Modified logarithmic Sobolev inequalities for Abelian quantum double models

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.

• The paper establishes a system-size-independent modified logarithmic Sobolev inequality for Davies semigroups of two-dimensional Abelian quantum double models at every positive temperature, improving mixing-time bounds from polynomial to polylogarithmic in system size.
• The authors derive explicit Davies conditional expectations, characterize Lindbladian kernels, and combine exact infinite-temperature factorization with exponentially accurate finite-temperature estimates to prove a strong martingale condition.
• The proof extends beyond CSS codes to general ergodic, translation-invariant single-edge jump operators, while identifying non-Abelian models, higher-dimensional phase transitions, and zero-temperature behavior as important open problems.

Overview and main result

The paper establishes a modified logarithmic Sobolev inequality (MLSI) for the Davies Markov semigroups associated with two-dimensional Abelian quantum double models (Kitaev's quantum doubles based on an arbitrary finite Abelian group $G$) at every strictly positive temperature. Concretely, for any inverse temperature $\beta > 0$ there exists an absolute constant $\alpha_\beta > 0$, independent of system size, such that the MLSI constant of the Davies Lindbladian on an $N \times N$ torus satisfies $$\alpha(\mathcal{L}_\Lambda^{(\beta)}) \geq \alpha_\beta$$. Via the standard Pinsker-type bound $$\|e^{t\mathcal{L}^*_\Lambda}(\sigma) - \rho\|_1 \leq e^{-\alpha t}\sqrt{\ln(1/\rho_{\min})}$$, this yields rapid mixing — mixing time $O(\mathrm{polylog}|\Lambda|)$ — improving on the previously known fast mixing ($O(\mathrm{poly}|\Lambda|)$) results obtained from spectral gap bounds (2605.19640). The result also extends the framework developed for CSS codes in prior work by the same authors to a broader class of jump operators.

The proof strategy has two components. The first, which constitutes the main technical novelty here, derives explicit expressions for the Davies conditional expectations $$\mathbb{E}_R^\beta = \lim_{t\to\infty} e^{t\mathcal{L}_R^{(\beta)}}$$ and verifies an approximate factorization property on overlapping rectangles — the strong martingale condition. The second component is a multi-scale analysis, largely imported from the CSS-code work, showing that the strong martingale condition implies a uniformly positive MLSI constant.

Model, dynamics, and mixing notions

The models live on $N\times N$ square lattices with periodic boundary conditions; each edge carries the Hilbert space $\beta > 0$0. The Hamiltonian is

$\beta > 0$1

with star operators built from left regular representations $\beta > 0$2 and plaquette operators from character operators $\beta > 0$3. All local terms commute, and the generalized projectors $\beta > 0$4, $\beta > 0$5 form resolutions of the identity. The Davies generator is assembled from single-edge bare jump operator sets that are self-adjoint closed, ergodic (trivial commutant), and translation invariant, with rates satisfying detailed balance $\beta > 0$6 and a uniform lower bound $\beta > 0$7. The generator is self-adjoint with respect to all weighted scalar products $\beta > 0$8, so its infinite-time limits are orthogonal projections onto $\beta > 0$9 in any of these inner products.

The distinction between fast and rapid mixing is quantitatively sharp: a uniform spectral gap only gives $\alpha_\beta > 0$0 because $\alpha_\beta > 0$1 for commuting Hamiltonians, whereas a uniform MLSI constant removes the dimension-dependent prefactor entirely and yields $\alpha_\beta > 0$2.

Gibbs marginals and the Dobrushin–Shlosman condition

A structural fact underpinning everything is Proposition 3.1: for star- and plaquette-connected regions $\alpha_\beta > 0$3, partial traces of Gibbs factors admit the explicit form

$\alpha_\beta > 0$4

with $\alpha_\beta > 0$5 powers of $\alpha_\beta > 0$6. Consequently, all marginals of the Gibbs state lie in a common Abelian algebra generated by the $\alpha_\beta > 0$7 and $\alpha_\beta > 0$8, so any two marginals of the Gibbs state commute — a property heavily exploited throughout. Moreover, marginals are bounded within factors $\alpha_\beta > 0$9 of $N \times N$0, where $N \times N$1 decays exponentially in the minimal number of stars or plaquettes touching $N \times N$2.

