Entanglement Trajectories

One Schmidt-spectrum path, many entanglement metrics.

This repository is the corrected computational companion and follow-up evidence package for:

Ruge Lin, “Entanglement Trajectory and its Boundary,” Quantum 8, 1282 (2024). DOI: 10.22331/q-2024-03-14-1282.

The journal article introduced an initial version of the idea. This repository preserves that insight, provides explicit mathematical corrections and scope clarifications, and tests the upgraded claim across four dynamical families used to probe scrambling, recurrence, disorder, and spectral complexity, six system sizes, and several non-equivalent functions of the Schmidt spectrum.

Central result. For a fixed bipartition of a pure state, standard spectrum-based entanglement measures are nonlinear projections of one ordered Schmidt-spectrum path. Across the tested models, the normalized projections share a dominant common mode and preserve substantial coarse morphology, while local disagreements expose spectral redistributions that no single scalar measure can order completely.

The object that unifies the metrics

For a pure state and a fixed bipartition, write the ordered reduced-density-matrix spectrum as

$$ \Gamma:t\longmapsto \boldsymbol{\lambda}(t) =\bigl(\lambda_1(t),\ldots,\lambda_d(t)\bigr). $$

A spectrum functional $E$ gives a projected trajectory

$$ \Gamma_E(t)=\bigl(\lambda_{\max}(t),E[\boldsymbol{\lambda}(t)]\bigr). $$

The family ${\Gamma_E}$ is the entanglement-trajectory atlas. In the supplied follow-up data, the main fixed-cut lenses represent four separated Rényi sectors:

Spectral lens	Implemented coordinate	What it emphasizes
$q=\tfrac12$	pure-state logarithmic negativity	small and intermediate Schmidt weights
$q=1$	von Neumann entropy	the spectrum in aggregate
$q=2$ class	linear entropy / purity	larger Schmidt weights
$q=\infty$	$\lambda_{\max}$ / min-entropy / geometric class	the leading Schmidt weight

These are not four unrelated definitions. They are different compressions of the same spectrum. They are also not redundant with one another, except within the explicitly documented equivalence classes.

Exact geometry before random-matrix theory

At fixed $p=\lambda_{\max}$, every compatible $d$-dimensional spectrum lies between two majorization extremizers.

The equal-tail spectrum

$$ \boldsymbol u(p)=\left(p,\frac{1-p}{d-1},\ldots,\frac{1-p}{d-1}\right) $$

maximizes every Schur-concave metric used here. The concentrated spectrum

$$ \boldsymbol c(p)= (\underbrace{p,\ldots,p}_{k\ \mathrm{times}},r,0,\ldots,0), \qquad k=\left\lfloor\frac1p\right\rfloor, \quad r=1-kp, $$

minimizes it. These spectra define exact finite-dimensional feasible envelopes and a common boundary-relative coordinate for comparing different metrics.

Random matrix theory enters only afterward, as a family of Haar/Wishart or spiked-Wishart reference ensembles inside the exact arena. It does not define the exact boundary.

What the follow-up study establishes

The table and committed figures below preserve the v1.0.0 analysis snapshot. Current per-path PCA and correlation calculations exclude unresolved near-constant coordinates under a declared numerical floor; fine rank statistics need not reproduce old roundoff orderings. See numerical foundations and the freeze audit. The global common-mode and later chronology-control conclusions are unchanged.

The included deterministic designed dataset contains 5,856 observations from 96 trajectories: four dynamical families, four declared conditions per family, and sizes $n=10,12,14,16,18,20$. The conditions are controlled examples, not independent draws from a population.

