Temporal Entanglement Entropy

• Temporal Entanglement Entropy (TEE) is the extension of conventional spatial entanglement to time- or Euclidean-time-separated regions, often resulting in complex-valued measures referred to as pseudo entropy.
• TEE techniques include analytic continuation in 2D CFT, branch-point twist field constructions in massive QFT, holographic prescriptions, and history-space reductions in quantum circuits.
• TEE provides practical insights into quantum anomalies, the interplay between space-time dualities, and the influence of temporal coarse-graining on entanglement dynamics.

Temporal Entanglement Entropy (TEE) denotes a family of entanglement-like constructions in which the relevant bipartition is associated with time-separated or Euclidean-time-separated regions rather than a purely spatial subregion. Across the literature, the term covers several non-equivalent objects: a timelike analytic continuation of ordinary interval entanglement entropy in relativistic QFT, a Euclidean “temporal” extension defined by tracing over a Euclidean-time interval, a branch-point-twist-field construction using timelike correlators in massive $1+1$-dimensional QFT, holographic prescriptions based on complex or piecewise extremal surfaces, a history-space von Neumann entropy obtained by tracing over time slots, and an entanglement measure of influence matrices in space-time-dual quantum circuits. A common structural feature is that the resulting quantity is often complex, which is why several works relate it to pseudo entropy rather than to the von Neumann entropy of an ordinary spatial reduced density matrix (Doi et al., 2023, Guo et al., 2024, Grieninger et al., 2023, Castro-Alvaredo, 21 Mar 2026).

1. Competing definitions and conceptual scope

TEE does not have a single universal definition. In two-dimensional conformal field theory, one influential definition takes timelike entanglement entropy to be the analytic continuation of the twist-field correlator from Euclidean to Lorentzian signature,

$\tau\to it+\epsilon,\qquad \tau'\to it'+\epsilon',$

so that timelike-separated endpoints are allowed and the resulting entropy is generally complex-valued (Guo et al., 2024). A closely related formulation describes timelike entanglement entropy as a Wick rotation that changes a spacelike boundary subregion to a timelike one, yielding a complex-valued measure that the authors argue should be interpreted as pseudo entropy, namely the von Neumann entropy of a reduced transition matrix (Doi et al., 2023).

A distinct Euclidean notion is introduced by defining “temporal entanglement entropy” as the entropy associated with tracing over a Euclidean-time interval. In this construction, one Wick rotates timelike entanglement entropy to Euclidean time and interprets tracing over a larger Euclidean time interval as temporal coarse-graining, with an explicit holographic implementation in cutoff AdS$_3$ and $T\bar T$-deformed theories (Grieninger et al., 2023). In non-relativistic holography, temporal entanglement entropy is likewise presented as the Euclidean-signature extension of timelike entanglement entropy, with the explicit warning that “the temporal entanglement entropy is not equal to the tEE” (Afrasiar et al., 2024).

Other works define TEE directly from a timelike two-point function of branch-point twist fields in massive $1+1$D QFT,

$S_n(t)=\frac{1}{1-n}\log \left[ \frac{\varepsilon^{4\Delta_n}\, {}_n\!\langle \Psi|\mathcal{T}(0,0)\tilde{\mathcal{T}(0,t)|\Psi_n\rangle} {}_n\!\langle \Psi_n|\Psi_n\rangle} \right],$

or from a time-reduced history density matrix in a multi-time history Hilbert space,

$$S_{\mathrm{temp}}=-\operatorname{Tr}(\rho^{\{j\}}\log\rho^{\{j\}}),$$

or from the entanglement of influence matrices regarded as states on a temporal slice in chaotic quantum circuits (Castro-Alvaredo, 21 Mar 2026, Castellani, 2021, Foligno et al., 2023). This suggests that “TEE” is best understood as an umbrella term for several non-identical temporal generalizations of entanglement.

