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arxiv: 2605.22424 · v1 · pith:7HSMKQOBnew · submitted 2026-05-21 · 🪐 quant-ph · cond-mat.stat-mech· math-ph· math.MP

Long-range nonstabilizerness of topologically encoded states from mutual information

David Aram Korbany , Tyler D. Ellison , David T. Stephen , Lorenzo Piroli This is my paper

Pith reviewed 2026-05-22 05:55 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mechmath-phmath.MP
keywords long-range nonstabilizernessmutual informationtopological ordertoric codemodular transformationsfault-tolerant gatesencoded quantum states
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The pith

Mutual information between non-contractible loops diagnoses long-range nonstabilizerness in topologically encoded states on a torus.

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

This paper shows that mutual information between non-overlapping regions containing non-contractible loops, combined with how that information transforms under modular real-space cuts, can diagnose long-range nonstabilizerness in states encoded in two-dimensional topological systems. Long-range nonstabilizerness is the obstruction to turning a nonstabilizer state into a stabilizer state using only shallow-depth local circuits. In the toric code the method gives a complete classification that confirms the obstruction for every encoded non-stabilizer state. In the doubled-Fibonacci string-net model it identifies the obstruction for all states except a finite subset with special modular properties. The results limit which logical gates can be realized fault-tolerantly without deep circuits.

Core claim

In topologically ordered systems defined on a torus, the mutual information between regions that contain non-contractible loops together with its change under modular real-space transformations serves as a diagnostic for long-range nonstabilizerness. For the toric code this diagnostic fully classifies the property and certifies its presence in every encoded non-stabilizer state. For the non-abelian string-net model with doubled Fibonacci topological order the same quantities detect long-range nonstabilizerness in all states except a finite subset that transforms in a special way under the modular group. These findings directly constrain the set of logical gates that admit fault-tolerant, low

What carries the argument

Mutual information between non-overlapping regions containing non-contractible loops, and the change of this mutual information under modular real-space transformations, serving as a proxy for the obstruction to removing nonstabilizerness by shallow local circuits.

If this is right

  • Every encoded non-stabilizer state in the toric code possesses long-range nonstabilizerness.
  • The same mutual-information diagnostic applies to both abelian and non-abelian topological orders on the torus.
  • The presence of long-range nonstabilizerness restricts the logical gates that can be implemented fault-tolerantly without deep circuits.
  • States with special modular transformation properties may evade detection in certain non-abelian models.

Where Pith is reading between the lines

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

  • The same diagnostic quantities could be applied to other topological codes or lattice geometries to test for long-range nonstabilizerness.
  • The classification may help identify which magic states can be injected fault-tolerantly into topological codes.
  • Numerical checks of modular invariance on small tori could provide an independent test of the claimed classification.

Load-bearing premise

That the change in mutual information under modular real-space transformations is a faithful indicator of whether nonstabilizerness can be removed by any shallow-depth local circuit.

What would settle it

An encoded non-stabilizer state in the toric code for which the mutual information between regions containing non-contractible loops remains invariant under every modular transformation yet can still be converted to a stabilizer state by a shallow local circuit.

Figures

Figures reproduced from arXiv: 2605.22424 by David Aram Korbany, David T. Stephen, Lorenzo Piroli, Tyler D. Ellison.

Figure 1
Figure 1. Figure 1: Left: vertex, plaquette, and string operators in [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Three inequivalent curves γα on the torus related by modular transformations. For each curve γα, we consider disjoint regions A and B containing curves that are topologi￾cally equivalent to it. So far, we have discussed the properties of MES for the vertical loop γy (γ ′ y ). However, the same properties hold for more general non-contractible loops. In the following, we will consider three types of inequiv… view at source ↗
Figure 3
Figure 3. Figure 3: Left: support of the WLO Z(γxy) on the dual lat￾tice, corresponding to the loop γxy in [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Pictorial representation of the string-net lattice. [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Dehn twist on the torus implemented by a shear [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Pictorial derivation of Eq. (56). We only show the “ket” layer. The horizontal double lines (colored) indicate a trace over C˜ on the left, and over C in the middle pannel. automorphism of the lattice. This is for example the case for a 2π/3 rotation on the honeycomb lattice. Let γ˜ be any non-contractible loop that is topologically in￾equivalent to γy, and let ζ be a modular transformation mapping ˜γ to γ… view at source ↗
read the original abstract

We study long-range nonstabilizerness (LRN), namely the obstruction to remove nonstabilizerness with shallow-depth local quantum circuits. In one-dimensional settings, the mutual information between disconnected spatial regions has proven to be a powerful tool to diagnose LRN. In this work, we focus on encoded states of two-dimensional topologically-ordered systems, and explore the ability of the mutual information to serve as a diagnostic of LRN. Focusing on the concrete setting of lattice models defined on a torus, we show that information about LRN can be gained from the analysis of the mutual information between non-overlapping regions containing non-contractible loops, and of the change of such mutual information under modular real-space transformations. We exemplify this idea in the toric code and the non-abelian string-net model with doubled Fibonacci topological order. In the former case, we show that the mutual information provides a full classification, certifying LRN for all encoded non-stabilizer states. In the latter case, instead, our approach does not lead to a full classification, as it detects LRN for all states except from a finite subset with special transformation properties under the modular group. Finally, we discuss how our results on LRN constrain the logical gates that can be implemented fault-tolerantly on the torus.

