Timelike Entanglement Density

• Timelike entanglement density is defined as the rate of quantum entanglement across temporal intervals, computed via analytic continuation and replica techniques in various QFT and holographic models.
• It exhibits universal features such as a constant imaginary component, scale-invariant behavior in CFTs, and nontrivial phase transitions in confining backgrounds.
• Methodologies include field-theoretic replica tricks, holographic extremal surface prescriptions, and operator formalism, providing insights into causality, non-Hermiticity, and dual gravity connections.

Timelike entanglement density is a quantitative measure of quantum entanglement associated with purely timelike-separated regions or intervals in relativistic quantum systems. Unlike the conventional notion of spatial entanglement density—entropy per unit length of a spatial interval—timelike entanglement density encodes the rate or distribution of entanglement across a temporal subregion, and is relevant in quantum field theory, holography, modern quantum information, and studies of causal structure. It is well defined and computable across a variety of settings, from free models to interacting conformal field theories (CFTs) and holographic frameworks, and is associated with distinctive complex phasing and universal dynamical features.

1. Formal Definitions and Fundamental Formulas

Given a quantum system or quantum field theory, the timelike entanglement entropy $S_{\mathrm{EE}}^{\rm timelike}(T)$ of a time interval of duration $T$ (possibly with fixed spatial position or within a spatial volume) is defined either as the analytic continuation of the entanglement entropy from spacelike to timelike configurations, or directly via an appropriate construction of the reduced density operator along the time direction. The timelike entanglement density $\rho_{\mathrm{EE}}^{\rm timelike}(T)$ is then

$$\rho_{\mathrm{EE}}^{\rm timelike}(T) = \frac{1}{T} S_{\mathrm{EE}}^{\rm timelike}(T)\ ,$$

or, more generally for a region of volume $V$,

$$\rho_{\mathrm{EE}}^{\rm timelike}(T) = \frac{1}{V} \frac{\partial S_{\mathrm{EE}}^{\rm timelike}(T)}{\partial T}\ .$$

For unitary 2D CFTs with central charge $c$, using the Rindler-method analytic continuation, the result is

$$S_{\mathrm{EE}}^{\rm timelike}(T) = \frac{c}{3} \log \left(\frac{T}{\varepsilon}\right) + i \frac{c\pi}{6}, \quad \rho_{\mathrm{EE}}^{\rm timelike}(T) = \frac{c}{3T} \log \left(\frac{T}{\varepsilon}\right) + i \frac{c\pi}{6 T}$$

with $\varepsilon$ the UV cutoff. The real part gives a scale-dependent entropic density, while the universal imaginary component arises from causal structure and commutators of twist operators (He et al., 2023, Guo et al., 23 Dec 2025).

In $d$-dimensional holographic conformal theories for slab regions (timelike strip of width $T$ and transverse area $L^{d-2}$),

$\rho_t(T) \sim T^{1-d},$

with the constant of proportionality fixed by the holographic central charge and geometric data (Nunez et al., 26 May 2025, Nunez et al., 18 Aug 2025).

2. Construction Methodologies Across Models

Field-Theoretic and Replica Approaches

In continuum relativistic QFTs, the timelike entanglement entropy is constructed via analytic continuation of the replica trick, where the Euclidean manifold of $n$-sheeted replicas for spacelike intervals is smoothly deformed to accommodate Lorentzian (timelike) separations. In 1+1D CFTs, the trace of the $n$-th power of the reduced density matrix for a timelike interval is computed via twist correlators and results in analytic expressions with a universal imaginary shift (Guo et al., 23 Dec 2025, He et al., 2023, Milekhin et al., 17 Feb 2025). In lattice models, the entropy of a set of times at a fixed spatial site is computed from reduced two-point correlators (Liu et al., 2022).

