---
title: Lorentzian Entanglement Growth Dynamics
url: https://www.emergentmind.com/topics/lorentzian-entanglement-growth
type: topic
---

# Lorentzian Entanglement Growth Dynamics

Lorentzian entanglement growth denotes several related but non-identical problems concerning entanglement in real time or Lorentzian signature. In holographic quench dynamics it refers to the time evolution of entanglement entropy computed by covariant extremal surfaces in time-dependent asymptotically AdS spacetimes [1311.1200]. In free scalar field theory it refers to the real-time growth of von Neumann and Rényi entropies after a global quench [1609.00872]. In a more specialized usage for the double Wick-rotated rotating BTZ black hole, it is defined by the coefficient of linear growth of **time-like entanglement entropy**, \(\lambda_{\mathrm{TL}}=\frac{c}{6}r_+\) [2604.15720]. The literature also treats operator-space entanglement under Heisenberg evolution, lightcone-modified correlation growth from initially entangled states, and several adjacent notions in which “Lorentzian” refers to a reservoir spectral density, a Lorentz transformation, or Lorentzian network histories rather than to a single canonical growth law [2603.25387], [1507.00529], [2202.06313], [1402.2605]. This suggests that the subject is best organized by observable and physical setting rather than by terminology alone.

## 1. Scope, observables, and meanings of the term

The central ambiguity is that the literature does not attach the phrase to one invariant quantity. In some works the object is ordinary spatial entanglement entropy \(S_{\mathcal A}(t)\) of a subregion; in others it is a time-like analogue, a Rényi entropy in Liouville space, or an equal-time connected correlator used as a proxy for information spreading. The distinction is substantive because the corresponding growth laws, bounds, and causal interpretations are different.

| Usage in the literature | Quantity | Representative result |
|---|---|---|
| Global-quench entanglement | \(S_{\mathcal A}(t)\), \(S_q(t)\) | Linear regime and geometry-dependent saturation [1609.00872] |
| Holographic thermalization | \(S_\Sigma(t)=\mathcal A_\Sigma(t)/(4G_N)\) | Quadratic, linear, memory-loss, and saturation regimes [1311.1200] |
| Time-like BTZ entropy | \(S_{\mathrm{TL}}\) | \(\lambda_{\mathrm{TL}}=\frac{c}{6}r_+\) [2604.15720] |
| Local operator entanglement | \(S_2[\rho^{(\mathcal O)}_{\mathscr A}(t)]\) | Late-time volume law in chaotic systems [2603.25387] |
| Initially entangled correlation spreading | \(\langle A(t)B(t)\rangle_{\mathrm c}\) | LR-type bound with explicit initial-state term [1507.00529] |

A second distinction concerns whether the observable diagnoses entanglement **build** or entanglement **move**. In the many-body localized setting, the Wehrl-Rényi entropy is introduced precisely because it is invariant under qubit permutations/SWAPs and therefore isolates entanglement **build** from mere entanglement **move** [2605.20656]. That separation does not appear in the usual spatial entanglement entropy of a subregion.

## 2. Real-time growth after quenches and the fate of lightcone structure

In free scalar field theory after a global quench, the subtracted entropy \(\hat S_{\mathcal A}(t)=S_{\mathcal A}(t)-S_{\mathcal A}(0)\) is studied for strips in \(1,2,3\) spatial dimensions and for spheres in \(2,3\) spatial dimensions. In the scaling regime the leading growth takes the form
\[
\hat S_{\mathcal A}(t)= v_E\, s\, \mathrm{area}(\partial\mathcal A)\, t
\qquad (\beta\ll t\ll R),
\]
with
\[
v_E= \frac{\Gamma(d/2)}{\sqrt{\pi}\,\Gamma((d+1)/2)}.
\]
For spheres the leading extensive contribution saturates at \(t_S^{(\mathrm{sphere})}=R\), whereas for strips \(t_S^{(\mathrm{strip})}=\infty\). At subleading order the numerics show an anomalous logarithmic growth \(\frac12\log t\) from the zero mode of the scalar [1609.00872].

Long-range interacting spin systems modify the same real-time picture in a different way. In a transverse-field Ising chain with algebraically decaying couplings \(J_{i,j}=\bar J/|i-j|^\alpha\), the half-chain entropy grows linearly in time for relatively short-range interactions, roughly \(\alpha\gtrsim 1\) and especially clearly for \(\alpha\gtrsim 2\), but for most of the long-range regime \(\alpha\lesssim 1\) it grows only logarithmically, \(S_{\rm vN}(t)\sim \log t\). Mutual information then shows the transition from delayed, front-like propagation to almost immediate short-time structure at long distance [1305.6880]. A central lesson is that the breakdown of a strict light cone does not imply faster bipartite entanglement growth.