Controlled result	Value	Interpretation
Common normalized metric mode	90.26% of variance	a strong shared trajectory component
Model-stratified design-cluster 95% interval	86.26%–93.41%	sensitivity to the declared conditions
Median per-trajectory common-mode fraction	94.75%	the shared mode is not produced by only a few paths
Boundary-normalized rank agreement	0.692–0.947	robustness is strong but metric-pair dependent
Metric-competitive scalar steps	808/5,760 = 14.03%	disagreement is real, not excluded by the framework
Competition in selected full-spectrum audit	50 events	all 50 occur on majorization-incomparable transitions
Exact turn counts equal across all three metrics	2/96 trajectories	fine projected topology is not invariant
Held-out-size model-centroid full-path accuracy	0.875	model-level morphology remains reproducible after gap-aware interpolation
Size-and-condition-held-out individual full-path accuracy	0.330	universal individual fingerprinting remains preliminary

Agreement and disagreement have one mechanism

If two successive Schmidt spectra are comparable by majorization, all Schur-concave entropies must order them consistently. If they are incomparable, different Rényi sectors may legitimately move in opposite directions.

The selected full-spectrum audit contains 400 transitions:

• 111 majorization-compatible entanglement increases;
• 9 majorization-compatible decreases;
• 280 incomparable transitions;
• 50 metric-competition events, all in the incomparable sector.

Incomparability permits disagreement but does not require it: 230 incomparable transitions still show metric consensus.

What the quoted “topological invariant” means

The phrase is retained as a historical conceptual label, but it is used only in the following operational sense:

A metric-robust trajectory class is the coarse path morphology that remains recognizable when one declared Schmidt-spectrum metric is replaced by another.

No formal topological invariant has been proved.

It does not currently mean equality of coordinates, preservation of every turn or crossing, a homeomorphism, a homotopy class, a winding-number theorem, persistent-homology invariance, or universality over all cuts, states, dynamics, and notions of entanglement.

The preferred technical terms are metric-robust trajectory class and projection-stable trajectory morphology.

Geometry versus temporal order

The chronology and matched-spectrum controls distinguish the shared instantaneous metric component from order-sensitive behavior. Joint shuffling preserves the 90.26% level common mode exactly, and the four specified static matched-spectrum references give still higher values. The observed chronological paths are instead smoother and exhibit more local raw-metric competition than reordered versions of the same data. These controls qualify the interpretation of the common mode; they do not establish a formal invariant or universal chaos fingerprint. Machine-readable results and draw-level records accompany the reproducible study. The frozen release datasets and figures are unchanged.

Conceptual literature bridge

This project sits at the intersection of entanglement-spectrum dynamics, reduced-density-matrix diagnostics of quantum chaos, multi-Rényi entanglement evolution, majorization, and the limits of spectral universality. The conceptual-neighbor map identifies ten especially close papers and states both the shared idea and the important scope difference for each. A machine-readable version is provided in metadata/conceptual_neighbors.json.

The map is intended for literature discovery, not priority claims: earlier neighboring papers are not described as citing this project, and conceptual similarity is not treated as equivalence.

Corrections to the 2024 paper

The paper remains the journal version of record. This repository supplies an explicit author-correction layer. The central trajectory idea survives, but several statements require correction or narrowing. The most important are:

1. The paper’s three simple curves are not the exact counterexample-free boundary; the exact piecewise envelopes follow from majorization.
2. The stated Page expression is asymptotic; the exact finite-dimensional mean uses harmonic numbers.
3. The deterministic mean part of the noncentral Wishart construction is rank one, with one nonzero eigenvalue.
4. A global quantum Fourier transform does not generally preserve a Schmidt spectrum; the observed overlap is family-specific and numerical.
5. Random-matrix curves are conditional references, not exact boundaries or universal attractors.
6. “Topological invariant,” fingerprint, entanglement-gap, continuity, and computational-usefulness claims must be read with the narrower scope documented here.

Read the correction summary, the author clarification, and the location-specific correction ledger.

Reproduce the results

Current numerical implementation. New simulations use stable spectrum extraction. The data and figures committed here remain the preserved v1.0.0 evidence; they are not silently replaced by new calculations. See Numerical foundations for independent accuracy tests, the explicit historical-compatibility mode, and conditioning limits of boundary-relative coordinates. Run make numerical-check for the full maintenance verification.