From this, the paper proves that the DS-condition holds at all positive temperatures: for overlapping rectangles $N \times N$3, $N \times N$4 with $N \times N$5,

$N \times N$6

with explicit constants $N \times N$7 and $N \times N$8. The proof factorizes the ratio of marginals into four commuting partial traces, each controlled via the marginal estimates; the exponent 10 arises from the worst case where the overlap $N \times N$9 splits into two disconnected rectangles. Since the 2D double model exhibits no static phase transition, the DS regime covers the entire positive-temperature range — a stronger statement than in generic commuting systems, where DS-type clustering typically holds only above a critical temperature.

Kernel of the Lindbladian

The central structural result is an exact characterization of the kernel: for star- and plaquette-connected $$\alpha(\mathcal{L}_\Lambda^{(\beta)}) \geq \alpha_\beta$$0 with $$\alpha(\mathcal{L}_\Lambda^{(\beta)}) \geq \alpha_\beta$$1,

$$\alpha(\mathcal{L}_\Lambda^{(\beta)}) \geq \alpha_\beta$$2

i.e., operators supported outside $$\alpha(\mathcal{L}_\Lambda^{(\beta)}) \geq \alpha_\beta$$3 that commute with all generalized star and plaquette projectors touching $$\alpha(\mathcal{L}_\Lambda^{(\beta)}) \geq \alpha_\beta$$4. The proof combines two ingredients: the general fact that the kernel equals the commutant of the jump operators (via the discrete convolution theorem for operator Fourier transforms $$\alpha(\mathcal{L}_\Lambda^{(\beta)}) \geq \alpha_\beta$$5), and an inclusion–exclusion ("telescope") argument recovering each individual star projector $$\alpha(\mathcal{L}_\Lambda^{(\beta)}) \geq \alpha_\beta$$6 from normalized partial traces of the collective star Hamiltonian over subregions. This simplifies the corresponding step in the CSS-code treatment by eliminating the "partition by support" machinery, and it holds for arbitrary finite sets of ergodic single-edge jump operators rather than only $$\alpha(\mathcal{L}_\Lambda^{(\beta)}) \geq \alpha_\beta$$7.

Conditional expectations as pinchings and traces

At infinite temperature, the conditional expectation admits a fully explicit formula:

$$\alpha(\mathcal{L}_\Lambda^{(\beta)}) \geq \alpha_\beta$$8

where $$\alpha(\mathcal{L}_\Lambda^{(\beta)}) \geq \alpha_\beta$$9 and $$\|e^{t\mathcal{L}^*_\Lambda}(\sigma) - \rho\|_1 \leq e^{-\alpha t}\sqrt{\ln(1/\rho_{\min})}$$0 are products of single-operator pinchings onto the eigenspaces of the $$\|e^{t\mathcal{L}^*_\Lambda}(\sigma) - \rho\|_1 \leq e^{-\alpha t}\sqrt{\ln(1/\rho_{\min})}$$1 and $$\|e^{t\mathcal{L}^*_\Lambda}(\sigma) - \rho\|_1 \leq e^{-\alpha t}\sqrt{\ln(1/\rho_{\min})}$$2. The proof rests on pairwise commutation of traces and pinchings (verified through the conjugation relations $$\|e^{t\mathcal{L}^*_\Lambda}(\sigma) - \rho\|_1 \leq e^{-\alpha t}\sqrt{\ln(1/\rho_{\min})}$$3 or $$\|e^{t\mathcal{L}^*_\Lambda}(\sigma) - \rho\|_1 \leq e^{-\alpha t}\sqrt{\ln(1/\rho_{\min})}$$4 depending on orientation) plus uniqueness of orthogonal projections. At finite temperature, the conditional expectation is obtained by conjugation with the marginal Gibbs state:

$$\|e^{t\mathcal{L}^*_\Lambda}(\sigma) - \rho\|_1 \leq e^{-\alpha t}\sqrt{\ln(1/\rho_{\min})}$$5

shown to be independent of $$\|e^{t\mathcal{L}^*_\Lambda}(\sigma) - \rho\|_1 \leq e^{-\alpha t}\sqrt{\ln(1/\rho_{\min})}$$6 by uniqueness of the projection onto the temperature-independent kernel. Crucially, commutation of $$\|e^{t\mathcal{L}^*_\Lambda}(\sigma) - \rho\|_1 \leq e^{-\alpha t}\sqrt{\ln(1/\rho_{\min})}$$7 with all stars and plaquettes makes this formula well defined and enables the subsequent factorization analysis.

