Entanglement Dynamics and Information Transport

Entanglement dynamics tracks how a prepared state shares information across space. Entropy growth, mutual information, conserved-charge transport, and operator spreading have different definitions and need not propagate with one common velocity.

Enter this chapter

Begin with State Preparation, Quench, and Regulator Contracts to specify the initial state, quench, subsystem, and regulator.

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.

Topics

1. State Preparation, Quench, and Regulator Contracts — Fixes the preparation and regulated observables of a quench.
2. Entanglement Growth after Quenches — Interprets entropy growth and saturation within a finite time window.
3. Quasiparticle Pictures of Entanglement Growth — Connects propagation of entangled quasiparticles to interval entropy.
4. Membrane, Hydrodynamic, and Coarse-Grained Entanglement Laws — Develops coarse-grained variational descriptions of entanglement growth.
5. Entanglement, Correlation, and Hydrodynamic Fronts — Defines fronts before comparing velocities and causal bounds.
6. Operator Entanglement and Channel–State Maps — Constructs operator entanglement through a regulated channel–state map.
7. Mutual Information and Correlation Spreading — Compares spreading total correlations with connected correlators.
8. Entanglement in Open and Monitored Field Dynamics — Distinguishes unconditioned open states from measurement trajectories.
9. Conservation Laws and Information Transport — Relates conserved quantities to information-transport timescales.
10. Integrable versus Chaotic Entanglement Dynamics — Tests integrable and chaotic descriptions with independent diagnostics.
11. Continuum Extrapolation and Finite-Time Windows — Identifies time windows compatible with size and continuum limits.
12. Entanglement-Dynamics Diagnostic Comparison — Chooses a dynamical diagnostic that answers the intended question.

How the ideas fit together

In suitable integrable quenches, entangled quasiparticles emitted by the preparation account for entropy growth across a boundary Calabrese and Cardy 2005, §§ 2–4. Other systems require different effective descriptions, such as a coarse-grained entanglement membrane.

Neither picture makes entropy velocity identical to signal velocity or charge diffusion. In monitored dynamics, conditioning on the retained measurement record changes the state whose entanglement is being computed. Common preparation, observable, regulator, and observation windows are therefore essential before comparing growth curves.

The comparison below separates the assumptions and tests needed for these different conclusions.

Entanglement-dynamics diagnostic comparison

Distinct information-dynamics diagnostics, their controlled regimes, and decisive comparisons

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 a common-cut subtraction	Initial curvature, growth slope, crossover, or plateau	Can alter leading behavior for inhomogeneous states and subleading behavior more generally	Unitary unless a channel and mixed-state diagnostic are declared	Exact covariance, tensor network, or controlled circuit	After the preparation transient and before truncation, saturation, or recurrence	Change region, cutoff, subtraction, and numerical bond dimension
Quasiparticle prediction	Mode-production entropy, species, and post-quench dispersion	Pair separation 2|vα(k)|t and entropy-weighted growth	Stationary occupations and dressed velocities in integrable models	Normally closed; scattering or loss needs a kinetic extension	Free or integrable space–time scaling limit	After pair production and before boundary return or strong inelastic scattering	Predict exact covariance data without refitting the mode entropy
Membrane law	Coarse-grained entropy profile and final entangling surface	Tension ℰ(v) and derived vE	Conserved hydrodynamic fields may couple to the entropy profile	Usually closed, chaotic, and locally equilibrated	Random circuit, holographic, or model-specific hydrodynamic scaling	Local-equilibration scale much smaller than region and observation scales	Fit one geometry and predict a distinct geometry
Connected correlator	Named local operators and a connected two-point function	Threshold arrival, peak, decay, and broadening	Strongly dependent on whether the operator overlaps a conserved mode	Heisenberg or channel evolution must be explicit	Free propagation, hydrodynamics, or converged numerics	Signal above the error floor and before reflection from boundaries	Choose an operator blind to the transported charge and vary the threshold
Mutual information	Two separated algebras, or reduced density matrices in a type-I regulator	Onset, peak, or decay of total correlations	Includes classical and quantum correlations in all retained sectors	Computed from the reduced state after the declared channel	Gaussian states, finite systems, or controlled replica calculations	Both regions resolved and regulator cancellation stable	Compare with several matched two-point functions
Operator entanglement	Normalized channel state for U(t) or a quantum channel	Growth across a specified input–output spatial partition	Symmetry blocks and reference measure must be retained	Unitary or channel normalization stated	Finite dimension or converged energy-constrained truncation	Cutoff converged at fixed reference energy, then reference profile varied	Identity and SWAP controls; change the channel-state reference
Monitored trajectory	Conditioned state and complete measurement record	Trajectory entropy growth or measurement-induced scaling	Symmetry of both unitary and measurement layers matters	Unraveling, efficiency, and postselection explicit	Hybrid circuits or specified continuously monitored models	Trajectory ensemble and finite-size scaling converged	Compare two unravelings of one unconditional master equation
Unconditional open-system state	Density operator averaged over unobserved outcomes	Mixing, decoherence, negativity, or recoverability rate	Bath may preserve or break the charge explicitly	Lindblad or microscopic environment model declared	Controlled weak-coupling, collision model, or finite simulation	Master-equation approximation and numerical truncation valid	Retain the record and compare average nonlinear diagnostics
Charge transport	Density profile, current, structure factor, or full counting statistics	Ballistic speed, diffusion constant, or anomalous exponent	Central defining input	Bath symmetry and continuity equation stated	Hydrodynamic or integrable scaling limit	After microscopic relaxation and before finite-volume return	Weakly break the conservation law and mix initial sectors
Recovery diagnostic	Encoding, noise channel, retained output, and recovery map	Fidelity, relative entropy loss, or recoverable information	Recovery may be sector constrained	Environment and accessible record are part of the task	Finite-dimensional code or energy-constrained channel	Optimization and truncation errors below the claimed effect	Entropy grows while an explicit recovery still succeeds

Where to go next

Continue to Scrambling, Decoupling, and Recoverability to define the encoded input and the recovery task.

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.