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

2 major / 2 minor

Summary. The manuscript claims that mutual information between non-overlapping regions containing non-contractible loops, together with its transformation under modular real-space cuts, diagnoses long-range nonstabilizerness (LRN) in encoded states of 2D topological models on a torus. In the toric code this yields a complete classification that certifies LRN for every encoded non-stabilizer state; in the doubled Fibonacci string-net model the same diagnostic detects LRN for all states except a finite subset possessing special modular transformation properties. The results are further used to constrain the set of logical gates that can be realized fault-tolerantly.

Significance. If the central mapping holds, the work supplies a concrete, topologically motivated diagnostic for LRN that extends earlier one-dimensional mutual-information techniques to two-dimensional codes. The toric-code classification is a strong, falsifiable outcome, and the explicit link to restrictions on fault-tolerant logical gates is directly relevant to quantum error correction. The modular-group analysis adds a distinctive topological ingredient that is not present in prior LRN studies.

major comments (2)
  1. [Abstract and toric-code section] Abstract and toric-code analysis: the claim that mutual information together with its modular transformations certifies LRN for every encoded non-stabilizer state rests on the unproven assertion that no constant-depth local circuit applied to a stabilizer state can reproduce the observed MI values (or their modular changes) while leaving logical nonstabilizerness intact. An explicit bound or counter-example argument showing that shallow circuits cannot adjust the MI signature without eliminating the logical nonstabilizerness is required to support the 'full classification' statement.
  2. [Fibonacci-model analysis] Fibonacci-model section: the identification of a finite subset of states that evade detection is presented as a limitation of the diagnostic rather than a statement about the absence of LRN in those states. Clarification is needed on whether these states truly lack LRN or whether the modular-MI probe is simply incomplete for them.
minor comments (2)
  1. Notation for the modular real-space cuts and the precise definition of the regions containing non-contractible loops should be made uniform across figures and text to improve readability.
  2. A short comparison table contrasting the toric-code and Fibonacci outcomes would help readers quickly grasp the difference in classification power.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for their careful reading and constructive comments on our manuscript. We address each of the major comments in detail below, indicating where revisions have been made to strengthen the presentation and rigor of our results.

read point-by-point responses

  1. Referee: [Abstract and toric-code section] Abstract and toric-code analysis: the claim that mutual information together with its modular transformations certifies LRN for every encoded non-stabilizer state rests on the unproven assertion that no constant-depth local circuit applied to a stabilizer state can reproduce the observed MI values (or their modular changes) while leaving logical nonstabilizerness intact. An explicit bound or counter-example argument showing that shallow circuits cannot adjust the MI signature without eliminating the logical nonstabilizerness is required to support the 'full classification' statement.

    Authors: We agree that an explicit argument is required to rigorously justify why the observed MI values (and their modular transformations) cannot be reproduced by a constant-depth local circuit applied to a stabilizer state while preserving logical nonstabilizerness. In the revised manuscript we add a new paragraph in the toric-code section that supplies this justification. We argue that any constant-depth circuit has a finite light-cone; consequently, when the non-contractible regions are taken sufficiently large compared with the circuit depth, the mutual information between them is invariant under the circuit action. Stabilizer states on the toric code yield a strictly smaller MI value than the non-stabilizer encoded states we consider. Therefore, no such shallow circuit can match the MI signature without either changing the topological sector or introducing nonstabilizerness that would itself constitute LRN. We also note that the modular transformations further constrain the possible circuit actions because they mix the non-contractible cycles. This addition directly supports the full-classification claim. revision: yes

  2. Referee: [Fibonacci-model analysis] Fibonacci-model section: the identification of a finite subset of states that evade detection is presented as a limitation of the diagnostic rather than a statement about the absence of LRN in those states. Clarification is needed on whether these states truly lack LRN or whether the modular-MI probe is simply incomplete for them.

    Authors: We thank the referee for highlighting the need for explicit clarification on this point. In the original text we already frame the finite subset as states that evade detection by the modular-MI diagnostic, without asserting that they lack LRN. To make this unambiguous we have revised the Fibonacci-model section to state explicitly that the probe is incomplete for these states: their special transformation properties under the modular group render the MI between non-contractible regions indistinguishable from that of stabilizer states, yet this does not preclude the presence of LRN that would be visible under other diagnostics. We emphasize that determining whether these states actually possess LRN lies beyond the scope of the mutual-information approach developed here and remains an open question. revision: yes

Circularity Check

0 steps flagged

No circularity: MI diagnostic derived from explicit model calculations

full rationale

The paper defines LRN independently as the obstruction to removing nonstabilizerness via shallow-depth local circuits. It then computes mutual information between regions containing non-contractible loops on the torus and tracks its change under modular real-space transformations. These quantities are evaluated directly on the toric-code and doubled-Fibonacci string-net ground states and their logical encodings using the standard definition of von Neumann mutual information. The classification statements follow from comparing the resulting MI values (and their modular orbits) between stabilizer and non-stabilizer logical states; no equation equates the MI diagnostic to the circuit-depth obstruction by construction, no parameter is fitted and then relabeled as a prediction, and no load-bearing step rests on a self-citation whose content is itself unverified. The derivation therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard properties of quantum mutual information and the definition of long-range nonstabilizerness as an obstruction to shallow local circuits; no free parameters, invented entities, or ad-hoc axioms are visible in the abstract.

axioms (2)
  • domain assumption Mutual information between disconnected regions diagnoses long-range nonstabilizerness in one-dimensional systems
    Invoked as the starting point for extending the diagnostic to two dimensions (abstract opening sentence).
  • domain assumption Topological order on the torus is captured by states defined on non-contractible loops
    Used to select the spatial regions whose mutual information is analyzed.

pith-pipeline@v0.9.0 · 5781 in / 1577 out tokens · 31874 ms · 2026-05-22T05:55:26.824602+00:00 · methodology

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