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

# Quantum Entanglement Growth

Entanglement growth quantifies the dynamical increase of quantum correlations between subsystems in a composite quantum many-body system. Following global or local quenches of the Hamiltonian, the bipartite or multipartite entanglement—most commonly measured by Rényi or von Neumann entropies—evolves from its initial value, revealing fundamental properties of thermalization, localization, operator spreading, and information scrambling. The scaling forms, fluctuations, and mechanisms of entanglement growth are strongly dependent on dimensionality, conservation laws, integrability, interaction range, initial-state structure, and the unitary or non-unitary nature of the dynamics. The subject occupies a central place in quantum statistical mechanics and quantum information theory.

## 1. Foundational Mechanisms of Entanglement Growth

Entanglement entropy growth is deeply connected to the spreading of initially local operators under Heisenberg evolution and to the propagation of correlations constrained by the Lieb–Robinson bound. In generic chaotic/ergodic systems, local operators spread ballistically, and their overlap with local product states decays rapidly, leading to an extensive increase in bipartite entanglement entropy. The growth rate is then bounded by the 'wavefront area'—the boundary between regions A and B—times an entanglement velocity $v_E$ set by the microscopic model [1508.03784]. The minimal-cut picture formalizes this, bounding $S(t)$ by the minimal number of spacetime gates crossed by a causal path (“entanglement tsunami” interpretation) [1608.06950, 2306.04764].

In contrast, in many-body localized (MBL) systems with emergent local integrals of motion, operator spreading is logarithmically slow: local integrals dephase with their environment, generating only logarithmic growth of entropy. The presence of conservation laws (e.g., $U(1)$ symmetry) or long-range interactions further modulates whether growth is linear, sublinear, or even saturates rapidly [1912.03645, 1305.6880].

## 2. Universal Growth Laws and Scaling Exponents

The time dependence and scaling of entanglement entropy under various dynamical protocols have been systematically classified.

### 2.1 Ballistic (Linear) Growth

For clean, chaotic, unconstrained systems:
- The entanglement entropy across a cut grows linearly with time,
  $$
  S_A(t) \simeq v_E \,\mathrm{Area}(\partial A)\, t,
  $$
  until it saturates to a volume law at late times [1508.03784, 1608.06950, 1311.1200, 2306.04764].

### 2.2 Diffusive or Sub-ballistic Regimes

- With a single diffusive $U(1)$ symmetry in 1D qubit chains ($q=2$):
  $$
  S_2(t) \sim c\sqrt{t}, \quad c = \sqrt{2/\pi},
  $$
  with the window of $\sqrt{t}$ growth persisting until finite-size saturation [1912.03645, 1907.00869]. In higher dimensions, this sub-ballistic window is parametrically narrow—for most practical times, growth remains linear.

- In monitored, weakly measured harmonic chains, the competition between ballistic quasiparticle motion and measurement-induced decay yields
  $$
  S_A(t) \propto t^{1/2}
  $$
  before ultimate area-law saturation on timescales $t\sim L^2$ [2403.04022].

- For quasi-long-range interactions, $S(t) \sim \log t$ replaces the linear regime [1305.6880].

### 2.3 Power-law and Logarithmic Growth

- In disordered, ergodic (pre-MBL) spin chains with no extensive conserved densities, entanglement grows as
  $$
  S_A(t) \propto t^{\alpha(W)},\quad 0<\alpha(W)<1,
  $$
  with $\alpha(W)$ vanishing as disorder crosses the MBL threshold; eventually, in the MBL phase,
  $$
  S_A(t)\sim \alpha\,\ln t,
  $$
  where $\alpha$ depends on the localization length [1908.07010].

- In MBL systems, entanglement growth for initial product states is universally logarithmic in time, a robust diagnostic of localization [2510.08344, 1508.03784].