Initial entanglement changes the propagation problem even when the Hamiltonian is local. For arbitrary initial states in quantum lattice systems, the connected correlator obeys a bound of the form
\[
\frac{|\langle A(t)B(t)\rangle_{\mathrm c}|}{\|A\|\|B\|}\le \mathscr B_r(t),
\]
where \(\mathscr B_r(t)\) contains a Lieb-Robinson-type dynamical term and an explicit initial-state term \((S_X(r):S_Y(r))\). In this framework, pre-existing entanglement can make distant correlations appear much earlier than for product states, but the enhancement does not increase the Lieb-Robinson velocity itself [1507.00529].

## 3. Holographic Lorentzian growth and speed limits

For homogeneous quenches in strongly coupled CFTs with gravity duals, the bulk geometry is AdS-Vaidya in the thin-shell limit, and entanglement entropy is computed by the covariant Hubeny-Rangamani-Takayanagi prescription. In the large-distance limit the extremal surface probes the geometry around and inside the event horizon, and the evolution exhibits four regimes: pre-local-equilibration quadratic growth, post-local-equilibration linear growth, a memory-loss regime, and a saturation regime [1311.1200]. The early-time result is
\[
\Delta S_\Sigma = \frac{\pi}{d-1}\,\mathcal E\,A_\Sigma\, t^2+\cdots,
\]
while for large regions after local equilibration
\[
\Delta S_\Sigma(t)= s_{\rm eq}\,A_\Sigma\,v_E\, t+O(1).
\]
In this setting the linear regime is controlled by critical extremal surfaces behind the horizon, and the slope is shape independent.

A later development turns these Lorentzian growth statements into rigorous speed limits in spatially uniform holographic states. In \(2d\) CFT, for a region that is a union of \(n\) finite intervals,
\[
\left|\frac{dS_R}{dt}\right| \le n\sqrt{\frac{8\pi c}{3}\Big(\langle T_{tt}\rangle-\langle T_{tt}\rangle_{\rm vac}\Big)}.
\]
For higher dimensions, thin-shell planar-symmetric spacetimes give corresponding area-type and volume-type bounds for strips and balls, and for small subregions one obtains
\[
\left| \frac{dS_R}{dt} \right| \le \kappa_d\,\mathrm{Vol}[R]\, \langle T_{tt}\rangle
\left[1+\mathcal O\!\left( \frac{\langle T_{tt}\rangle \ell^d}{c_{\rm eff}} \right) \right].
\]
The key structural result is the momentum-entanglement correspondence,
\[
\frac{dS_R}{dt}=\int_X G\, \mathcal T_{ab} n^a t^b,
\]
which identifies entanglement growth with bulk momentum crossing the HRT surface. The same analysis proves sharp bounds on the smallest radius an extremal surface can probe and shows that the tips of boundary-anchored extremal surfaces cannot lie in trapped regions [2211.07654].

## 4. Time-like entanglement entropy and the rotating BTZ definition

A distinct and narrower definition appears in the analysis of the double Wick rotation of the rotating BTZ black hole. There the field-theory object dual to the double Wick-rotated geometry is not an ordinary density matrix but a **transition matrix** with the usual shape at an imaginary chemical potential, and the geometric entropy is defined by tracing over a subsystem of that transition matrix [2604.15720]. The crucial Lorentzian continuation exchanges space and time and leads to a **time-like interval** rather than a spatial interval on a constant-time slice.

The corresponding time-like entanglement entropy is
\[
S_{\mathrm{TL}}=\dfrac{c}{6}\log \Big[ \dfrac{\beta^{\prime 2}(1+\Omega_E^{\prime 2})}{\pi^2\epsilon^{\prime 2}}
\sinh \Big(\dfrac{\pi\Delta \tilde{t}}{\beta'(1+i\Omega_E')}\Big)
\sinh \Big(\dfrac{\pi\Delta \tilde{t}}{\beta'(1-i\Omega_E')}\Big) \Big]
+\dfrac{\pi ic}{6}.
\]
At late times,
\[
S_{\mathrm{TL}}(t\to\infty)\simeq \frac{c\pi t}{3\beta'(1+\Omega_E'^2)}=\frac{c}{6}r_+\, t,
\]
and the coefficient
\[
\lambda_{\mathrm{TL}}=\frac{c}{6}r_+
\]
is what the paper calls the **new Lorentzian entanglement growth** [2604.15720]. The comparison with the usual Lyapunov exponent,
\[
\lambda_L=2\pi T,
\]
is central: \(\lambda_{\mathrm{TL}}\) remains finite in the extremal limit where \(\lambda_L\to 0\). The construction is therefore not a quench calculation in the usual spatial-entanglement sense, but a Lorentzian, analytically continued, time-like entropy growth law tied to complexified modular flow.

