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arxiv: 2604.27053 · v1 · submitted 2026-04-29 · 🪐 quant-ph · cond-mat.str-el· math.QA

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Resolving spurious topological entanglement entropy in stabilizer codes

Peilun Han , Zijian Liang , Yifei Wang , Bowen Yang , Yingfei Gu , Yu-An Chen
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Pith reviewed 2026-05-07 08:51 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.str-elmath.QA
keywords topological entanglement entropystabilizer codesspurious contributionsLevin-Wen prescriptionconcave partitionbivariate bicycle codestopological frustrationanyons
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The pith

A concave partition for Levin-Wen calculations eliminates spurious contributions to topological entanglement entropy in translation-invariant stabilizer codes of prime-dimensional qudits.

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

Standard calculations of topological entanglement entropy can include extra terms that overestimate the true topological order in gapped quantum phases. The paper identifies the microscopic origin of these spurious contributions and introduces a concave partition for the Levin-Wen formula. It rigorously proves that this partition yields the exact topological contribution without overcounting for translation-invariant stabilizer codes on prime-dimensional qudits. Complementary analysis of bivariate bicycle codes on cylinders shows that entanglement entropy varies with circumference, indicating frustration among the underlying anyons.

Core claim

For translation-invariant stabilizer codes of prime-dimensional qudits, the Levin-Wen topological entanglement entropy computed via a concave partition is free of spurious contributions, as proven rigorously in the work. As a complementary probe, the entanglement entropy in bivariate bicycle codes on a bipartite cylinder depends sensitively on the cylinder circumference, revealing topological frustration of the anyons.

What carries the argument

The concave partition, a specific division of the lattice into regions that avoids boundary-induced errors in the Levin-Wen formula for topological entanglement entropy.

Load-bearing premise

The stabilizer codes must be translation-invariant and defined on prime-dimensional qudits for the proof of no spurious terms to hold.

What would settle it

Apply the concave partition to a known translation-invariant prime-qudit stabilizer code that previously showed spurious TEE and verify that the measured value matches the expected topological contribution exactly.

Figures

Figures reproduced from arXiv: 2604.27053 by Bowen Yang, Peilun Han, Yifei Wang, Yingfei Gu, Yu-An Chen, Zijian Liang.

Figure 1
Figure 1. Figure 1: FIG. 1. Two tripartitions for computing the Levin-Wen topo view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) Decomposition of a stabilizer view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) By multiplying an appropriate stabilizer view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Bipartite entanglement entropy on an infinite cylin view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. sTEE of the shifted toric code. (a) The stabilizers of the shifted toric code obtained from the standard toric code by view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Left: The geometry of the Levin-Wen prescription. Right: An anyon view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. An example of slicing view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8 view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Using three anyon string operators: view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. In a view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Construction of a Pauli operator, fully supported in view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Decomposition of a stabilizer view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13. Restriction of view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Eliminating residual support in region view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. Decomposing view at source ↗
read the original abstract

Topological entanglement entropy (TEE) is a key diagnostic of long-range entanglement in two-dimensional gapped phases of matter, but it can suffer from spurious contributions that overestimate the total quantum dimension of the underlying topological order. In this work, we identify the microscopic origin of spurious TEE and introduce a concave partition for computing the Levin-Wen TEE of translation-invariant stabilizer codes of prime-dimensional qudits. We rigorously prove that this prescription is free of spurious contributions. As a complementary probe, we study bivariate bicycle codes on a bipartite cylinder and show that the entanglement entropy depends sensitively on the cylinder circumference, revealing topological frustration of the underlying anyons.

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

1 major / 2 minor

Summary. The manuscript identifies the microscopic origin of spurious contributions to the Levin-Wen topological entanglement entropy (TEE) and introduces a concave partition for computing TEE in translation-invariant stabilizer codes of prime-dimensional qudits. It rigorously proves that this prescription eliminates spurious terms and complements the proof with a numerical study of bivariate bicycle codes on a bipartite cylinder, showing that entanglement entropy depends sensitively on cylinder circumference and revealing topological frustration of the underlying anyons.

