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arxiv: 2605.19640 · v1 · pith:34PA472Snew · submitted 2026-05-19 · 🪐 quant-ph · math-ph· math.MP· math.PR

Modified logarithmic Sobolev inequalities for Abelian quantum double models

Sebastian Stengele , \'Angela Capel , Li Gao , Angelo Lucia , David P\'erez-Garc\'ia , Antonio P\'erez-Hern\'andez , Cambyse Rouz\'e , Simone Warzel This is my paper

Pith reviewed 2026-05-20 06:32 UTC · model grok-4.3

classification 🪐 quant-ph math-phmath.MPmath.PR
keywords Abelian quantum double modelsDavies Markov semigroupsmodified logarithmic Sobolev inequalityrapid mixingDobrushin-Shlosman conditionmartingale conditionquantum spin systems
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The pith

Davies semigroups for 2D Abelian quantum double models satisfy modified logarithmic Sobolev inequalities at any positive temperature, yielding rapid mixing.

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

The paper shows that the Davies Markov semigroups associated with 2D Abelian quantum double models mix rapidly at every positive temperature. A Dobrushin-Shlosman type condition is first established and shown to hold independently of temperature. This condition implies a modified logarithmic Sobolev inequality for the Davies Lindbladian once a strong martingale condition is verified for the local conditional expectations. The inequality then controls the rate at which the semigroup approaches its unique equilibrium state.

Core claim

We establish rapid mixing for Davies Markov semigroups associated with 2D Abelian quantum double models at any positive temperature. A condition of Dobrushin-Shlosman (DS) type holds at any temperature, and we show that the latter implies a modified logarithmic Sobolev inequality for the Davies Lindbladian. A key step in the argument is to verify a strong martingale condition for the local conditional expectations of the Davies semigroup in the regime of validity of the DS condition.

What carries the argument

The strong martingale condition on local conditional expectations of the Davies semigroup, which converts the Dobrushin-Shlosman condition into a modified logarithmic Sobolev inequality.

If this is right

  • Rapid mixing holds for the Davies semigroup at any positive temperature.
  • The modified logarithmic Sobolev inequality is satisfied by the Davies Lindbladian.
  • The Dobrushin-Shlosman condition remains valid at all temperatures for these models.
  • The strong martingale condition is satisfied in the regime where the Dobrushin-Shlosman condition applies.

Where Pith is reading between the lines

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

  • The same strategy may extend to other 2D quantum spin models for which a Dobrushin-Shlosman condition can be checked directly.
  • Results of this form would imply that thermalization remains efficient in topologically ordered systems even when temperature is varied.
  • Explicit constants in the inequality could be extracted by tracking the dependence on the local interaction strength.

Load-bearing premise

The strong martingale condition holds for the local conditional expectations of the Davies semigroup whenever the Dobrushin-Shlosman condition is valid.

What would settle it

Explicit computation or simulation of the mixing time for a sequence of finite-size 2D Abelian quantum double models at fixed positive temperature showing that the time grows with system size.

Figures

Figures reproduced from arXiv: 2605.19640 by \'Angela Capel, Angelo Lucia, Antonio P\'erez-Hern\'andez, Cambyse Rouz\'e, David P\'erez-Garc\'ia, Li Gao, Sebastian Stengele, Simone Warzel.

Figure 1
Figure 1. Figure 1: Left: the N × N lattice on the torus. Right: The orientation of the edges. Let us now fix an arbitrary finite Abelian group G with ∣G∣ elements. We will denote by g (instead of g −1 ) the inverse of any g ∈ G, and by 1 ∈ G the identity. The character group of G is the set Gˆ of all homomorphisms χ ∶ G → U(1), where U(1) denotes the set of complex numbers of modulus one, endowed with the pointwise multiplic… view at source ↗
Figure 2
Figure 2. Figure 2: Three sets of overlapping rectangles. (i) The full torus is split into [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗
read the original abstract

We establish rapid mixing for Davies Markov semigroups associated with 2D Abelian quantum double models at any positive temperature. A condition of Dobrushin-Shlosman (DS) type holds at any temperature, and we show that the latter implies a modified logarithmic Sobolev inequality for the Davies Lindbladian. A key step in the argument is to verify a strong martingale condition for the local conditional expectations of the Davies semigroup in the regime of validity of the DS condition.

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. This manuscript aims to establish rapid mixing for Davies Markov semigroups associated with 2D Abelian quantum double models at any positive temperature. The approach involves showing that a Dobrushin-Shlosman (DS) type condition holds at arbitrary temperatures and using this to derive a modified logarithmic Sobolev inequality (mLSI) for the Davies Lindbladian, with a key verification of a strong martingale condition for the local conditional expectations in the DS regime.

Significance. If the results are correct, this would represent a notable contribution to the study of quantum mixing times and thermalization in many-body systems with topological order. It successfully adapts classical Dobrushin-Shlosman methods to the quantum setting and provides evidence that rapid mixing holds broadly for these models. The explicit checks of the martingale property add credibility to the logical chain from DS condition to mLSI to rapid mixing.

major comments (1)
  1. [§4] §4: The strong martingale condition for the local conditional expectations is verified using the Abelian stabilizer commutation relations, but the argument would be strengthened by explicitly showing in Eq. (20) or the surrounding text how this property combines with the DS condition to yield the mLSI bound without additional assumptions on the temperature or locality.
minor comments (2)
  1. [Abstract] The abstract is concise but could specify the dimension (2D) more prominently in the first sentence for clarity.
  2. [References] Ensure all relevant works on quantum Dobrushin conditions or modified LSIs are cited, such as recent papers on quantum spin systems.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive feedback. We address the single major comment below.

read point-by-point responses

  1. Referee: [§4] §4: The strong martingale condition for the local conditional expectations is verified using the Abelian stabilizer commutation relations, but the argument would be strengthened by explicitly showing in Eq. (20) or the surrounding text how this property combines with the DS condition to yield the mLSI bound without additional assumptions on the temperature or locality.

    Authors: We agree that an explicit step-by-step account of the combination would improve clarity. In the revised manuscript we will insert a short paragraph immediately after Equation (20) that spells out how the strong martingale property (already verified via the Abelian stabilizer relations) together with the Dobrushin-Shlosman condition directly produces the mLSI bound. The added text will make plain that the derivation relies only on the properties established for the model at any positive temperature and on the locality of the interactions, without invoking further assumptions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper's chain proceeds by establishing a Dobrushin-Shlosman-type condition at arbitrary positive temperature for 2D Abelian quantum double models, verifying an explicit strong martingale property on the local conditional expectations of the Davies semigroup within that regime (using the Abelian stabilizer commutation relations), and then deriving the modified logarithmic Sobolev inequality to conclude rapid mixing. Each step consists of direct verification and implication rather than fitting parameters to data, renaming known results, or reducing the target inequality to a self-citation whose content is itself unverified or defined in terms of the conclusion. The argument supplies the requisite commutation checks internally and does not invoke load-bearing uniqueness theorems or ansatzes from overlapping prior work that would collapse the derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The proof rests on standard domain assumptions from open quantum systems and functional analysis; no free parameters are fitted and no new entities are postulated in the abstract.

axioms (1)
  • domain assumption Standard properties of Davies generators and quantum Markov semigroups
    The argument invokes the usual construction of the Davies Lindbladian and its contractivity properties without deriving them from scratch.

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Works this paper leans on

25 extracted references · 25 canonical work pages · 1 internal anchor

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