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arxiv: 2606.28054 · v1 · pith:6D2T64R7new · submitted 2026-06-26 · 🪐 quant-ph · cond-mat.stat-mech· math-ph· math.MP

Static features from mixing in short- and long-range Lindbladians: Markov property and correlations

Paul Rosa-Ruiz , Matteo Scandi , \'Angela Capel , \'Alvaro M. Alhambra This is my paper

Pith reviewed 2026-06-29 04:06 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mechmath-phmath.MP
keywords Lindbladiansrapid mixingmutual informationconditional mutual informationMarkov propertylong-range interactionsfrustration-freenessGibbs states
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The pith

Lindbladians with rapid mixing and frustration-freeness produce fixed points whose conditional mutual information decays with shielding distance, polynomially under long-range power-law interactions.

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

The paper shows that static correlation measures in the steady states of open quantum systems follow from the dynamical properties of their Lindbladian generators. Rapid mixing together with frustration-freeness implies that the conditional mutual information of the fixed point decays according to the shielding distance. When the interactions are long-range and decay as a power law with rate alpha, both the conditional mutual information and the mutual information decay polynomially rather than exponentially. The same dynamical conditions also guarantee a local Markov property for the Gibbs states of long-range non-commuting Hamiltonians at any temperature. These relations are confirmed numerically for the long-range Ising model in regimes where the polynomial bound holds.

Core claim

Local Lindbladians satisfying global rapid mixing and frustration-freeness have fixed points whose conditional mutual information decays with the shielding distance. Local rapid mixing together with primitivity and regularity implies global decay of mutual information. For long-range interactions decaying with power-law rate alpha, both quantities decay polynomially rather than exponentially. Gibbs states of long-range non-commuting Hamiltonians satisfy a local Markov property at any temperature.

What carries the argument

The rapid mixing property of the Lindbladian generator together with frustration-freeness, which together control how quickly the system converges to its fixed point and thereby determine the decay of correlations in that fixed point.

If this is right

  • Fixed points of short-range Lindbladians exhibit exponential decay of conditional mutual information and thus a finite Markov length, while long-range cases exhibit polynomial decay.
  • The classification of mixed-state phases for long-range systems must use polynomial rather than exponential decay of mutual information.
  • Gibbs states of long-range non-commuting Hamiltonians obey a local Markov property at arbitrary temperatures.
  • Numerical checks on the long-range Ising model with and without transverse field confirm the polynomial decay of conditional mutual information in the regimes predicted by the bounds.

Where Pith is reading between the lines

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

  • The polynomial decay bounds suggest that steady states in long-range platforms such as trapped ions or Rydberg arrays may require adjusted scaling for correlation-based diagnostics compared with short-range theory.
  • The link between mixing rates and static correlations could be used to engineer dissipative systems whose steady-state correlation lengths are tuned by control parameters that affect the mixing time.
  • The results indicate that the Markov length concept must be generalized to allow power-law rather than exponential tails when long-range interactions are present.

Load-bearing premise

The Lindbladians are assumed to satisfy rapid mixing (global or local), frustration-freeness, primitivity and regularity, with these dynamical properties taken as given rather than derived from concrete models.

What would settle it

A local Lindbladian that satisfies global rapid mixing and frustration-freeness yet possesses a fixed point whose conditional mutual information fails to decay with shielding distance, or a long-range power-law system whose fixed point shows exponential rather than polynomial decay of the conditional mutual information.

Figures

Figures reproduced from arXiv: 2606.28054 by \'Alvaro M. Alhambra, \'Angela Capel, Matteo Scandi, Paul Rosa-Ruiz.

Figure 1.1
Figure 1.1. Figure 1.1: Geometry of the tripartition A ⊔ B ⊔ C = Λ. makes the system behave akin to a short-range system (for which an exponential decay of the CMI is expected [45]). The details of the numerical simulations are in Section 4.4. 1.1 Summary of Main results First of all, consider a general dissipative process generated by a local Lindbladian L = X Z⊂Λ LZ, ∥LZ∥cb ≲ F(diam(Z)), for an F-function F (see Definition 2.… view at source ↗
Figure 3.1
Figure 3.1. Figure 3.1: Geometry considered for the MI: A and C and their frontier ∂AC Theorem 3.7. Let (Γ, d) be a countable metric space equipped with the F-function Fα(r) := 1 (1 + r) α , for r ≥ 0 and α > 3D − 1. Let L = {LZ}Z∈P0(Γ) be a local and primitive dissipative interaction with ∥L∥F < ∞, satisfying local rapid mixing. Fix A, C ∈ P0(Γ) with A ∩ C = ∅ (see [PITH_FULL_IMAGE:figures/full_fig_p024_3_1.png] view at source ↗
Figure 4.1
Figure 4.1. Figure 4.1: Decay exponent q(α) of the conditional mutual information at β = 5 for different system sizes N. The dashed line indicates the linear fit for N = 18 (α ≥ 2). whose Gibbs state is diagonal in the computational basis and therefore identical to the classical distribution above. For each interaction exponent α and system size N, we compute numerically the conditional mutual information I(A : C|Bℓ) as a funct… view at source ↗
Figure 4.2
Figure 4.2. Figure 4.2: CMI decay I(A : C|Bℓ) as a function of buffer size ℓ for selected values of α and transverse field h, at inverse temperature β = 5, N = 10. Power-law (PL) and exponential (EXP) fits are shown; R2 values quantify the relative quality of each ansatz. The running effective exponent qeff(ℓ) is shown in the legend [PITH_FULL_IMAGE:figures/full_fig_p041_4_2.png] view at source ↗
read the original abstract