Significance. If the proof holds, the work supplies a concrete, provably clean prescription for extracting TEE in a well-defined and practically relevant class of codes, removing a known source of overestimation of total quantum dimension. The explicit scoping to translation-invariant prime-dimensional stabilizer codes permits a rigorous argument, and the cylinder numerics usefully illustrate geometric sensitivity even if they are presented only as illustration. These elements together strengthen the reliability of TEE as a diagnostic for topological order in stabilizer models.

major comments (1)
  1. [Proof of absence of spurious contributions] The central proof (abstract and the section presenting the concave partition) is scoped exactly to translation-invariant stabilizer codes of prime-dimensional qudits; the manuscript should state whether the absence of spurious terms relies on primality or translation invariance in an essential way, and whether a counter-example exists outside this class, because this determines how load-bearing the restriction is for the claim.
minor comments (2)
  1. [Cylinder study] The numerical section on bivariate bicycle codes should specify the exact code parameters, cylinder circumferences, and whether error bars or multiple disorder realizations are included, so that the claimed sensitivity to circumference can be reproduced.
  2. [Introduction and definitions] Notation for the concave partition function and the Levin-Wen combination should be introduced with a single equation or figure early in the text and then used consistently, to improve readability for readers unfamiliar with the construction.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment and the recommendation for minor revision. We address the major comment below.

read point-by-point responses

  1. Referee: [Proof of absence of spurious contributions] The central proof (abstract and the section presenting the concave partition) is scoped exactly to translation-invariant stabilizer codes of prime-dimensional qudits; the manuscript should state whether the absence of spurious terms relies on primality or translation invariance in an essential way, and whether a counter-example exists outside this class, because this determines how load-bearing the restriction is for the claim.

    Authors: We agree that clarifying the dependence on these assumptions strengthens the manuscript. The proof that the concave partition removes spurious contributions relies on both translation invariance and primality in essential ways. Translation invariance enables a uniform, lattice-periodic definition of the concave regions that avoids boundary-induced artifacts in the entropy calculation. Primality is required for the finite-field structure of the qudit Pauli operators, which underpins the exact algebraic cancellation of the spurious terms. We do not construct or exhibit a counter-example outside this class, as the work is deliberately scoped to the setting where a fully rigorous proof is achievable. In the revised version we will add an explicit paragraph in the introduction and in the section defining the concave partition that states these dependencies and notes that the proof technique does not extend immediately beyond prime-dimensional, translation-invariant stabilizer codes. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper's central claim is a rigorous proof that a newly introduced concave partition eliminates spurious contributions to the Levin-Wen TEE, scoped precisely to translation-invariant stabilizer codes of prime-dimensional qudits. This rests on direct analysis of the code properties and the partition definition rather than any fitted parameters, self-referential predictions, or load-bearing self-citations that reduce the result to prior inputs by construction. The bivariate cylinder study is presented only as a complementary illustration of geometric sensitivity, not as part of the proof. No step in the provided derivation chain equates a claimed result to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on standard definitions of stabilizer codes, the Levin-Wen formula for TEE, and translation invariance; no free parameters or invented entities are introduced beyond the concave partition itself, which is a methodological choice rather than a new physical entity.

axioms (2)
  • domain assumption Stabilizer codes on prime-dimensional qudits admit a well-defined Levin-Wen TEE that can be computed from region entropies.
    Invoked in the statement of the problem and the proof claim.
  • domain assumption Translation invariance allows a clean separation between bulk and boundary contributions in the entanglement entropy.
    Required for the concave partition to be well-defined and for the cylinder analysis.

pith-pipeline@v0.9.0 · 5417 in / 1390 out tokens · 43982 ms · 2026-05-07T08:51:06.744520+00:00 · methodology

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Reference graph

Works this paper leans on

13 extracted references

  1. [1]

    Pauli operators are characterized by their commutation relations

    Notation In a two-dimensional translationally invariant stabilizer code, we are concerned with three kinds of objects: Pauli operators, stabilizers, and anyons, as well as the relations among them. Pauli operators are characterized by their commutation relations. Stabilizers are generated by products of translated stabilizer generators. Anyons are defined...