### 2.4 KPZ and Pinned Membrane Universality

- For noisy random unitary dynamics (random circuits with large $q$), the bipartite von Neumann entropy $S(x,t)$ obeys the Kardar–Parisi–Zhang (KPZ) equation:
  $$
  \partial_t S = \nu\,\partial_x^2 S - \frac{\lambda}{2}(\partial_x S)^2 + \eta(x,t) + c,
  $$
  in 1D, leading to
  $$
  \langle S(x,t) \rangle \sim v_E t + B t^{1/3}
  $$
  with fluctuations and spatial correlations scaling as $t^{1/3}$ and $t^{2/3}$, respectively [1608.06950, 1806.04686]. In $d$ dimensions, the problem maps to a $d$-dimensional random elastic membrane, with universal exponent structure (see Table 1) [2306.04764].

**Table 1: Universal Entanglement Growth Exponents in Random Circuit Dynamics**

| $d$ | Fluctuation Exponent $\theta$ | Roughness Exponent $\zeta$ | Growth Exponent $\beta$ | Universality           |
|-----|------------------------------|----------------------------|------------------------|------------------------|
| 1   | $1$                          | $2/3$                      | $1/3$                  | KPZ                    |
| 2   | $\approx0.86$                | $\approx0.44$              | $\approx0.36$          | Pinned membrane in 3D  |
| 3   | $\approx1.4$                 | $\approx0.30$              | $\approx0.40$          | Pinned membrane in 4D  |
| 4   | $2$                          | —                          | $0.5$                  | Gaussian (mean-field)  |

## 3. Deterministic Protocols, Dual Unitarity, and Quenches

### 3.1 Free and Interacting Models

- For free scalar field theory following a global quench, analytic quasiparticle models—where each entangled pair at $t=0$ contributes only when its members are spatially separated across the cut—predict explicit scaling functions:
  $$
  \hat S_A(t) = s\,\mathrm{Vol}(A) f(t/R),
  $$
  with volume-law saturation as $t \to \infty$ and ballistic early growth with dimension-dependent velocity $v_E$ [1609.00872].

- Holographic CFTs, via extremal surfaces in Vaidya–AdS backgrounds, reproduce: (i) quadratic early-time growth, (ii) linear growth in the post-local-equilibration regime,
  $$
  \Delta S_A(t) \sim s_{\mathrm{eq}}\,A_{\Sigma} v_E t,
  $$
  (iii) “memory loss” scaling near the saturation point, and (iv) continuous/discontinuous saturation with critical exponents depending on subsystem shape [1311.1200].

### 3.2 Dual-Unitary Circuits

- Dual-unitary circuits (those satisfying unitary evolution upon exchange of space and time) permit exact solutions: for generic initial states, the entanglement increment $\Delta S(t)$ at each time step is bounded by $\log d$ (local Hilbert space dimension), and for “highly entangling” gates the velocity
  $$
  v_E = \limsup_{t\to\infty} S_A(t)/(4t\log d) = 1
  $$
  is maximal, saturating the bound imposed by the circuit geometry [2208.00030].

## 4. Multipartite and Structured Entanglement Growth

- Beyond bipartite measures, genuine multipartite entanglement, quantified by measures such as the generalized geometric measure (GGM), grows rapidly under random unitary circuit dynamics, saturating to near-maximality after $O(N)$ layers for $N$ qubits [2003.12546]. Clifford circuits exhibit slower, but eventually comparable, multipartite scrambling.

- Initial-state structure is crucial in non-ergodic (especially MBL) systems. Initial states prepared with tunable entanglement exhibit two-regime behavior: for small initial entanglement, the net entanglement growth increases as single-site memories are degraded; for large initial entanglement, growth is suppressed due to inter-site correlation dominance. This produces a non-monotonic peak in the net growth as initial entanglement is varied (see "Entanglement Growth from Structured Initial States in Many-Body Localized Systems" [2605.20656]). This phenomenon is quantitatively described by local integrals of motion (LIOM) and can be captured via generalized Gibbs or “Scrooge” ensemble descriptions.

