Entanglement-Dynamics Diagnostic Comparison

Different dynamical diagnostics answer different operational questions. Entropy measures bipartite mixedness, mutual information measures total regional correlation, channel-state quantities characterize an evolution map, conserved densities measure transport, and recovery tests retained information. Agreement is informative; disagreement is often the scientific result rather than a defect.

Required background. The information-measure domain atlas supplies definition domains, and continuum finite-time windows supply the common comparison interval.

Helpful background. Front extraction, operator entanglement, and open and monitored dynamics provide the principal specialized diagnostics.

Choose the question first

Use subregion entropy when asking how a specified bipartition becomes entangled under a known global state. Use mutual information when asking how much total correlation connects two regional algebras. Use connected correlators to identify which observable carries a signal. Use operator entanglement when asking how nonlocal the evolution map is under a declared doubled partition. Use density and current profiles for conserved transport. Use recovery fidelity or conditional mutual information when asking whether apparently dispersed information remains reconstructible.

OTOCs and circuit complexity are neighboring diagnostics, but they own operator growth and computational structure rather than subregion entanglement. They may correlate with entropy growth in a model without becoming substitutes for it.

A matched comparison suite

Run one regulated preparation in four dynamical classes: integrable, chaotic, localized, and open or monitored. On the common trustworthy time window, compute:

• $\Delta S_A(t)$ for several region sizes;
• $I(A{:}B;t)$ and at least two connected correlators;
• a conserved density and current when a symmetry exists;
• an energy-constrained operator-entanglement diagnostic;
• a recovery fidelity for a localized encoded perturbation;
• conditioned and unconditional entanglement in the monitored case.

Normalize neither axes nor amplitudes separately until the raw physical units are recorded. Then ask which qualitative statements survive a matched change of cutoff, size, threshold, and dynamical class.

The comparison begins from one preparation and microscopic evolution, then reads several direct diagnostics. An effective picture is retained only if it predicts more than the diagnostic used to fit it. The map is schematic.

Interpreting disagreements

Entropy can grow while recovery remains possible because information is encoded nonlocally rather than destroyed. Mutual information between two small regions can fall while a larger decoder succeeds. A connected correlator can vanish because the chosen operator is blind. In an open system, an apparent revival or “backflow” in a coarse observable may result from changing resolution or conditioning rather than non-Markovian information return.

To establish recoverability, specify an encoder, accessible region, decoder class, energy constraint, and error. To establish non-Markovian backflow, use a measure with the required contractivity and compare the same pair of initial states under the same channel; a single entropy dip is insufficient.

Falsifying comparisons

The strongest negative controls are deliberately mismatched cases. Logarithmic entanglement growth with suppressed transport is established for an interacting localized chain by Bardarson, Pollmann, and Moore 2012, Eqs. (1)–(3), whereas Nahum et al. 2017, §§ II–IV provide a random-unitary chaotic comparator:

• an integrable model where a membrane fit should fail out of sample;
• a chaotic model where bare quasiparticle occupations should not predict all cuts;
• a localized model with entropy growth but suppressed particle transport;
• two unravelings of one Lindblad equation with equal unconditional but unequal trajectory entanglement;
• a recoverable code whose local mutual information is small.

Observable choice, finite window, dynamical class, and record conditioning can each reverse an inference. A comparison is credible only when these changes are controlled independently. The map is schematic.

Reporting rule

State the direct observable first, the extracted quantity second, and the physical interpretation third. For example: “the half-maximum contour of $\Delta I(A{:}B)$ is ballistic over this window” is evidence; “information propagates at $v$” is a broader inference that needs intervention, decoder, and causal controls.

References

• Bardarson, Jens H., Frank Pollmann, and Joel E. Moore. “Unbounded Growth of Entanglement in Models of Many-Body Localization.” Physical Review Letters 109 (2012): 017202. DOI.
• Nahum, Adam, Jonathan Ruhman, Sagar Vijay, and Jeongwan Haah. “Quantum Entanglement Growth under Random Unitary Dynamics.” Physical Review X 7 (2017): 031016. DOI.