Quantum Entanglement Growth

• Entanglement Growth is the dynamic increase of quantum correlations between subsystems, quantifiable via entropic measures like von Neumann and Rényi entropies.
• It exhibits various regimes—ballistic, diffusive, logarithmic, and KPZ scaling—depending on factors such as dimensionality, conservation laws, and interaction range.
• Research on entanglement growth provides critical insights into thermalization, many-body localization, operator spreading, and the scrambling of quantum information.

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 (Ho et al., 2015). The minimal-cut picture formalizes this, bounding $S(t)$ by the minimal number of spacetime gates crossed by a causal path (“entanglement tsunami” interpretation) (Nahum et al., 2016, Sierant et al., 2023).

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 (Znidaric, 2019, Schachenmayer et al., 2013).

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 (Ho et al., 2015, Nahum et al., 2016, Liu et al., 2013, Sierant et al., 2023).

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 (Znidaric, 2019, Rakovszky et al., 2019). 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 $t\sim L^2$ (Young et al., 2024).

• For quasi-long-range interactions, $S(t)$0 replaces the linear regime (Schachenmayer et al., 2013).

2.3 Power-law and Logarithmic Growth

• In disordered, ergodic (pre-MBL) spin chains with no extensive conserved densities, entanglement grows as

$S(t)$1

with $S(t)$2 vanishing as disorder crosses the MBL threshold; eventually, in the MBL phase,

$S(t)$3

where $S(t)$4 depends on the localization length (Lezama et al., 2019).

• In MBL systems, entanglement growth for initial product states is universally logarithmic in time, a robust diagnostic of localization (Zhang et al., 9 Oct 2025, Ho et al., 2015).

2.4 KPZ and Pinned Membrane Universality

• For noisy random unitary dynamics (random circuits with large $S(t)$5), the bipartite von Neumann entropy $S(t)$6 obeys the Kardar–Parisi–Zhang (KPZ) equation:

$S(t)$7

in 1D, leading to

$S(t)$8

with fluctuations and spatial correlations scaling as $S(t)$9 and $U(1)$0, respectively (Nahum et al., 2016, Knap, 2018). In $U(1)$1 dimensions, the problem maps to a $U(1)$2-dimensional random elastic membrane, with universal exponent structure (see Table 1) (Sierant et al., 2023).

Table 1: Universal Entanglement Growth Exponents in Random Circuit Dynamics

$U(1)$3	Fluctuation Exponent $U(1)$4	Roughness Exponent $U(1)$5	Growth Exponent $U(1)$6	Universality
1	$U(1)$7	$U(1)$8	$U(1)$9	KPZ
2	$S_A(t) \simeq v_E \,\mathrm{Area}(\partial A)\, t,$0	$S_A(t) \simeq v_E \,\mathrm{Area}(\partial A)\, t,$1	$S_A(t) \simeq v_E \,\mathrm{Area}(\partial A)\, t,$2	Pinned membrane in 3D
3	$S_A(t) \simeq v_E \,\mathrm{Area}(\partial A)\, t,$3	$S_A(t) \simeq v_E \,\mathrm{Area}(\partial A)\, t,$4	$S_A(t) \simeq v_E \,\mathrm{Area}(\partial A)\, t,$5	Pinned membrane in 4D
4	$S_A(t) \simeq v_E \,\mathrm{Area}(\partial A)\, t,$6	—	$S_A(t) \simeq v_E \,\mathrm{Area}(\partial A)\, t,$7	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 $S_A(t) \simeq v_E \,\mathrm{Area}(\partial A)\, t,$8 contributes only when its members are spatially separated across the cut—predict explicit scaling functions:

$S_A(t) \simeq v_E \,\mathrm{Area}(\partial A)\, t,$9

with volume-law saturation as $U(1)$0 and ballistic early growth with dimension-dependent velocity $U(1)$1 (Cotler et al., 2016).

