase-transition structure in non-conformal theories.

Temporal entanglement entropy, the Euclidean counterpart obtained by tracing over a Euclidean time interval, has been linked to renormalization-group flow and to momentum-space entanglement. In cutoff AdS$\mathcal D(A)$9 with $\mathcal D(A_t)$0 deformation, increasing the UV cutoff enhances the resolution of finer time intervals, while tracing over a larger Euclidean time interval is formally equivalent to integrating out more UV degrees of freedom or lowering the temperature (Grieninger et al., 2023).

In non-relativistic holography, timelike entropy is highly sensitive to Lorentz invariance breaking. For three-dimensional Lifshitz spacetime with anisotropic exponent $\mathcal D(A_t)$1, several holographic prescriptions agree and give

$\mathcal D(A_t)$2

so both the real and imaginary parts scale as $\mathcal D(A_t)$3 (Jena et al., 2024). In more general hyperscaling-violating and Lifshitz-like theories, timelike entanglement can diagnose Fermi surfaces. When $\mathcal D(A_t)$4, one finds

$\mathcal D(A_t)$5

so the logarithmic real part and the constant imaginary part both signal the Fermi-surface regime (Afrasiar et al., 2024).

6. Black holes, interior probes, and current controversies

Black-hole applications have pushed timelike entanglement entropy beyond the constant-imaginary-part paradigm of $\mathcal D(A_t)$6D vacuum CFT. In a proposal for Hawking radiation, the Lorentzian black-hole geometry is analytically continued to a Euclidean section, and the entropy of a Euclidean-time interval becomes

$\mathcal D(A_t)$7

with Page times determined by

$\mathcal D(A_t)$8

Schwarzschild, Reissner-Nordström, Kerr, Myers-Perry, and higher-dimensional black holes then exhibit periodic or quasi-periodic timelike Page times governed by surface gravity and rotation (Ladghami et al., 6 Feb 2026).

A Lorentzian holographic treatment in BTZ and AdS-Schwarzschild backgrounds realizes timelike entropy through a spacelike branch for the real part and a timelike branch for the imaginary part. In BTZ the result is

$\mathcal D(A_t)$9

while in higher-dimensional AdS-Schwarzschild black holes there is a dimension-dependent critical turning point, a large-subsystem volume-plus-area structure, and exponential near-horizon growth $2$00 for both branches (Afrasiar et al., 24 Dec 2025).

For Schwarzschild-AdS and hairy black holes, timelike entropy has been proposed as a single-boundary probe of the interior. In planar Schwarzschild-AdS, after subtraction of the vacuum divergence, the real part grows linearly at large temporal width,

$2$01

while the imaginary part in $2$02 scales as

$2$03

so it carries non-trivial physical information rather than being a pure regulator (Li et al., 26 Jan 2026). In hairy black holes, a critical temporal width $2$04 separates a “time-like entanglement phase” dominated purely by timelike contributions from a regime in which spacelike contributions re-emerge; the existence of a Cauchy horizon drives $2$05 (Li et al., 26 Jan 2026).

Several controversies remain active. One concerns uniqueness: mixed Lorentzian surfaces, CWES prescriptions, and complexified extremal surfaces do not automatically pick the same saddle (Li et al., 2022). A second concerns generality: naive analytic continuation works in many $2$06D and AdS$2$07 examples, but can fail in non-conformal confining theories (Afrasiar et al., 2024). A third concerns the status of the imaginary part: in much of the pseudo-entropy literature it is a universal signal of timelike kinematics, whereas the operator-algebraic program argues for a real-valued timelike entropy because the relevant algebra is that of the causal diamond rather than a non-Hermitian transition matrix (Jiang et al., 25 Mar 2025). A plausible implication is that “time-like entanglement entropy” currently names a class of related observables rather than a single settled invariant.