2. Analytic continuation in relativistic QFT and $2$d CFT

In the analytic-continuation approach, the starting point is the standard replica expression for interval entanglement entropy in $2$d CFT. For a vacuum interval with endpoints $P=(t_P,x_P)$, $\tau\to it+\epsilon,\qquad \tau'\to it'+\epsilon',$0, one obtains

$\tau\to it+\epsilon,\qquad \tau'\to it'+\epsilon',$1

which, after continuation to a timelike interval, becomes

$\tau\to it+\epsilon,\qquad \tau'\to it'+\epsilon',$2

For a purely timelike interval with $\tau\to it+\epsilon,\qquad \tau'\to it'+\epsilon',$3 and $\tau\to it+\epsilon,\qquad \tau'\to it'+\epsilon',$4,

$\tau\to it+\epsilon,\qquad \tau'\to it'+\epsilon',$5

The same additive imaginary term appears for a CFT on a circle and for a thermal CFT on the infinite line (Doi et al., 2023).

Within the twist-operator formalism of two-dimensional CFT, the vacuum Lorentzian Rényi entropy is

$\tau\to it+\epsilon,\qquad \tau'\to it'+\epsilon',$6

with

$\tau\to it+\epsilon,\qquad \tau'\to it'+\epsilon',$7

For timelike separation with $\tau\to it+\epsilon,\qquad \tau'\to it'+\epsilon',$8, $\tau\to it+\epsilon,\qquad \tau'\to it'+\epsilon',$9, this becomes

$_3$0

and the $_3$1 limit gives a constant imaginary contribution $_3$2 in the vacuum timelike entanglement entropy (Guo et al., 2024).

The same analytic-continuation logic extends to anomalous $_3$3d CFTs with $_3$4. There the twist field carries scaling dimension

$_3$5

and spin

$_3$6

For a pure timelike interval, the paper finds

$_3$7

so that the real part depends on $_3$8 while the imaginary part depends asymmetrically on the chiral central charges and, in the conventions used there, only on $_3$9. This asymmetric imaginary part is proposed as a probe of gravitational anomaly (Chu et al., 28 Apr 2025).

A further development formulates temporal entanglement as a systematic continuation of ordinary entanglement entropy by rigidly transporting a spatial subregion on a flat slice through boosted slices across the light cone. In holographic theories, the prescription is to analytically continue all codimension-two bulk extremal surfaces satisfying the homology constraint and then pick the continuation with the smallest real part of the area as the leading saddle. One explicit conclusion is that analytic continuation and saddle minimization do not commute (Heller et al., 23 Jul 2025).

3. Twist fields, operator reconstruction, and temporal correlators

A central result in two-dimensional CFT is the relation between timelike and spacelike entanglement entropy established for a “diverse range of states.” Using lightcone coordinates

$T\bar T$0

the timelike entropy is expressed as a linear combination of spacelike entropy on the $T\bar T$1 slice, first-order time derivatives, and one mixed second derivative: $T\bar T$2 The derivation uses operator reconstruction inside the causal domain and an exact decomposition in the $T\bar T$3d massless free scalar, then identifies scalar correlators with $T\bar T$4, $T\bar T$5, and $T\bar T$6 (Guo et al., 2024).

The same work gives a twist-operator interpretation of the first-order derivatives by defining

$T\bar T$7

For all states explicitly analyzed, only four operator types contribute in the causal-domain expansion: $T\bar T$8 The imaginary part of timelike entanglement entropy is then attributed to the non-commutativity between the twist operator and its first-order temporal derivative. Concretely,

$T\bar T$9

is written as an integral of equal-time commutators such as

$1+1$0

and the expected local form

$1+1$1

reproduces the constant imaginary piece $1+1$2 in the entanglement-entropy limit (Guo et al., 2024).

A different branch-point-twist-field program in massive $1+1$3D QFT defines temporal Rényi entropies directly through the timelike correlator

$1+1$4

In that framework the temporal von Neumann entropy has the asymptotic form

$1+1$5

and, more generally,

$1+1$6

Because $1+1$7 is evaluated at imaginary argument, the entropy is complex, oscillatory, and power-law damped. The paper argues that spatial and temporal entropies are “two sides of the same coin,” related by $1+1$8, and that both encode universal data such as the UV central charge and the mass spectrum (Castro-Alvaredo, 21 Mar 2026).