Strong martingale condition

The strong martingale condition requires, for overlapping rectangles with $$\|e^{t\mathcal{L}^*_\Lambda}(\sigma) - \rho\|_1 \leq e^{-\alpha t}\sqrt{\ln(1/\rho_{\min})}$$8 between $$\|e^{t\mathcal{L}^*_\Lambda}(\sigma) - \rho\|_1 \leq e^{-\alpha t}\sqrt{\ln(1/\rho_{\min})}$$9 and $O(\mathrm{polylog}|\Lambda|)$0,

$O(\mathrm{polylog}|\Lambda|)$1

in the completely positive order. Two facts make this derivable. First, at infinite temperature the conditional expectations factorize exactly on arbitrary rectangle unions, $O(\mathrm{polylog}|\Lambda|)$2, since the constituent traces and pinchings commute. Second, the finite-temperature defect between $O(\mathrm{polylog}|\Lambda|)$3 and the composition is governed precisely by the ratio $O(\mathrm{polylog}|\Lambda|)$4, which the DS-condition controls exponentially in $O(\mathrm{polylog}|\Lambda|)$5. Because the intervening states commute with the image of the relevant pinching-composition, the scalar ordering of states lifts to the complete positivity order of maps. Hence the DS-condition implies the strong martingale condition, and the exponential decay rate transfers directly.

From the martingale condition to the MLSI constant

The final step follows the multi-scale scheme of the CSS-code paper. Defining $O(\mathrm{polylog}|\Lambda|)$6 over rectangles of size at most $O(\mathrm{polylog}|\Lambda|)$7, the approximate tensorization of relative entropy induced by the strong martingale condition yields the recursion

$O(\mathrm{polylog}|\Lambda|)$8

valid once $O(\mathrm{polylog}|\Lambda|)$9. Iterating from a fixed scale $O(\mathrm{poly}|\Lambda|)$0 gives $O(\mathrm{poly}|\Lambda|)$1 for all $O(\mathrm{poly}|\Lambda|)$2. The base constant $O(\mathrm{poly}|\Lambda|)$3 is bounded below using Gao–Rouzé's complete entropic inequalities, which reduce the problem to a spectral gap bound on small rectangles; there, the uniform jump-rate lower bound $O(\mathrm{poly}|\Lambda|)$4 and translation invariance reduce the infimum over rectangles to a finite minimum. The combination closes the argument and proves the main theorem.

Limitations and open questions

Several restrictions are acknowledged explicitly. Translation invariance of the bare jump operators is assumed for convenience of the proof and is not fundamental, but it is used. The extension to $O(\mathrm{poly}|\Lambda|)$5-dimensional Abelian doubles is described as mostly notational, but there the DS-condition would hold only up to a potential phase transition rather than at all temperatures. Most significantly, the non-Abelian case remains open even in two dimensions: the argument relies essentially on commutativity of the Gibbs marginals and on the dual-group structure available only for Abelian $O(\mathrm{poly}|\Lambda|)$6, and no tractable expression for the conditional expectations is known without these properties. Whether the MLSI constant can be tracked down to zero temperature, as was possible for the 3D toric code in the CSS setting via the star/plaquette splitting omitted here, is not addressed.

Conclusion

The paper shows that 2D Abelian quantum double models satisfy a modified logarithmic Sobolev inequality with a size-independent constant at every positive temperature, thereby establishing rapid mixing of their Davies dynamics. The proof isolates a reusable pipeline — explicit kernels, pinching formulas for conditional expectations, exact infinite-temperature factorization, and DS-controlled finite-temperature corrections feeding a strong martingale condition — and demonstrates that this pipeline tolerates general ergodic single-edge jump operators. The remaining obstacles, non-Abelian groups and temperature ranges beyond the DS regime in higher dimensions, are clearly delineated by the structure of the argument itself.