• Holographic CFTs, via extremal surfaces in Vaidya–AdS backgrounds, reproduce: (i) quadratic early-time growth, (ii) linear growth in the post-local-equilibration regime,

$U(1)$2

(iii) “memory loss” scaling near the saturation point, and (iv) continuous/discontinuous saturation with critical exponents depending on subsystem shape (Liu et al., 2013).

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 $U(1)$3 at each time step is bounded by $U(1)$4 (local Hilbert space dimension), and for “highly entangling” gates the velocity

$U(1)$5

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 $U(1)$6 layers for $U(1)$7 qubits (Bera et al., 2020). 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" (Xu et al., 20 May 2026)). 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 (Zhang et al., 9 Oct 2025).

5. Role of Conservation Laws and Measurement Dynamics

• Conservation laws limit entanglement growth: $U(1)$8-symmetric, charge-conserving systems with strictly diffusive transport show $U(1)$9 entropy growth for $q=2$0, $q=2$1; in $q=2$2 or $q=2$3 settings, conservation laws are generically ineffective at suppressing linear-in-$q=2$4 entropy accumulation (Znidaric, 2019).
• In systems subject to local monitoring (projective or weak measurements), the interplay between coherent evolution and quantum jumps yields rich dynamics:
  • Local monitoring of a single site in a chain of free fermions induces a steady-state volume law in entanglement. This is rooted in the statistics of quantum jumps: long Poissonian “dark” intervals enable extended non-Hermitian evolution, which builds up extensive entanglement before “reset” jumps (Fresco et al., 2024).
  • In monitored harmonic chains, the diffusive nature of weak, spatially coarse-grained measurements ($q=2$5) ensures $q=2$6 growth, eventually saturating to an area law (Young et al., 2024).

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 (Wang et al., 2022).

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 (Pitsios et al., 2016), and large-scale simulations via Clifford circuits (Sierant et al., 2023, Bera et al., 2020).
• 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 (Ho et al., 2015).

• "Entanglement Growth from Structured Initial States in Many-Body Localized Systems" (Xu et al., 20 May 2026)
• "Entanglement growth in diffusive systems" (Znidaric, 2019)
• "Entanglement growth after inhomogenous quenches" (Rakovszky et al., 2019)
• "Entanglement growth in the dark intervals of a locally monitored free-fermion chain" (Fresco et al., 2024)
• "On Locality, Growth and Transport of Entanglement" (Omnès, 2012)
• "Entanglement Growth from Entangled States: A Unified Perspective on Entanglement Generation and Transport" (Zhang et al., 9 Oct 2025)
• "Diffusive entanglement growth in a monitored harmonic chain" (Young et al., 2024)
• "Power-law entanglement growth from typical product states" (Lezama et al., 2019)
• "Universal Entanglement Growth along Imaginary Time in Quantum Critical Systems" (Shen et al., 29 Dec 2025)
• "Entanglement production and information scrambling in a noisy spin system" (Knap, 2018)
• "Entanglement growth in quench dynamics with variable range interactions" (Schachenmayer et al., 2013)
• "Quantum Entanglement Growth Under Random Unitary Dynamics" (Nahum et al., 2016)
• "Entanglement dynamics and classical complexity" (Wang et al., 2022)
• "Growth of genuine multipartite entanglement in random unitary circuits" (Bera et al., 2020)
• "Photonic Simulation of Entanglement Growth After a Spin Chain Quench" (Pitsios et al., 2016)
• "Entanglement dynamics in quantum many-body systems" (Ho et al., 2015)
• "Entanglement growth during thermalization in holographic systems" (Liu et al., 2013)
• "Entanglement Growth and Minimal Membranes in $q=2$7 Random Unitary Circuits" (Sierant et al., 2023)
• "Entanglement Growth after a Global Quench in Free Scalar Field Theory" (Cotler et al., 2016)
• "Growth of entanglement of generic states under dual-unitary dynamics" (2208.00030)