4. Holographic realizations across relativistic, non-relativistic, and black-hole settings

One holographic formulation defines timelike entanglement entropy as the total complex-valued area of a stationary combination of spacelike and timelike extremal surfaces homologous to a timelike boundary region. In AdS$1+1$9, this reproduces

$S_n(t)=\frac{1}{1-n}\log \left[ \frac{\varepsilon^{4\Delta_n}\, {}_n\!\langle \Psi|\mathcal{T}(0,0)\tilde{\mathcal{T}(0,t)|\Psi_n\rangle} {}_n\!\langle \Psi_n|\Psi_n\rangle} \right],$0

for the vacuum, and in BTZ yields

$S_n(t)=\frac{1}{1-n}\log \left[ \frac{\varepsilon^{4\Delta_n}\, {}_n\!\langle \Psi|\mathcal{T}(0,0)\tilde{\mathcal{T}(0,t)|\Psi_n\rangle} {}_n\!\langle \Psi_n|\Psi_n\rangle} \right],$1

The same paper interprets timelike entanglement entropy as pseudo entropy and relates AdS timelike EE to dS holographic pseudo entropy by double Wick rotation (Doi et al., 2023).

Later holographic work extends timelike entanglement entropy beyond $S_n(t)=\frac{1}{1-n}\log \left[ \frac{\varepsilon^{4\Delta_n}\, {}_n\!\langle \Psi|\mathcal{T}(0,0)\tilde{\mathcal{T}(0,t)|\Psi_n\rangle} {}_n\!\langle \Psi_n|\Psi_n\rangle} \right],$2d CFTs. A Lorentzian RT-like prescription is formulated using a codimension-two extremal surface $S_n(t)=\frac{1}{1-n}\log \left[ \frac{\varepsilon^{4\Delta_n}\, {}_n\!\langle \Psi|\mathcal{T}(0,0)\tilde{\mathcal{T}(0,t)|\Psi_n\rangle} {}_n\!\langle \Psi_n|\Psi_n\rangle} \right],$3 and the functional

$S_n(t)=\frac{1}{1-n}\log \left[ \frac{\varepsilon^{4\Delta_n}\, {}_n\!\langle \Psi|\mathcal{T}(0,0)\tilde{\mathcal{T}(0,t)|\Psi_n\rangle} {}_n\!\langle \Psi_n|\Psi_n\rangle} \right],$4

with $S_n(t)=\frac{1}{1-n}\log \left[ \frac{\varepsilon^{4\Delta_n}\, {}_n\!\langle \Psi|\mathcal{T}(0,0)\tilde{\mathcal{T}(0,t)|\Psi_n\rangle} {}_n\!\langle \Psi_n|\Psi_n\rangle} \right],$5 in Lorentzian signature. For AdS$S_n(t)=\frac{1}{1-n}\log \left[ \frac{\varepsilon^{4\Delta_n}\, {}_n\!\langle \Psi|\mathcal{T}(0,0)\tilde{\mathcal{T}(0,t)|\Psi_n\rangle} {}_n\!\langle \Psi_n|\Psi_n\rangle} \right],$6 slabs, the renormalized entropy scales as

$S_n(t)=\frac{1}{1-n}\log \left[ \frac{\varepsilon^{4\Delta_n}\, {}_n\!\langle \Psi|\mathcal{T}(0,0)\tilde{\mathcal{T}(0,t)|\Psi_n\rangle} {}_n\!\langle \Psi_n|\Psi_n\rangle} \right],$7

while for timelike interval $S_n(t)=\frac{1}{1-n}\log \left[ \frac{\varepsilon^{4\Delta_n}\, {}_n\!\langle \Psi|\mathcal{T}(0,0)\tilde{\mathcal{T}(0,t)|\Psi_n\rangle} {}_n\!\langle \Psi_n|\Psi_n\rangle} \right],$8,

$S_n(t)=\frac{1}{1-n}\log \left[ \frac{\varepsilon^{4\Delta_n}\, {}_n\!\langle \Psi|\mathcal{T}(0,0)\tilde{\mathcal{T}(0,t)|\Psi_n\rangle} {}_n\!\langle \Psi_n|\Psi_n\rangle} \right],$9

For purely timelike slabs $$S_{\mathrm{temp}}=-\operatorname{Tr}(\rho^{\{j\}}\log\rho^{\{j\}}),$$0, the dependence is $$S_{\mathrm{temp}}=-\operatorname{Tr}(\rho^{\{j\}}\log\rho^{\{j\}}),$$1 with dimension-dependent phases, and in $$S_{\mathrm{temp}}=-\operatorname{Tr}(\rho^{\{j\}}\log\rho^{\{j\}}),$$2 the behavior becomes logarithmic (Nunez et al., 18 Aug 2025).

