Support-size entanglement trajectories of random subset states

Code, data, released figures, and validation for the manuscript “Support-size entanglement trajectories of random subset states.” The manuscript updates the existing preprint record arXiv:2501.06292.

Equal-positive-amplitude subset states exhibit a rise–peak–fall entanglement trajectory as their computational-basis support grows. This repository contains the fixed-cardinality ensemble calculations, exact-moment checks, numerical figure scripts, and constrained reference ensembles used in the paper.

Key results represented in this repository

• Exact ensemble-mean reduced state for uniformly random supports of fixed size.
• Exact average purity across a balanced bipartition, yielding a rigorous power-law support window, an optimal interior balance, and a dense-side boundary.
• Numerical rise–peak–fall trajectories and retained peak estimates through $n=30$.
• Exact hypergeometric diagonal-entropy and residue-class entropy bounds.
• Cardinality-, parity-, mod-4-, and mod-8-matched reference ensembles for almost-prime supports, evaluated before and after a quantum Fourier transform.

Analytic scale hierarchy

Let $N=2^n$ and write a power-law support size as $M=cN^\gamma$, with $c>0$. The exact fixed-cardinality average-purity formula has the leading structure

$$\overline{P}_{N,M} = 2N^{-1/2} +c^{-1}N^{-\gamma} +c^2N^{2\gamma-2} +\text{lower-order terms}.$$

Near-maximal window. Every fixed exponent $\frac{1}{2}<\gamma<\frac{3}{4}$ gives

$$\overline{S}_{N,M} \geq \frac{n}{2}-1-o(1).$$

Purity-optimal interior scale. The exponent $\gamma=\frac{2}{3}$ uniquely balances the sparse $N^{-\gamma}$ correction and the dense rectangle $N^{2\gamma-2}$ correction. Optimizing the remaining prefactor gives

$$M = 2^{-1/3}N^{\frac{2}{3}}+O(1).$$

Dense-side boundary. The exponent $\gamma=\frac{3}{4}$ is the point at which the rectangle contribution enters at the same $N^{-1/2}$ order as the balanced-cut background.

The numerical maximum of the mean von Neumann entropy is a separate quantity. Over $n=10,\ldots,30$, the retained estimates give

$$\log_2 \widehat{M}_n = 0.703541\,n-0.357734.$$

The pointwise effective exponent is

$$\gamma_n^{\mathrm{eff}} = \frac{\log_2 \widehat{M}_n}{n}, \qquad \gamma_{10}^{\mathrm{eff}}\approx 0.674, \qquad \gamma_{30}^{\mathrm{eff}}\approx 0.694.$$

It increases over the retained range. This motivates comparison with the $\frac{3}{4}$ dense-side boundary but does not establish an asymptotic peak law.

Repository map

subset_states/     reusable scientific routines
scripts/           figure, validation, and local-verification entry points
data/              Table I and released matched-null datasets
outputs/            manuscript-ready figures and generated outputs
tests/              lightweight and exhaustive small-system regression tests
validation/         final independent validation reports
exploratory/        historical experiments not used in the manuscript
docs/               audit and release documentation

Quick start

python3 -m venv .venv
source .venv/bin/activate
python3 -m pip install -r requirements.txt
python3 scripts/run_smoke_tests.py

Run the independent publication validation:

python3 scripts/final_scientific_validation.py

Regenerate the fast deterministic figures and verify that all seven released PDFs are present:

python3 scripts/reproduce_publication_figures.py

A reduced computational smoke run is available through:

python3 scripts/reproduce_publication_figures.py --smoke

The full publication computations can be launched with --full, but several runs are intentionally expensive.

Publication figures

Figure	Scientific content	Main script	Released output
1	Support-size trajectory and cut comparison	scripts/fig1_concentration.py	outputs/fig1/fig1_concentration.pdf
2	Peak entropy and support-size scaling	scripts/fig2_peak_scaling.py	outputs/fig2/fig2_peak_scaling.pdf
3	Reduced-state spectrum near the peak	scripts/fig3_spectral.py	outputs/fig3/fig3_spectral_bulk.pdf
4	Sparse and dense approximations	scripts/fig4_approximation.py	outputs/fig4/fig4_approximation.pdf
5	Cardinality and residue controls	scripts/fig5_qft_residue_controls.py	outputs/fig5/fig5_qft_residue_controls.pdf
6	Rényi trajectories	scripts/fig6_renyi.py	outputs/fig6/fig6_renyi.pdf
7	Entropy over balanced cuts	scripts/fig7_partitions.py	outputs/fig7/fig7_partitions.pdf

Figure 5 can be redrawn exactly from the released CSV files. The matched-null raw dataset contains 12,000 paired computational/Fourier samples.

Table I and local verification

data/table_i_peaks.csv stores the retained peak estimates used in the paper. The command

python3 scripts/peak_scaling_verification.py --n-values 10 12 14

performs a local neighbourhood check around selected tabulated values. It is not a reconstruction of the original global search. Full technical provenance and the robustness comparison are documented in PROVENANCE.md.

Figure 5 constrained ensembles

The released files are:

data/fig5_random_qft_summary.csv
data/fig5_almost_prime_unions.csv
data/residue_matched_summary.csv
data/residue_matched_samples.csv
data/residue_entropy_bounds.csv

Regenerate the null samples with:

python3 scripts/run_residue_controls.py
python3 scripts/fig5_qft_residue_controls.py

Reproducibility levels

1. Fast: unit tests, exact small-system checks, Figure 2, and Figure 5 redraw.
2. Standard: regenerate selected numerical figures with reduced or custom workloads.
3. Computational: use publication defaults for the full sampling runs and local high-dimensional peak checks.

All stochastic scripts expose seeds. The tested software environment is recorded in requirements-tested.txt.

Citation

Use the citation metadata in CITATION.cff. The preferred paper citation is the updated arXiv record and, once available, the journal version.

License

The repository is released under the MIT License. See LICENSE.