Technical Summary: "Is rapid mixing stable?"

Problem Statement

The paper investigates the stability of rapid mixing in open quantum systems governed by Lindbladian dynamics. Rapid mixing is defined as a relaxation time to a steady state that scales at most polylogarithmically with the system size ($N$). This property is crucial for efficient dissipative state preparation, thermalization, and defining mixed-state phases via local dissipative equivalence.

While it is physically intuitive that local perturbations (e.g., implementation errors or noise) should not drastically alter mixing times—especially if the perturbation is weak and local—the authors question whether rapid mixing is inherently robust. Specifically, they ask: If a Lindbladian $L$ is rapidly mixing, is $L + K$ (where $K$ is a local perturbation) also rapidly mixing?

The authors distinguish between:

1. Rapid mixing: Mixing time $\tau_{mix} = O((\log N)^q)$.
2. Rapid mixing with constant decay rate: Trace-norm distance decays as $C_N e^{-\gamma t}$ where $C_N = O(N^c)$ and $\gamma$ is independent of $N$.
3. Uniform rapid mixing: Rapid mixing holds for restrictions to local lattice regions.

Methodology

The authors employ a dual approach combining counterexample construction and the derivation of sufficient stability conditions.

1. Counterexamples (Instability Results)

To demonstrate that rapid mixing is not generally stable, the authors construct specific families of Lindbladians where adding a local perturbation (even one that preserves the steady state or is arbitrarily weak) causes the mixing time to scale polynomially or exponentially with system size.

• Mechanisms explored:
  • Hamiltonian Cancellation: Two rapidly mixing generators with the same steady state can sum to a generator where order-one Hamiltonian rotations cancel, exposing a slow mode with a gap closing as $O(N^{-2})$.
  • Translation Invariance: Even with translation-invariant, fixed-coefficient fermionic generators, adding a second rapidly mixing generator can cancel local damping terms, leaving a long-wavelength mode with decay rate $O(N^{-2})$.
  • Dissipation-Only Systems: Two purely dissipative, translation-invariant generators with unique steady states can sum to a system with a degenerate steady-state subspace and mixing time scaling as $O(N)$.
  • Arbitrarily Weak Perturbations: A localized mode in a boundary-driven chain can be stabilized by an arbitrarily weak local potential, creating a bottleneck where relaxation requires exponential time in the physical chain length (which is logarithmic in the auxiliary parameter $N$), resulting in polynomial mixing time in $N$.
  • Low-Temperature Ising Model: Summing two rapidly mixing heat-bath dynamics (acting on checkerboard sublattices) can lead to mixing times scaling as $e^{\Omega(\sqrt{N})}$ due to free-energy barriers between magnetization phases.

2. Stability Criteria (Positive Results)

The authors identify specific structural conditions under which rapid mixing is stable under local perturbations. These rely on:

• Commutativity: If generators commute, mixing bounds are inherited.
• Strong Bulk Dissipation: If local regions are covered by dissipative terms that erase local degrees of freedom faster than interactions can reintroduce fluctuations (quantified by an "influence condition" $\kappa t_{loc} < 1/2$).
• Quasifree Fermionic Dynamics: Stability is guaranteed if the single-particle drift matrix has a spectral gap or if the dissipative part (matrix $M$) is uniformly positive definite.
• Functional Inequalities: Stability is proven for systems satisfying Modified Logarithmic Sobolev Inequalities (MLSI) or $\chi^2$-divergence contraction, provided the perturbation preserves the steady state or commutes with the stationary projection.

Key Contributions and Results

1. Negative Results: Instability of Rapid Mixing

The paper establishes that rapid mixing is fragile and does not imply stability under local perturbations, even under strong assumptions:

• Non-additivity: The sum of two rapidly mixing generators is not necessarily rapidly mixing (Lemma 3.2, 3.4, 3.6).
• Failure of Uniqueness: Even if both summands have a unique, full-rank steady state, their sum may not mix rapidly.
• Failure of MLSI: Rapid mixing with a constant decay rate does not imply a positive MLSI constant (Lemma 3.12). Conversely, MLSI alone does not guarantee stability under local perturbations (Example A.4).
• Arbitrarily Weak Perturbations: An arbitrarily weak local Hamiltonian can destroy rapid mixing, turning a polylogarithmic mixing time into a polynomial one (Lemma 3.10). This occurs via the localization of a mode away from boundary dissipation.

2. Positive Results: Sufficient Conditions for Stability

The paper provides rigorous criteria for when rapid mixing is stable:

• Commuting Generators: If $[L, K] = 0$, the mixing time of $L+K$ is bounded by the maximum of the individual mixing times (Proposition 4.3).
• Strong Bulk Dissipation: If the system admits a "bulk-dissipative cover" where local terms relax faster than non-commuting interactions can perturb them, rapid mixing is stable under small local perturbations (Proposition 4.4). This applies to high-temperature Gibbs samplers and injective Matrix Product Density Operator (MPDO) parent dynamics.
• Quasifree Fermions:
  • If the dissipative matrix $M$ satisfies $\text{Re}(M) \succeq \mu I$ uniformly, the system is rapidly mixing and stable under arbitrary quasifree additions (Proposition 4.10).
  • For translation-invariant systems, a uniform spectral gap in the single-particle sector implies rapid mixing and stability under weak perturbations (Proposition 4.11).
• Functional Inequalities:
  • MLSI: If stationary projections commute or if the perturbation's kernel is contained within the unperturbed kernel, the MLSI constant of the sum is bounded from below (Propositions 4.13, 4.14).
  • $\chi^2$ Divergence: If a common faithful stationary state exists, the $\chi^2$ contraction rate is additive, ensuring stability for fast mixing with constant decay rate (Proposition 4.15).

Significance and Claims

The paper claims that robustness of rapid mixing is not a generic property of local Lindbladians. The favorable time scaling required for efficient dissipative state preparation and the dynamical definition of mixed-state phases can be destroyed by local implementation errors unless specific structural conditions are met.

• Motivational Context: The work addresses a gap in the literature where Hamiltonian gap stability is well-understood, but the stability of mixing times in open systems was an open question.
• Practical Implication: For experimental implementations of dissipative state preparation (e.g., for thermal states or tensor network states), one cannot assume that small local errors will preserve the polylogarithmic scaling. One must verify specific structural properties (like bulk dissipation or commutativity) to guarantee stability.
• Theoretical Contribution: The authors provide a comprehensive map of when rapid mixing is stable, distinguishing between "fragile" rapid mixing (which relies on fine-tuned cancellations or specific boundary conditions) and "robust" rapid mixing (guaranteed by strong local dissipation or functional inequalities).

The paper concludes by posing open questions, such as whether rapid mixing is generically stable (i.e., if the counterexamples require fine-tuning) and whether uniform rapid mixing implies a size-independent positive MLSI constant. It explicitly states that the boundary-driven counterexamples do not yet settle the question of stability under uniform rapid mixing, leaving that as a direction for future research.