Information clustering and geometric pressure in AdS3

Feiyu Deng

doi:10.5281/zenodo.23534563

Start with cooperative storage

Correlations can reduce the memory needed to store quantum systems jointly, while bringing laboratories into a cooperating group also costs resources. The paper specifies component algebras at a fixed finite factorizing regulator. Each block must recover its source from its own memory, and every additional laboratory in that block has a per-source-use access price τ.

The objective is the net memory saving after this cooperation charge. For a partition P of the component labels, the paper uses the established entropy-partition objective

G_\tau(\rho_A;\{A_i\})=\max_{\mathcal P\vdash[n]}\left\{\sum_{B\in\mathcal P}T(B)-\tau(n-|\mathcal P|)\right\}.

Here T(B) is the total correlation of block B, n is the number of laboratories, and |P| counts the blocks. All set partitions are allowed, including blocks of nonconsecutive labels. This information task is defined using actual marginal-state entropies, together with the specified algebras, operations and resource accounting.

The area–volume representation

On a static spatial slice, the geometric problem minimizes free perimeter plus a positive homology-area penalty, with area source λ/R and overall normalization 1/(4G). This is the area–volume structure in the two-dimensional slice. The reference value is the sum of single-interval caps separately reoptimized at the same source λ.

For fixed finitely many separated intervals in the leading Einstein–AdS₃ vacuum RT entropy model, the public version establishes

\begin{aligned} F_\lambda^\epsilon-F_{\lambda,\mathrm{disc}}^\epsilon&=-qG_\tau^{\mathrm{RT}}+O(c\epsilon/\delta),\\ q&=\sqrt{1-\lambda^2},\\ \tau&=\frac{c\pi\lambda}{3q}. \end{aligned}

The factors q and τ relate the geometric source λ to the information task's merger tariff, with 0 ≤ λ < 1. The ultraviolet error depends on the cutoff ε and separation scale δ and is uniform on compact subintervals of that range. The admissible positive regions have finite perimeter, exact trace and compact essential support. Entropy-preserving component refinement and a global calibration establish the correspondence.

Examples and scope

Partition structure: collective transitions, nested groups and free-Dirac examples distinguish different partition effects.
Area response: controlled first-derivative estimates determine the geometric area jump by taking the ultraviolet limit before approaching the phase boundary from either side; finite-c actual-state derivatives require separate analysis.
Thermal states: the noncompact thermal-line representation includes a charged horizon contribution.
Compactness: a compact-BTZ annulus has a finite mismatch at the unchanged merger tariff.

Replacing RT entropies with actual-state entropies requires additional error bounds. These model, regulator and region assumptions delimit the result.

Reading the main article

Begin with component algebras and the storage task in section 2, then the partition geometry and pressure representation in sections 3–4. Section 5 covers collective phases, nested groups and area response; section 6 compares the thermal line with the compact annulus. Appendices C–H develop refinement, global minimization, exact trace and ultraviolet estimates. Appendix I records numerical methods and checks.

The companion note

The 17-page companion is titled Field constructions and their composition limits. It retains five separately scoped investigations: a complementary-spectrum obstruction, an interval-entropy construction and its composition limit, exact transport with supplied conformal probes, density-matrix geometry, and replica channel weights. Each uses its own field data and admissible operations. The main article owns the storage objective, positive homology pressure problem, calibration and ultraviolet correspondence.

The main article, companion and reproducibility source belong to the same public Zenodo record. The local PDFs preserve the original bytes, acknowledgments and AI-use disclosures. The record is licensed CC BY 4.0; third-party style files in the source archive retain their original notices.

Source abstract

Joint quantum storage saves memory through correlations, while cooperation requires access to several laboratories. We formulate this competition at a fixed finite factorizing regulator, with specified component algebras, independently recoverable block memories and a nonnegative merger price. The optimal net saving is the established entropy-partition objective evaluated on actual component marginals. In the leading Einstein-AdS $_3$ vacuum RT entropy model, this optimum has a geometric representation. For fixed finitely many separated intervals, the minimum of free perimeter plus positive homology area, relative to separately optimized single-interval caps, satisfies $F_\lambda^\epsilon-F_{\lambda,\mathrm{disc}}^\epsilon=-qG_\tau^\mathrm{RT}+O(c\epsilon/\delta)$ , where $q=\sqrt{1-\lambda^2}$ and $\tau=c\pi\lambda/(3q)$ . The ultraviolet estimate is uniform on compact subcritical source intervals, and the minimization is over positive finite-perimeter regions with exact trace and compact essential support. Entropy-preserving component refinement and a global calibration establish the representation. Replacing RT entropies by actual-state entropies requires separate error bounds. Collective, nested and free-Dirac examples distinguish the partition effects. First-derivative estimates determine the geometric area jump, without extending the result to finite- $c$ derivatives. The representation extends to the noncompact thermal line with a charged horizon. A compact-BTZ annulus instead gives a finite mismatch at the unchanged tariff.

Paper files

Original version record and checksums ↗

Citation

Citation / BibTeX

@misc{Deng:2026InformationClustering,
  author = {Deng, Feiyu},
  title = {Information clustering and geometric pressure in AdS3},
  year = {2026},
  month = {oct},
  doi = {10.5281/zenodo.23534563},
  url = {https://zenodo.org/records/23534563},
  note = {Preprint, Zenodo; published online 2026-10-22}
}