  2. [2]

    In this subsection, we derive an explicit support bound and thereby prove Lemma 5

    Proof of Lemma 5 By definition, every trivial anyon can be created by a finitely supported Pauli operator, but for a given trivial anyon there is generally no obvious bound on the size of such an operator. In this subsection, we derive an explicit support bound and thereby prove Lemma 5. Lemma 5.In a two-dimensional translation-invariant topological stabi...

  3. [3]

    Proof of Lemma 4 This subsection develops both a qualitative and a quantitative understanding of the generation of boundary gauge operators. By definition, a boundary gauge operator is a Pauli operator supported near the boundary that commutes with all bulk stabilizers, although different boundary gauge operators need not commute with one another. We firs...

  4. [4]

    bulk stabilizer generators in the upper half-plane

  5. [5]

    path” along which anyons are moved by anyon strings. To show the “l down sites down

    stabilizer generators of the infinite plane, truncated to their support iny≥0. Moreover, every factor is supported in y≤8r 3q4 + 5r+ 3h .(E128) Letb G be a finitely supported boundary gauge operator with height at mosth. Then there existsθ∈ bSsuch that bG =πw(θ).(E129) We splitθinto the part already lying in the upper half-plane and the part extending bel...

  6. [6]

    Theorem 1.Consider a translation-invariant topological stabilizer code on a square lattice ofZ p (with primep) qudits withqqudits per site

    Proof of Theorem 1 With the preparations above, we can now finally prove Theorem 1. Theorem 1.Consider a translation-invariant topological stabilizer code on a square lattice ofZ p (with primep) qudits withqqudits per site. Assume that the stabilizer generators have ranger, i.e., each generator is supported within anr×rsquare. Then, for the concave partit...

  7. [7]

    suppw(k ixni ymi ej)⊂A∪B∪C

  8. [8]

    suppw(k ixni ymi ej)̸⊂A∪B∪C, but suppw(k ixni ymi ej)⊂A∪B∪C∪D

  9. [9]

    suppw(k ixni ymi ej)̸⊂A∪B∪C, but suppw(k ixni ymi ej)⊂A∪B∪C∪E. We denote byθ Si the sum of all monomial terms in theith class, and define sSi :=w(θ Si), i= 1,2,3.(E189) By construction, onlys S2 can have support intersectingD, but this support cannot be canceled bys S1 ors S3. Hence sS2 cannot have supports inD. By the same argument,s S3 cannot have suppo...

  10. [10]

    (E42), combining with the fact thatt (ς) h itself commutes with all stabilizers gives that: xnymgµ ∗π x −∞:x1 t(ς) h =π x −∞:x1 xnymgµ ∗t (ς) h =x nymgµ ∗t (ς) h = 0.(E200)

    it is cut by the projectionπ x −∞:x1, namely, xmin(xnymgµ)< x 1 ≤x max(xnymgµ); (E199) otherwise, the identity in Eq. (E42), combining with the fact thatt (ς) h itself commutes with all stabilizers gives that: xnymgµ ∗π x −∞:x1 t(ς) h =π x −∞:x1 xnymgµ ∗t (ς) h =x nymgµ ∗t (ς) h = 0.(E200)

  11. [11]

    it overlaps the support oft (ς) h , which by Eq. (E197) requires ymin(xnymgµ)< r, y max(xnymgµ)≥0.(E201) Using locality of the generators, these conditions imply (n, m)∈[x 1 −r, x 1)×[−r, r).(E202) Thus, every violated generator lies inside ar×2rbox around the endpointx=x 1. In later proofs, using a larger box with size 2r×2ris sufficient. Therefore, to t...

  12. [12]

    Since it is supported nearQ, it can be created by a local Pauli operator supported inD

    Ifαis trivial, then P i η(ςi) 0 is also trivial. Since it is supported nearQ, it can be created by a local Pauli operator supported inD. Henceαitself can be created by a Pauli operator supported inD

  13. [13]

    Ifαis nontrivial, then P i η(ςi) 0 has the opposite anyon type toα. By attaching a semi-infinite thin string operator whose endpoint cancels this reference anyon inQ, and multiplying it by the Pauli operator supported inDconstructed above, we obtain a semi-infinite string operator supported inA∪D∪Ewhose endpoint createsα. 42 Therefore, the two conditions ...