Holographic and Extremal Surface Prescriptions

In holographic QFTs, timelike entanglement entropy is given by the area of codimension-2 extremal surfaces in Lorentzian AdS or black hole backgrounds, with proper analytic continuation and extremization over surfaces homologous to the timelike region (Heller et al., 23 Jul 2025, Nunez et al., 26 May 2025, Afrasiar et al., 24 Dec 2025). The density is computed by differentiating the finite, renormalized area with respect to the temporal width or volume,

$\rho_t(T) = \frac{1}{V} \frac{d S_{tEE}(T)}{dT}$

where $V$ is the transverse volume and $S_{tEE}(T)$ is the renormalized area functional (Nunez et al., 18 Aug 2025, Nunez et al., 26 May 2025). For general dimensions,

$\rho_t(T) \sim c_{\rm hol} T^{1-d}$

with $c_{\rm hol}$ the holographic central charge (Nunez et al., 18 Aug 2025). In confining backgrounds, $\rho_t(T)$ can undergo first-order transitions and become double-valued.

Operator Formalisms and Non-Hermitian Density Matrices

Alternative frameworks employ the reduction of time-ordered transition operators, pseudo-density operators, or the history-based density operator, leading to generally non-Hermitian reduced density matrices whose entropy can be complex-valued. The density is defined either by rate of entropy growth per unit time or as the normalized negativity between events separated by $\Delta t$,

$\eta_N = \frac{N(R_{AB})}{\Delta t},$

where $N(R_{AB})$ is the negativity of the pseudo-density operator $R_{AB}$ (Marletto et al., 2019). The Schatten-2 norm (imagitivity) of the non-Hermitian density acts as a measure of causal influence in time-like entanglement (Harper et al., 15 Dec 2025, Milekhin et al., 17 Feb 2025).

3. Universal Features and Distinctive Regimes

Timelike entanglement density exhibits several robust universalities:

• Complex entanglement entropy: The ubiquity of a constant imaginary component, e.g., $i (c\pi/6) / T$ in 2D CFTs, reflects causality and the nontrivial commutator structure for timelike separations. This trait is observed both in vacuum and quenched states and emerges generically in replica-based and path-integral calculations (He et al., 2023, Guo et al., 23 Dec 2025, Milekhin et al., 17 Feb 2025).
• Scaling: In CFTs, the real part of $\rho_t(T)$ is scale-invariant for large $T$ in thermal states (plateau), but for short intervals, it diverges as $T\to 0$, following $T^{1-d}$ power-law scaling. In free fermion chains, an area-to-volume crossover is observed: spacelike intervals yield vanishing $\rho(\ell)/\ell$ (logarithmic correction), while timelike slices yield a volume law, $\rho(t)\sim\mathrm{const}$ for discrete times with $\tau\gg\tau_0=2\pi/E_0$ (Liu et al., 2022).
• Entanglement first law: The timelike first law states that for a time interval of length $T$, small perturbations satisfy

$$\delta S_{t} = \delta \langle K_{t} \rangle = \int_{-T/2}^{T/2} dt\; s(t)\, \delta\langle T_{00}(t)\rangle,$$

with the local entanglement density

$s(t) = 2\pi\,\frac{(T/2)^2 - t^2}{T}$

being maximal at the center and vanishing at endpoints. This smearing kernel encodes sensitivity to energy insertions and is in one-to-one correspondence with linearized bulk Einstein’s equations in AdS$_3$ (Li et al., 21 Nov 2025).

• Quasiparticle picture: In dynamical CFT quenches, the real part of $\rho_t$ is interpreted as counting quasiparticle pairs with only one member intersecting the timelike interval (Guo et al., 23 Dec 2025).

4. Holographic, High-dimensional, and Non-Conformal Extensions

Holographic methods supply explicit formulas and scaling laws for timelike entanglement density in a wide class of theories:

• Exact and approximate formulas: For a slab region in $d$ dimensions, $S_{tEE}(T)\simeq c_d / |T|^{d-2}$, leading to $\rho_t(T)\propto |T|^{1-d}$. In confining backgrounds, $S_{tEE}(T)$ and $\rho_t(T)$ display non-monotonic, double-valued, or even discontinuous behavior at a critical $T_c$ beyond which the connected geometric configuration ceases to exist and a disconnected phase dominates (Nunez et al., 26 May 2025, Nunez et al., 18 Aug 2025).
• Signatures and phases: In Lorentzian signature, timelike entanglement entropy generically acquires a phase $e^{-i\pi(d-2)/2}$; in mass-gapped/confining backgrounds, phase transitions in $S_{tEE}$ and $\rho_t(T)$ signal changes in the dominant geometric saddle, associated with the IR "wall" in the bulk (Nunez et al., 18 Aug 2025, Nunez et al., 26 May 2025).
• Black hole backgrounds: For AdS-Schwarzschild and higher-dimensional black holes, the near-horizon regime (large $T$) yields exponential approach to the horizon with Lyapunov exponent $2\pi/\beta$, and the density exhibits a volume-plus-area decomposition and universal late-time behavior. Subleading $1/T$ corrections to the density inform the fate of a "timelike area theorem," which is violated in certain large-$d$ limits (Afrasiar et al., 24 Dec 2025).

