Entanglement covariance of Haar-random states under fixed boundary gates, with operator Schmidt formulas, proofs, and reproducible calculations.
How much of a state's entanglement fluctuation survives a gate acting across a small boundary?
Take a random pure state of two large quantum systems. Apply a fixed gate to a small part of each system, then compare the entanglement before and after the gate. Averaging over Haar-random input states gives the same mean entropy at both times. The question is whether a state that starts above the mean tends to remain above it.
The central result relates that correlation to the gate's operator Schmidt spectrum. For balanced growing systems and fixed boundary access, this spectrum determines the limiting covariance of every fixed positive-order Rényi entropy, including von Neumann entropy and pure-state logarithmic negativity.
Three consequences make the relation useful:
- A gate with fixed spatial access leaves a positive limiting same-order entropy correlation. For a boundary qubit pair, the minimum Rényi-2 correlation is 1/4.
- Gates with the same operator purity can retain different correlations at other entropy orders.
- An ideal hierarchy of integer-order entropy covariances identifies the operator Schmidt probabilities, although its poor conditioning limits practical reconstruction.
The absolute entropy fluctuations shrink as the half-dimension grows. The theorem concerns their rescaled covariance. Its input ensemble is complex Haar; the tested Floquet-eigenstate extension did not follow the same law.
The curves are analytical large-d predictions for a boundary ZZ gate. Horizontal lines give the minima attained by an active-qubit SWAP. Understand this example, or regenerate the figure. The figure adds no random-state samples.
One background tutorial: James A. Mingo and Roland Speicher, Free Probability and Random Matrices (2017). The selected passages and reading map use the verified author PDF, not published-book page numbers. The repository supplies the quantum-information dictionary and the gate-dependent argument. Primary research sources remain credited proof inputs, not additional prerequisite tutorials.
| Route | Where to go |
|---|---|
| LEARN | Physical setup and selected reading → book-to-project bridge → complete gate calculation → results and evidence |
| CHECK | Go directly to the theorem, derivation and hypotheses, scope and claims, primary attribution, and evidence. No educational detour is required. |
| REPRODUCE | Run the focused commands to check the formula implementation and reference data and regenerate the analytical figure. |
For a specific question:
| Question | Read |
|---|---|
| What is being compared, and why is it interesting? | Start here |
| How does Wishart fluctuation theory become a quantum entropy prediction? | Tutorial bridge |
| What happens for an actual two-qubit gate? | Worked example |
| What exactly is the theorem? | Setting and result |
| How do product gates, SWAP, local basis changes, and different times fit? | Gate controls and limiting cases |
| Why does the operator Schmidt spectrum appear? | Technical derivation and proof dependencies |
| What supports each claim, and where does the theory fail? | Results and evidence, scope and claims |
| Which parts build on prior work? | References and attribution |
The repository contains the theorem and its derivation, worked gate examples, supporting calculations, and the code and data for the numerical comparisons. The scope is the fixed-support Haar covariance law. Its limitations and the unsuccessful extension to Floquet eigenstates are part of the scientific account.
Use Python 3.12:
python -m pip install -r requirements.txt
python scripts/reproduce.pyThe command checks the maintained calculations against saved exact results, verifies the scientific reference files, runs the six existing deterministic checks, and generates the reader figure and table under build/reproduction/. Reference data and study implementations are checked by SHA-256 before and after the run. Reproduction details distinguish this default calculation from optional sampling and figure scripts.
| Location | Contents |
|---|---|
| docs/ | Physical explanation, tutorial bridge, results, scope and references |
| theory/ | Covariance theorem and proof, gate controls, moment inversion, finite-purity identities, two-cut modes and gate designs |
| gate_covariance/ | Reusable gate realignment and entropy-covariance formulas |
| checks/ | Six deterministic calculations with reference results |
| studies/ | Haar samples, non-diagonal gate tests and the Floquet comparison, with methods, code and data |
| figures/, results/ | Analytical reader figure and its numerical values |
The repository map identifies the document or calculation for each question. The reproduction guide separates the quick deterministic run from optional sample regeneration.
llms.txt is a concise relevance and source map for research assistants. It connects questions about Haar entropy correlations, operator Schmidt spectra, boundary gates and moment reconstruction to the theorem, proof, examples and code, with the assumptions and limitations needed to use them correctly.
This repository serves as a record of the work and a guide for the author’s self-directed learning. For discussion or potential collaboration, please contact Ruge Lin at gogoko699@gmail.com.
The original code, documentation, figures and accompanying data in this repository are available under the MIT License, copyright 2026 Ruge Lin. External publications and separately installed dependencies retain their own licenses; the references credit the scientific sources.
To cite this work, use CITATION.cff and identify the commit used for your calculations. Scientific attribution and the license terms are separate: the citation request adds no condition to the MIT License.