## 5. Operator-space growth and structured initial states

Local Operator Entanglement extends Lorentzian entanglement growth into operator space. A traceless local Hermitian operator \(\mathcal O\) is evolved in the Heisenberg picture, vectorized into a doubled Hilbert space, and the \(2\)-Rényi entropy of the reduced operator-state is computed:
\[
S_2\!\left[\rho^{(\mathcal O)}_{\mathscr A}(t)\right]
= -\ln\!\left[\Tr\!\left(\rho^{(\mathcal O)}_{\mathscr A}(t)\right)^2\right].
\]
Chaotic short-range systems show LOE growth at most linearly in time, while integrable systems show only logarithmic growth. The main analytical result is not a derivation of the ramp but of the late-time saturation value: under a 4-non-resonance condition, the ETH ansatz, and a Haar-random replacement of Hamiltonian eigenstates in the final expression, the late-time LOE exhibits volume-law scaling [2603.25387].

Structured initial states in many-body localized systems provide a different generalization. In the random-field XXZ chain, a product state is first evolved under a chaotic Hamiltonian for a preparation time \(T_0\), and only then quenched to the target Hamiltonian. In the localized phase the hallmark growth law is
\[
S_{\text{HC}}^{(2)}\propto \ln t,
\]
with half-chain saturation time of order \(O(e^L)\). The net late-time growth of the half-chain second Rényi entropy is non-monotonic as a function of \(T_0\) for \(z\)-polarized seeds: it first increases, then decreases, reflecting a crossover from LIOM magnetization relaxation to inter-site correlations. The same paper explicitly states that it does **not** provide a Lorentzian fitting function and does **not** claim Lorentzian time dependence of entanglement growth [2605.20656].

## 6. Adjacent usages, ambiguities, and non-equivalences

Several papers use nearby language in ways that should not be conflated with the real-time or time-like growth laws above. In continuous-variable open systems, “Lorentzian” refers to the reservoir spectral density, \(J(\omega)=1/(\omega^2+\lambda^2)\) or \(J(\omega)=\omega/(\omega^2+\lambda^2)\). The resulting non-Markovian dynamics can show entanglement sudden death, revivals, oscillatory backflow, and “always alive” behavior, but the mechanism is reservoir memory rather than a Lorentzian-signature entanglement law [2202.06313].

In relativistic two-qubit models, Lorentz transformation changes the spin-state entanglement through Wigner rotation. The boost can degrade entanglement already present in the initial state and can also create nonzero spin entanglement from some states that are initially separable; partially entangled states are reported to be more robust than maximally entangled states [1402.2605]. Here “Lorentzian” refers to frame dependence under boosts, not to entanglement growth in many-body real time.

Other Lorentzian settings are again different in kind. Quantum fields confined to Lorentzian histories of freely falling networks recover an area law for vacuum entanglement, but the calculation is static and kinematic rather than a study of \(S(t)\) after a quench [2312.17313]. Scalar cosmological perturbations in Lorentzian group field theory are sourced by nearest-neighbor two-body entanglement, but the work does not compute entanglement entropy growth, mutual information spreading, or a time-dependent entanglement measure [2308.13261]. Lorentzian AdS wormholes clarify the structure of same-boundary and cross-boundary correlators and the distinction between coupling and entanglement, but they do not compute time-dependent entanglement entropy growth [1012.4478].

Taken together, these results establish that Lorentzian entanglement growth is not a single universal law. The literature contains at least three technically distinct cores: real-time growth of spatial entanglement after quenches, Lorentzian holographic speed limits for HRT surfaces, and the time-like entropy growth defined in the rotating BTZ context. Around those cores lies a wider zone of adjacent usages in which “Lorentzian” names a reservoir, a boost, or a causal background rather than the same dynamical observable.

Source: https://www.emergentmind.com/topics/lorentzian-entanglement-growth