5. Experimental, Model, and Operational Aspects

Timelike entanglement density has been studied in explicit lattice, quantum circuit, and qubit models:

• Gaussian free fermion chains show discrete-time volume law $\rho(t)\sim\mathrm{const}$ for large time spacings, and logarithmically suppressed density in the quasi-continuous ($\tau\ll\tau_0$) regime (Liu et al., 2022).
• Quantum circuits in the history-vector formalism quantify temporal entanglement as von Neumann entropy of the reduced time-slice density matrix, with discrete and continuum definitions for density per time interval (Castellani, 2021).
• Experimental protocols using pseudo-density operator tomography for sequential timelike measurements yield a well-defined negativity-based density, and SWAP-test-based ancilla circuits access the entanglement in time in qubit arrays or on quantum hardware (Marletto et al., 2019, Milekhin et al., 17 Feb 2025).

6. Connections to Causality, Monogamy, and Gravity

• Causality and Imagitivity: The presence of nonzero imagitivity (Schatten-2 norm of $[\rho,\rho^\dagger]$) is both necessary and sufficient for the presence of timelike (causal) influence, and sets the bound for commutator expectation values in Hamiltonian-local systems (Milekhin et al., 17 Feb 2025, Harper et al., 15 Dec 2025).
• Monogamy and non-Hermiticity: Timelike entanglement densities in the pseudo-density operator formalism evade standard monogamy constraints, as they may add linearly and reach or even exceed the spatial maximum unless nonlinear corrections are imposed (Marletto et al., 2019).
• Duality with bulk gravity: In AdS/CFT, the first-law correspondence between timelike entanglement variation and the modular Hamiltonian of the boundary interval is equivalent to the linearized Einstein equations in the bulk; thus, the timelike entanglement density kernel is dual to the shape of bulk extremal surfaces (Li et al., 21 Nov 2025). In certain wormhole constructions, traversability is directly tied to timelike entanglement and the non-Hermitian structure of boundary operators (Harper et al., 15 Dec 2025).

7. Open Problems and Research Directions

• Extensions to general spacetimes: Timelike entanglement density has been generalized to de Sitter geometry and non-conformal theories, leading to real or complex values depending on the surface anchoring and initial state. There remain questions about the field-theoretic replica construction for timelike intervals in dS/CFT and the physical significance of the density in cosmological and quantum-gravitational contexts (Narayan, 2022, Narayan et al., 2023).
• Pseudo-entropy and generalizations: The connection to pseudo-entropy and the structure of the reduced "transition" operators suggests a broader unification of spacetime correlations and entropic measures, potentially encompassing both spacelike and timelike entanglement (Narayan et al., 2023, Milekhin et al., 17 Feb 2025).
• Area theorems and c-functions: The violation of a timelike area theorem in certain regimes poses conceptual puzzles for RG monotonicity and the definition of c-functions based on temporal intervals (Afrasiar et al., 24 Dec 2025).

Timelike entanglement density is thus a robust, operationally meaningful quantity that bridges quantum information, relativistic field theory, and holographic gravity, and encodes both universal and signature-dependent features of entanglement in spacetime. Its precise scaling laws, complex structure, and operational realizability make it a unique probe of causality, non-Hermiticity, and dynamical features in quantum many-body systems and their gravitational duals (He et al., 2023, Guo et al., 23 Dec 2025, Nunez et al., 18 Aug 2025, Li et al., 21 Nov 2025, Nunez et al., 26 May 2025, Milekhin et al., 17 Feb 2025).