A top-down Lorentzian prescription avoids analytic continuation by introducing a signature parameter $$S_{\mathrm{temp}}=-\operatorname{Tr}(\rho^{\{j\}}\log\rho^{\{j\}}),$$3 directly in the bulk metric and extremizing a Lorentzian generalized area. For strip geometries in holographic CFTs, the exact time-like separation satisfies

$$S_{\mathrm{temp}}=-\operatorname{Tr}(\rho^{\{j\}}\log\rho^{\{j\}}),$$4

and the finite regularized strip tEE scales as $$S_{\mathrm{temp}}=-\operatorname{Tr}(\rho^{\{j\}}\log\rho^{\{j\}}),$$5. The same work introduces a stability criterion

$$S_{\mathrm{temp}}=-\operatorname{Tr}(\rho^{\{j\}}\log\rho^{\{j\}}),$$6

with $$S_{\mathrm{temp}}=-\operatorname{Tr}(\rho^{\{j\}}\log\rho^{\{j\}}),$$7 interpreted as a stable embedding and $$S_{\mathrm{temp}}=-\operatorname{Tr}(\rho^{\{j\}}\log\rho^{\{j\}}),$$8 as an unstable one, and uses it to diagnose phase-transition-like behavior in confining backgrounds (Nunez et al., 26 May 2025).

In Lorentz-breaking holographic theories, timelike entanglement entropy is defined as a union of spacelike and timelike extremal surfaces, with the real and imaginary parts coming from the two sectors respectively. The Euclidean “temporal entanglement entropy” is then defined through a Euclidean extremal surface anchored on a strip extended along Euclidean time. In anisotropic Lifshitz-like geometries, the temporal entropy depends on orientation; for hyperscaling-violating metrics it scales as

$$S_{\mathrm{temp}}=-\operatorname{Tr}(\rho^{\{j\}}\log\rho^{\{j\}}),$$9

while in the Fermi-surface regime $2$0 it becomes logarithmic,

$2$1

matching the real part of the timelike entropy. In the same regime the Lorentzian tEE acquires a constant imaginary part

$2$2

which is proposed as an additional Fermi-surface diagnostic (Afrasiar et al., 2024).

A three-dimensional Lifshitz example provides an analytic comparison of four holographic prescriptions. All four yield

$2$3

so that both the real and imaginary parts depend on the anisotropic exponent $2$4. This result is presented as evidence that TEE is sensitive to Lifshitz anisotropy even when ordinary spatial EE in that setup is not (Jena et al., 2024).

In asymptotically AdS black holes with a spacelike singularity and no inner horizon, the Complex-valued Weak Extremal Surface prescription yields a late-time linear growth of the real part of time-like entanglement entropy. If

$2$5

and $2$6 is the location of the critical extremal surface defined by $2$7, then

$2$8

The imaginary part,

$2$9

is independent of the temporal width $2$0. The existence of the critical surface is tied to a Kasner-like interior and the null energy condition (Li et al., 19 Jun 2026).

5. Euclidean-time coarse-graining, history-space entropy, and circuit-based temporal entanglement

A Euclidean holographic formulation interprets temporal entanglement entropy as the entropy associated with tracing over a Euclidean-time interval of size $2$1. In Euclidean AdS$2$2, the induced line element along a bulk curve $2$3 gives the area functional

$2$4

and the interval size is related to the turning point by

$2$5

At finite cutoff, the entropy depends on $2$6 and the cutoff scale $2$7 through

$2$8

and a lower bound on the resolvable temporal interval is

$2$9

This is interpreted as a direct relation between UV cutoff and temporal resolution, supporting the claim that tracing over Euclidean time corresponds to coarse-graining and can be connected to RG flow and momentum-space entanglement (Grieninger et al., 2023).

A more literal temporal generalization appears in the history-vector formalism. There a history state is

$P=(t_P,x_P)$0

with $P=(t_P,x_P)$1 denoting tensor product over times. The history density operator

$P=(t_P,x_P)$2

can be partially traced over a subset of times $P=(t_P,x_P)$3, producing the time-reduced density matrix

$P=(t_P,x_P)$4

Temporal entanglement entropy is then defined as

$P=(t_P,x_P)$5

In the two explicit circuit examples discussed there, the entangler circuit gives TEE $P=(t_P,x_P)$6, while in the teleportation circuit

$P=(t_P,x_P)$7

ranging from $P=(t_P,x_P)$8 to $P=(t_P,x_P)$9 depending on the input state (Castellani, 2021).

