Entanglement Dynamics and Information Transport

Entanglement dynamics is not a single velocity measurement. A quench creates a regulated nonequilibrium state; microscopic evolution then produces entropy growth, correlation spreading, charge transport, operator entanglement, and record-dependent monitored dynamics. Quasiparticle, membrane, and hydrodynamic descriptions apply in different regimes and must be tested against direct observables over a controlled time window.

Helpful background. Regulated subregion entropy supplies the continuum quantity, while mutual information supplies a finite correlation diagnostic. Equilibration and dephasing, OTOC regularization, and operator spreading distinguish neighboring dynamical questions. Real-time lattice evolution and quench regimes supply numerical and many-body baselines.

Enter this chapter

Start by freezing the state-preparation, quench, and regulator contract. Then extract entanglement growth after quenches before interpreting it through either entangled quasiparticles or a membrane and hydrodynamic law. The fronts and velocities page keeps entropy, correlation, transport, butterfly, and causal speeds distinct.

The second sequence develops operator entanglement and channel–state maps, mutual-information and correlator spreading, and open or monitored entanglement. Then compare the effects of conservation laws and integrable, chaotic, and localized dynamics. Finish by establishing a continuum finite-time window and choosing a diagnostic through the comparison page.

Microscopic evolution, effective pictures, and measured diagnostics are different layers of a dynamical claim. Integrability, chaos, localization, openness, and the available size–time window decide which reduction is justified. The map is schematic and not to scale.

The dynamical contract

A reproducible calculation specifies an initial density operator $\rho_0^{(a,L)}$, a preparation map if $\rho_0$ is not an eigenstate, a post-quench Hamiltonian $H^{(a,L)}$, and an order of limits. For a region $A$,

$$\rho_A(t)=\operatorname{tr}_{\bar A} \!\left(e^{-iHt}\rho_0e^{iHt}\right), \qquad S_A(t)=-\operatorname{tr}\rho_A(t)\log\rho_A(t).$$

The continuum QFT question concerns a controlled quantity such as

$$\Delta S_A(t)=S_A(t)-S_A(0)$$

at fixed physical region and time, or a mutual information whose leading local divergence cancels. Taking $a\to0$ while holding the number of lattice sites fixed shrinks the physical region and does not answer the same question.

For a global quench in one-dimensional continuum systems, the exact and conformal calculations of Calabrese and Cardy 2005, §§ 2–4 establish the canonical quasiparticle growth and saturation picture together with its assumptions.

A linear growth fit,

$$\Delta S_A(t)\simeq s_{\rm eq}\,v_E\,|\partial A|\,t,$$

defines an entanglement velocity $v_E$ only in a coarse-grained window with a specified equilibrium entropy density $s_{\rm eq}$. It is generally distinct from the causal speed, a Lieb–Robinson velocity, the butterfly velocity, a quasiparticle group velocity, and a diffusive transport coefficient. The membrane construction and its velocity bounds are derived in Mezei and Stanford 2017, §§ 2–4, while Nahum et al. 2017, §§ II–IV derive entanglement growth and fluctuations in a random-unitary setting.

Diagnostic comparison

Information-dynamics diagnostics and the regimes in which they answer distinct questions

Diagnostic	Preparation and observable	Reported front or rate	Conservation input	Open-system assumption	Controlled limit	Required size–time window	Falsifying comparison
Subregion entropy	Pure or mixed quench; SA(t) or its change	Early curvature, linear slope, saturation time	May modify subleading growth	Unitary unless channel specified	Free covariance, tensor network, random circuit	After the ultraviolet transient and before saturation or recurrence	Change the region, cutoff, and subtraction
Quasiparticle prediction	Pair production density and dispersion	Distance 2|vk|t across an entangling cut	Mode occupations fixed in the integrable limit	Closed	Free or integrable models	Before finite-size returns and after the production transient	Compare with exact covariance data
Membrane law	Coarse-grained entropy profile	Tension ℰ(v) and entanglement velocity vE	Hydrodynamic coupling may be extra	Usually closed chaotic dynamics	Random circuits, large-scale chaotic systems	Local-equilibration time ≪ t, ℓ ≪ recurrence time	Fit one cut, predict another
Connected correlator	Chosen local operators	Threshold arrival and broadening	Strongly observable dependent	Declared Heisenberg or channel dynamics	Free fields, hydrodynamics, numerics	Signal above the error floor and below boundary return	Use an operator blind to the carried charge
Mutual information	Two separated subregions	Correlation front or peak	Includes all correlations between the chosen algebras	Depends on the reduced channel	Gaussian and tensor-network settings	Regions resolved; regulator cancellation stable	Compare with matched two-point functions
Operator entanglement	Regulated channel state for U(t)	Growth across the input–output partition	Symmetry blocks must be retained	Channel or unitary explicitly normalized	Finite dimension or energy-constrained Gaussian limit	Reference squeezing and truncation converged	Change the channel-state reference
Monitored trajectory entropy	Conditioned state and complete record	Trajectory growth or phase transition	Symmetry of the measurement circuit matters	Unraveling and efficiency explicit	Random or Clifford circuits, monitored Gaussian models	Trajectory ensemble converged; postselection cost stated	Compare two unravelings of one master equation
Charge transport	Density profile, current, or structure factor	Ballistic speed or diffusion constant	Central defining input	Bath must preserve or break charge explicitly	Hydrodynamics, integrable models	After local equilibration and before finite-volume return	Weakly break the conservation law

The table compares questions, not interchangeable estimators. Entropy can grow while a particular correlation decays; mutual information can be small while a recoverable channel remains; an unconditional mixed state can lose entanglement while conditioned trajectories stay highly entangled.

Four independent changes can overturn an apparent law: switch the diagnostic, refine the regulator and window, change the conservation or integrability class, or retain rather than discard the measurement record. The map is schematic.

Scope boundary

This chapter uses thermalization, hydrodynamics, operator growth, and open-system theory as inputs but does not replace their dedicated treatments. It asks what entanglement and information diagnostics directly establish. In particular, a fitted entropy slope does not establish a butterfly velocity, an OTOC front, or a causal speed. Claims about continuum evolution require a finite-time scaling window; long-time lattice saturation and recurrence are not continuum QFT predictions.

References

• Calabrese, Pasquale, and John Cardy. “Evolution of Entanglement Entropy in One-Dimensional Systems.” Journal of Statistical Mechanics: Theory and Experiment 2005 (2005): P04010. DOI.
• Mezei, Márk, and Douglas Stanford. “On Entanglement Spreading in Chaotic Systems.” Journal of High Energy Physics 2017, no. 5 (2017): 065. DOI.
• Nahum, Adam, Jonathan Ruhman, Sagar Vijay, and Jeongwan Haah. “Quantum Entanglement Growth under Random Unitary Dynamics.” Physical Review X 7 (2017): 031016. DOI.