The classification of mixed-state phases requires criteria beyond two-point correlation functions, such as the decay of the mutual information (MI) and the conditional mutual information (CMI), with the latter encapsulated in the notion of Markov length. Here we show how such static properties of the fixed point of a Lindbladian follow from natural dynamical features of its generators: rapid mixing and frustration-freeness. We focus on systems with long-range interactions, and prove (i) that local Lindbladians satisfying (global) rapid mixing and frustration-freeness have fixed-points whose CMI decays with the shielding distance, and (ii) that (local) rapid mixing together with primitivity and regularity implies global decay of MI. For long-range interactions decaying with a power law with rate $\alpha$, both quantities decay polynomially rather than exponentially, in contrast to the finite- and short-range regimes where exponential decay (a finite Markov length) is expected within a phase. We further show that Gibbs states of long-range, non-commuting Hamiltonians satisfy a local Markov property at any temperatures, extending the recent results (Chen--Rouz\'e, 2025) for short-range systems to the long-range regime relevant to a variety of experimental platforms. As a numerical example, we study the long-range Ising model both with and without a transverse field. We find regimes in which the polynomial decay of the CMI holds, in accordance with the bounds proven.

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 / 3 minor

Summary. The paper proves that local Lindbladians satisfying global rapid mixing and frustration-freeness have fixed points whose conditional mutual information (CMI) decays with shielding distance, and that local rapid mixing together with primitivity and regularity implies global decay of mutual information (MI). For long-range power-law interactions with exponent α, both quantities decay polynomially (rather than exponentially). It further shows that Gibbs states of long-range non-commuting Hamiltonians satisfy a local Markov property at any temperature, extending prior short-range results, and provides numerical support via the long-range Ising model (with and without transverse field) in regimes where the polynomial CMI decay is observed.

Significance. If the derivations hold, the work supplies a direct implication from standard dynamical assumptions (rapid mixing, frustration-freeness, primitivity, regularity) to static correlation decay, including the first extension of the local Markov property to long-range Gibbs states. This is relevant for mixed-state phase classification and for experimental platforms with power-law interactions. The conditional framing of all claims on the dynamical premises is explicit, and the implication structure shows no circularity between the fitted quantities and the decay statements. The polynomial-versus-exponential distinction for long-range cases is a clear, falsifiable prediction.

minor comments (3)
  1. [Numerical example (Ising model)] The abstract notes post-hoc regime selection for the Ising numerics; the main text should state the precise selection criterion and confirm that the chosen parameter windows lie inside the regime where the proven bounds apply (e.g., by referencing the relevant theorem on polynomial decay).
  2. [Preliminaries / Theorem statements] Notation for the shielding distance and the precise definition of the local Markov property should be introduced once in a dedicated preliminary section rather than re-defined inline in each theorem statement.
  3. [Introduction / Discussion] A short table or paragraph comparing the decay rates obtained here with the corresponding short-range results (Chen–Rouzé 2025) would help readers see the precise extension achieved for long-range interactions.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading, positive summary, and recommendation of minor revision. The report raises no specific major comments or criticisms, so we have no individual points requiring detailed rebuttal or clarification. We appreciate the recognition of the paper's contributions to linking dynamical assumptions with static correlation properties, including the extension to long-range systems.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper derives decay of CMI and MI in the fixed point from independent dynamical assumptions (rapid mixing, frustration-freeness, primitivity, regularity) that are taken as given inputs. The implication direction is dynamics to static properties, with no equations or steps that reduce the claimed results to the inputs by definition, no fitted parameters renamed as predictions, and no load-bearing self-citations. The extension to long-range interactions and the numerical Ising example follow the same non-circular structure. The derivation is self-contained against the stated premises.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract only; no explicit free parameters or invented entities are stated. The central claims rest on domain assumptions about the Lindbladian generators.

axioms (1)
  • domain assumption Lindbladian generators are local and satisfy rapid mixing (global or local), frustration-freeness, primitivity and regularity
    These properties are invoked as the starting point for the decay theorems and the Markov-property extension.

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discussion (0)

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