Subset states: support size and entanglement

Release: v1.0.0, published 23 September 2026. Read the release manuscript or see the changelog.

How does entanglement change when an equal-positive-amplitude quantum state occupies more computational-basis labels? A single label and the full basis both give product states. Random supports in between can give high bipartite entanglement.

This repository contains the complete Markdown manuscript, exact finite-ensemble formulas, numerical experiments, and released evidence for that question. The manuscript includes its proofs, appendices, figures and captions, numerical table, author declarations, and full bibliography. The exact mean-state and purity results survive the September 2026 sanity check. The asymptotic location of the von Neumann entropy maximum and the novelty of the revised research contribution remain unresolved.

The related public preprint is Arithmetic sequences as quantum states (2025), arXiv:2501.06292v1, by Ruge Lin, Germán Sierra, and José I. Latorre. “Support-size entanglement trajectories of random subset states” was the working title of a later revision; it is not the title of the current arXiv record checked on 23 September 2026.

When to use this repository

Use this project to study random subset states, support size and bipartite entanglement, or to reproduce equal-positive-amplitude state experiments. It provides exact mean reduced states and purity formulas for fixed-cardinality supports, entropy bounds, finite spectral and Rényi-entropy comparisons, and almost-prime state controls that match cardinality and low-bit residues.

Start with the manuscript for derivations, the reproducibility guide for experiments, and the provenance record before reusing numerical results. The reader and LLM guide maps these topics to the relevant source files and their limits.

Read the project

Document	Purpose
Full manuscript	Complete paper, including all four appendices and 33 references
Manuscript integration record	Source identity, content coverage, conversion choices, and audit annotations
Research notes	Concise mathematical guide and additional audit derivations
Sanity check	Findings, corrections, and limits of the validation
Final repository review	Final science, mathematical fidelity, visual layout, and citation checks
Provenance	What the released data can and cannot substantiate
Reproducibility	Commands, environments, and computational costs
References	Verified primary sources and the open novelty question
How to cite	Public arXiv citation and attribution for the exact code or data used

All research prose is Markdown with GitHub-native rendered mathematics: inline formulas and display equations use math syntax directly inside Markdown. There are no standalone TeX documents, bibliography builds, or PDF workflows. Python code, CSV evidence, PNG figures, and machine-readable metadata retain their useful formats.

What is established?

For an even number of qubits $n$, write $N = 2^n$ and $d = 2^{n/2}$. Choose $M$ distinct basis labels uniformly, assign each amplitude $1/\sqrt{M}$, and split the qubits into two fixed halves.

Result	Status
Exact ensemble-mean reduced state and average purity	Derived by counting; independently checked on all nonempty supports at $n = 4$
Mean entropy at least $\frac{n}{2} - 1 - o(1)$, for $M = cN^\gamma$ and $\frac{1}{2} < \gamma < \frac{3}{4}$	Consequence of the exact purity formula, for fixed $c > 0$
Purity-minimizing support $M = 2^{-1/3}N^{2/3} + O(1)$	Analytic result; does not locate the von Neumann entropy maximum
Rise–peak–fall curves and retained peak estimates through $n = 30$	Finite numerical evidence; original global-search records are incomplete
Hypergeometric diagonal-entropy bound and low-bit residue ceiling	Exact bounds with stated ensemble/cut assumptions
Almost-prime deficits after matching cardinality and residues	Finite comparisons at $n = 14$; no unique arithmetic fingerprint established

This figure summarizes sampled finite systems. It is not a concentration theorem or an asymptotic scaling result.

Run the checks

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

Redraw all seven figures from released CSV files, without repeating simulations:

python scripts/reproduce_publication_figures.py

The historical command name is retained for compatibility. New outputs go under generated/; the released evidence stays in data/ and outputs/. See reproducibility for reduced simulations and expensive runs.

Scope and citation

This is a research repository, not a submission-readiness certificate. The original global searches, especially $n = 22$–$30$, have not been reconstructed. Small changes in fitted slopes when dropping those rows do not resolve that provenance gap. Broader novelty claims require comparison with existing sparse-state and fixed-size subset-state results.

Please cite the public arXiv paper when using this work:

Ruge Lin, Germán Sierra, and José I. Latorre. Arithmetic sequences as quantum states. arXiv:2501.06292 [quant-ph] (2025). doi:10.48550/arXiv.2501.06292.

For this revised implementation, manuscript, or released data, also identify the release tag or repository commit and the files used. See How to cite for the v1.0.0 citation and CITATION.cff for machine-readable metadata. The existing MIT license and author attribution are preserved.