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arxiv: 2412.05484 · v2 · submitted 2024-12-07 · 🪐 quant-ph · cond-mat.str-el· hep-th

Topological entanglement entropy meets holographic entropy inequalities

Joydeep Naskar , Sai Satyam Samal This is my paper

Pith reviewed 2026-05-23 08:07 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.str-elhep-th
keywords topological entanglement entropyholographic entropy inequalitiesKitaev-Preskill probeLevin-Wen probetopological orderentanglement entropygapped quantum systemssubtraction schemes
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The pith

Subtraction schemes for topological entanglement entropy succeed because they meet explicit necessary conditions that isolate the topological contribution, and holographic entropy inequalities hold for the entanglement entropy of non-degree

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows why the subtraction schemes in the Kitaev-Preskill and Levin-Wen quantities extract topological entanglement entropy by spelling out the necessary conditions any such quantity must obey. These conditions separate the two original probes into distinct classes and support generalization to an arbitrary number of subregions through the cyclic quantities Q_{2n+1} and the multi-information I_n. The same framework leads to the claim that holographic entropy inequalities are satisfied by the entanglement entropy of non-degenerate gapped topologically ordered states in two dimensions. A reader cares because the conditions give a systematic test for whether a given information quantity captures topological order and link quantum-information probes to holographic principles.

Core claim

The subtraction schemes work because they satisfy explicit necessary conditions that isolate the topological contribution; these conditions differentiate Kitaev-Preskill and Levin-Wen probes and allow generalization to Q_{2n+1} and I_n; holographic entropy inequalities hold for the entanglement entropy of non-degenerate gapped topologically ordered 2D states.

What carries the argument

The necessary conditions an information quantity must satisfy to capture topological entanglement entropy, which differentiate the Kitaev-Preskill and Levin-Wen classes and enable the cyclic quantities Q_{2n+1} and multi-information I_n.

If this is right

  • Infinitely many information quantities can serve as TEE probes once they meet the necessary conditions.
  • Holographic entropy inequalities apply directly to the entanglement entropy of non-degenerate gapped topologically ordered 2D states.
  • The conditions allow systematic construction of new probes such as the cyclic quantities Q_{2n+1} and the multi-information I_n for any number of subregions.
  • The Kitaev-Preskill and Levin-Wen schemes belong to separate classes distinguished by which conditions they satisfy.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same conditions could be used to test candidate TEE probes in specific lattice models such as the toric code before numerical computation.
  • If the conditions extend beyond two dimensions they would supply a route to TEE detection in three-dimensional topological phases.
  • The link between TEE probes and holographic inequalities suggests that information quantities satisfying the conditions may obey additional inequalities from the holographic side.

Load-bearing premise

The ground state is assumed to be non-degenerate, gapped, and topologically ordered in two dimensions.

What would settle it

A concrete counter-example in which a non-degenerate gapped topologically ordered 2D ground state violates one of the holographic entropy inequalities for its entanglement entropy, or an information quantity that extracts TEE without obeying the stated conditions.

Figures

Figures reproduced from arXiv: 2412.05484 by Joydeep Naskar, Sai Satyam Samal.

Figure 1
Figure 1. Figure 1: (a) Geometry used by Kitaev and Preskill [ [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Geometry considered for computing the topological [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: A disk with 6−subregions obtained after smooth deformations which is used to obtain the value of the infor￾mation quantity Q6,1, Eq. (15) and Q6,2, Eq. (21). It turns out that the information quantity Q6,1 can not be used to probe the TEE however the latter quantity, Q6,2 can probe the TEE. balance, is both topological and sign-definite, on a 2d disk￾like geometry. Proof. By conjecture III.3, Qi is topolog… view at source ↗
read the original abstract

Topological entanglement entropy (TEE) is an efficient way to detect topological order in the ground state of gapped Hamiltonians. The seminal work of Kitaev and Preskill~\cite{preskill-kitaev-tee} and simultaneously by Levin and Wen~\cite{levin-wen-tee} proposed information quantities that can probe the TEE. In the present work, we explain why the subtraction schemes in the proposed information quantities~\cite{levin-wen-tee,preskill-kitaev-tee} work for the computation of TEE and generalize them for arbitrary number of subregions by explicitly noting the necessary conditions for an information quantity to capture TEE. Our conditions differentiate the probes defined by Kitaev-Preskill and Levin-Wen into separate classes. While there are infinitely many possible probes of TEE, we focus particularly on the cyclic quantities $Q_{2n+1}$ and multi-information $I_n$. We also show that the holographic entropy inequalities are satisfied by the quantum entanglement entropy of the non-degenerate ground state of a topologically ordered two-dimensional medium with a mass gap.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript derives necessary conditions that an information quantity must satisfy to isolate the topological entanglement entropy (TEE) contribution in gapped 2D systems. It uses these conditions to explain the success of the Kitaev-Preskill and Levin-Wen subtraction schemes, shows that the two schemes belong to distinct classes, generalizes the construction to the cyclic quantities Q_{2n+1} and the multi-information I_n, and proves that the entanglement entropy of non-degenerate gapped topologically ordered 2D states obeys the holographic entropy inequalities.

Significance. If the derivations of the necessary conditions are rigorous, the work supplies a systematic, condition-based framework for constructing TEE probes and establishes a direct link between TEE and holographic entropy inequalities for topological states. The explicit differentiation of the KP and LW classes together with the generalization to arbitrary numbers of subregions constitute a clear advance over the original subtraction schemes.

minor comments (2)
  1. [Abstract] The abstract states that the necessary conditions are derived and the inequalities are shown, yet the introduction would benefit from a concise enumerated list of those conditions to orient the reader before the technical sections.
  2. Notation for the generalized quantities Q_{2n+1} and I_n should be introduced with an explicit definition or reference to the standard multi-information expression at first appearance.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the summary of our derivations of necessary conditions for TEE probes, the differentiation of KP and LW schemes, the generalizations to cyclic quantities and multi-information, and the connection to holographic entropy inequalities. We appreciate the recommendation for minor revision.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper starts from the standard definitions of entanglement entropy, topological order, and the known Kitaev-Preskill and Levin-Wen subtraction schemes. It derives necessary conditions that isolate the topological contribution, differentiates the two classes, and generalizes to Q_{2n+1} and I_n quantities through explicit algebraic requirements on information quantities; these steps are self-contained mathematical statements that do not reduce to fitted parameters or prior results by the same authors. The claim that holographic entropy inequalities hold for the entanglement entropy of non-degenerate gapped 2D topologically ordered states follows directly from the stated premises (mass gap, non-degeneracy, topological order) without invoking self-citations or ansatzes that presuppose the target result. No load-bearing step equates a derived quantity to an input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities can be extracted. The central claims rest on standard assumptions of gapped topological order and information-theoretic definitions already present in the cited literature.

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