A separate circuit-based framework defines temporal entanglement as the entanglement of influence matrices arising in space-time duality. For a time-like path $\tau\to it+\epsilon,\qquad \tau'\to it'+\epsilon',$00, the reduced density matrix is

$\tau\to it+\epsilon,\qquad \tau'\to it'+\epsilon',$01

and the temporal Rényi entropies are $\tau\to it+\epsilon,\qquad \tau'\to it'+\epsilon',$02. The paper proves that temporal entanglement generically follows a volume law in time, but identifies two marginal cases: pure space evolution in generic chaotic systems and any space-like evolution in dual-unitary circuits. In those cases, Rényi entropies with index larger than one are sub-linear in time while the von Neumann entropy grows linearly. The stated mechanism is the existence of a product state with large overlap with the influence matrices, producing a temporal entanglement spectrum with a few large Schmidt values and many exponentially small ones (Foligno et al., 2023).

6. Regimes of validity, common features, and unresolved issues

Several recurring properties are shared across otherwise distinct definitions. First, many constructions yield a complex quantity. In the analytic-continuation and holographic timelike formulations, the imaginary part is often a constant proportional to a central charge or to a Lifshitz exponent, such as $\tau\to it+\epsilon,\qquad \tau'\to it'+\epsilon',$03, $\tau\to it+\epsilon,\qquad \tau'\to it'+\epsilon',$04, or $\tau\to it+\epsilon,\qquad \tau'\to it'+\epsilon',$05 (Doi et al., 2023, Chu et al., 28 Apr 2025, Afrasiar et al., 2024). Second, multiple works explicitly interpret temporal or timelike entanglement as pseudo entropy or as an entropy-like quantity rather than as the entropy of a standard reduced density matrix on a Cauchy slice (Doi et al., 2023, Guo et al., 2024, Nunez et al., 18 Aug 2025). Third, several proposals use analytic continuation from ordinary spatial entanglement entropy, whether through interval formulas, twist correlators, or the continuation of holographic extremal surfaces (Guo et al., 2024, Castro-Alvaredo, 21 Mar 2026, Heller et al., 23 Jul 2025).

At the same time, the literature repeatedly emphasizes non-universality. The relation between timelike and spacelike entanglement entropy in $\tau\to it+\epsilon,\qquad \tau'\to it'+\epsilon',$06d CFT is stated not to hold for arbitrary states; generic twist-operator OPE contributions can generate additional imaginary parts that are not captured by the linear-combination formula (Guo et al., 2024). The holographic dictionary for time-like or temporal entanglement entropy is described as “not settled” or “not yet settled” in work based on piecewise or complex extremal surfaces (Li et al., 19 Jun 2026, Nunez et al., 18 Aug 2025, Nunez et al., 26 May 2025). Euclidean temporal entanglement entropy is explicitly distinguished from Lorentzian timelike entanglement entropy in non-relativistic holography (Afrasiar et al., 2024). The top-down Lorentzian prescription likewise states that it is a robust computational framework without resolving the full conceptual issue of which surface should be regarded as the correct Lorentzian minimizer (Nunez et al., 26 May 2025).

A plausible implication is that the phrase “Temporal Entanglement Entropy” now names a research program rather than a single axiomatically fixed observable. Within that program, the best-developed technical strands are: analytic continuation in $\tau\to it+\epsilon,\qquad \tau'\to it'+\epsilon',$07d CFT and pseudo entropy, branch-point-twist-field constructions in massive $\tau\to it+\epsilon,\qquad \tau'\to it'+\epsilon',$08D QFT, Euclidean-time coarse-graining and cutoff holography, complex or piecewise holographic extremal-surface prescriptions across dimensions, history-space temporal reduction, and temporal entanglement of influence matrices in space-time-dual circuit dynamics (Doi et al., 2023, Castro-Alvaredo, 21 Mar 2026, Grieninger et al., 2023, Castellani, 2021, Foligno et al., 2023).