- A general classification: “build–move” decomposition—“build” generates new entanglement, “move” redistributes existing entanglement. MBL and random SWAP circuits are move-dominated, while chaotic (thermalizing) circuits are build-dominated. The non-monotonic dependence of net entanglement growth on initial state is a signature of move-dominated dynamics [2510.08344].

## 5. Role of Conservation Laws and Measurement Dynamics

- Conservation laws limit entanglement growth: $U(1)$-symmetric, charge-conserving systems with strictly diffusive transport show $\sqrt{t}$ entropy growth for $d=1$, $q=2$; in $d>1$ or $q>2$ settings, conservation laws are generically ineffective at suppressing linear-in-$t$ entropy accumulation [1912.03645].

- In systems subject to local monitoring (projective or weak measurements), the interplay between coherent evolution and quantum jumps yields rich dynamics:
  - Local monitoring of a single site in a chain of free fermions induces a steady-state volume law in entanglement. This is rooted in the statistics of quantum jumps: long Poissonian “dark” intervals enable extended non-Hermitian evolution, which builds up extensive entanglement before “reset” jumps [2411.13667].
  - In monitored harmonic chains, the diffusive nature of weak, spatially coarse-grained measurements ($R\gg 1$) ensures $S_A(t)\sim t^{1/2}$ growth, eventually saturating to an area law [2403.04022].

## 6. Classical and Quantum Complexity Perspectives

The initial quasiclassical regime of entanglement growth from separated coherent states can be mapped directly to the Kolmogorov–Sinai entropy of the corresponding classical dynamical evolution, making the connection between quantum entanglement production and classical chaotic complexity explicit [2211.11213].

## 7. Outlook and Experimental Realizations

- The theoretical predictions outlined above have been confirmed in a variety of experimental platforms, including photonic simulators that verify volumetric entanglement growth after engineered quenches [1603.02669], and large-scale simulations via Clifford circuits [2306.04764, 2003.12546].
- Measurement-based experimental protocols have been proposed and realized—utilizing quantum switches and Loschmidt echo measurements—to probe Renyi entropy growth directly with a minimal local probe [1508.03784].

## References

- "Entanglement Growth from Structured Initial States in Many-Body Localized Systems" [2605.20656]
- "Entanglement growth in diffusive systems" [1912.03645]
- "Entanglement growth after inhomogenous quenches" [1907.00869]
- "Entanglement growth in the dark intervals of a locally monitored free-fermion chain" [2411.13667]
- "On Locality, Growth and Transport of Entanglement" [1212.0331]
- "Entanglement Growth from Entangled States: A Unified Perspective on Entanglement Generation and Transport" [2510.08344]
- "Diffusive entanglement growth in a monitored harmonic chain" [2403.04022]
- "Power-law entanglement growth from typical product states" [1908.07010]
- "Universal Entanglement Growth along Imaginary Time in Quantum Critical Systems" [2512.23361]
- "Entanglement production and information scrambling in a noisy spin system" [1806.04686]
- "Entanglement growth in quench dynamics with variable range interactions" [1305.6880]
- "Quantum Entanglement Growth Under Random Unitary Dynamics" [1608.06950]
- "Entanglement dynamics and classical complexity" [2211.11213]
- "Growth of genuine multipartite entanglement in random unitary circuits" [2003.12546]
- "Photonic Simulation of Entanglement Growth After a Spin Chain Quench" [1603.02669]
- "Entanglement dynamics in quantum many-body systems" [1508.03784]
- "Entanglement growth during thermalization in holographic systems" [1311.1200]
- "Entanglement Growth and Minimal Membranes in $(d+1)$ Random Unitary Circuits" [2306.04764]
- "Entanglement Growth after a Global Quench in Free Scalar Field Theory" [1609.00872]
- "Growth of entanglement of generic states under dual-unitary dynamics" [2208.00030]

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