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 repository provides exact spectral geometry, a correction layer for the article, and a comparison of entanglement measures across four dynamical families, 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.

Results

The table and figures use the v1.0.0 analysis snapshot. Per-path PCA and correlation calculations in the current implementation exclude unresolved near-constant coordinates under a declared numerical floor; the snapshot's fine rank statistics are sensitive to roundoff. See numerical foundations.

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.

The QCA, kicked-Ising, quantum-baker, and XXZ-derived families probe scrambling, recurrence, disorder, and spectral complexity; the conditions carry no blanket classification as quantum chaotic. The coordinate $\tau=\mathrm{step}/n$ is a scaled iteration coordinate. The XXZ rows describe fixed one-substep product-formula circuits. Their refinement study shows that these rows are not a convergence-controlled approximation to continuous-time Hamiltonian evolution, while the global common-mode fraction is stable under refinement.

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 paper's phrase has the following operational meaning here:

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.

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 smoother and exhibit more local raw-metric competition than reordered versions of the same data. A large common mode alone therefore does not identify temporal organization. Machine-readable results and draw-level records accompany the study.

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.

For static entropy–largest-eigenvalue diagrams, noncentral Wishart states, outlier-and-bulk entropy, normalized matrix spectra, Schmidt-scale trajectories, and algorithmic entanglement, see Research questions and literature connections. This guide gives concrete answers and recent related reading while distinguishing the separate analytical note, the trajectory paper, and the repository’s corrected and extended results.

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

Simulations use stable spectrum extraction. The committed data and figures are the v1.0.0 snapshot; release-v1 extraction provides compatibility with that snapshot. See Numerical foundations for accuracy tests and conditioning limits of boundary-relative coordinates.

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 from its canonical sources:

make public-context

Run the automated tests and public metadata/link validation:

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

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

make rebuild-included

See Reproducibility for deterministic seeds, numerical tolerances, snapshot figure rebuilding (make public-figures), and the expensive state-vector regeneration through 20 qubits (make full).

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 details	Exact spectral geometry, Analysis methods, and Numerical foundations
Reproduction	Reproducibility and release environment
AI or automated research assistant	AI context, conceptual neighbors, and public claims JSON

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

The results concern pure-state dynamics, specified bipartitions or declared averages over cuts, and the implemented spectrum functionals. Limitations documents the sampling and numerical qualifications needed to interpret them.