Python 3.10 or later is supported for development. The canonical v1.0.0 numerical release uses CPython 3.11.15 and exact dependency locks documented in Canonical release environment.

Standard development installation:

python -m pip install -e '.[analysis,test]'

Rebuild the machine-facing context and five public figures from the canonical documents and included data:

make public-context
make public-figures

Run the automated tests and public metadata/link validation:

make test
make public-validate
make peer-review-check

The combined public-layer workflow is:

make public

Rebuild all analyses from the included trajectory and selected-spectrum data:

make rebuild-included

The complete state-vector regeneration through 20 qubits is separate and expensive:

make full

See Reproducibility for output locations, deterministic seeds, numerical tolerances, and the distinction between included-data reconstruction and full simulation.

Reading paths

Time	Recommended path
30 seconds	this summary and Figure 1
5 minutes	Results at a glance and Corrections
20 minutes	Scientific overview and Public figure story
Technical audit	Peer-review release audit, Exact spectral geometry, Analysis methods, and Release QA
Reproduction	Reproducibility and release environment
AI or automated research assistant	AI context, conceptual neighbors, and public claims JSON
Historical record	legacy/ and the paper-2024-original branch

Repository map

src/entanglement_trajectories/   canonical metrics, boundaries, models, and robustness tools
analysis/                        quantitative robustness and correction verification
scripts/                         reproducible simulations, figures, and validation workflows
data/                            canonical trajectories and compact spectrum/figure-input archives
figures/public/                  five GitHub-facing figures and the social preview
docs/                            scientific explanation, methods, limitations, and FAQ
paper/                           public author clarification for the published article
metadata/                        claims, definitions, metrics, figures, corrections, references, and discovery records
environment/                     machine-readable canonical release environment
requirements/                    exact release dependency locks
legacy/                          provenance archive and historical-branch instructions

Citation

The preferred citation is the published article:

@article{lin2024entanglement,
  title   = {Entanglement Trajectory and its Boundary},
  author  = {Lin, Ruge},
  journal = {Quantum},
  volume  = {8},
  pages   = {1282},
  year    = {2024},
  doi     = {10.22331/q-2024-03-14-1282}
}

Machine-readable citation records are provided in CITATION.cff and codemeta.json. See also the foundational and conceptual references, the conceptual-neighbor map, and the machine-readable reference registry.

Scope and nonclaims

This repository concerns pure-state dynamics, specified bipartitions or explicitly declared averages over cuts, and the implemented spectrum functionals. It does not silently extend the claim to mixed-state entanglement, genuine multipartite invariants, discord-like quantities, entanglement cost, every possible metric, or every notion of quantum chaos.

The complete public nonclaim list is maintained in AI_CONTEXT.md and Limitations.

Freeze-audit record

The September 2026 frozen-scope audit records the numerical/API checks, interpretation updates, and explicit limits of the rerun. It does not substitute automated checks for external peer review or extend the scope of the 2024 article.

Repository-edition status

Version 1.0.0 is the corrected public repository edition. It freezes the exact mathematical layer, the repaired follow-up computation, the quantitative metric-robustness result, the paper-correction record, and the human/AI discovery layer. A narrow formal journal corrigendum remains recommended, but none has yet been submitted.

Interpretation of the four families and the horizontal coordinate

The QCA, kicked-Ising, quantum-baker, and XXZ-derived examples are dynamical families used to probe scrambling, recurrence, disorder, and spectral complexity. The repository does not assert that every declared condition is independently established to be quantum chaotic.

The common coordinate $\tau=\mathrm{step}/n$ is a scaled iteration coordinate, not one universal physical time across circuits, maps, and product-formula dynamics. The released XXZ rows are fixed one-substep symmetric product-formula circuits generated from random-field XXZ terms. The convergence study in XXZ product-formula convergence shows that this one-substep circuit is not a convergence-controlled approximation to continuous-time XXZ evolution, although the global multi-metric common-mode result is stable under refinement.