Stability of mixed-state phases under weak decoherence

TL;DR

The paper proves that the Gibbs states of local, commuting-Pauli Hamiltonians are stable under weak local decoherence by showing the associated Markov length remains finite and that local recovery channels exist. The authors develop a three-tier approach—classical single-site noise, classical finite-depth noise, and quantum commuting-Pauli noise with stabilizer-mixing channels—tying stability to a finite Markov length via pinning lemmas and cluster expansions. A key mechanism is the reduction of the quantum stabilizer problem to an equivalent classical stabilizer distribution, which inherits exact Markov properties under suitable noise, thereby enabling local reversibility of decoherence. These results imply nonzero decoherence thresholds for thermally stable quantum memories near critical temperatures and suggest efficient local denoisers in diffusion-model dynamics, with broad implications for both quantum information and generative modeling. The framework clarifies when mixed-state phases remain locally reversible, even in the presence of long-range correlations, by providing explicit local decoders and quantitative bounds on recovery.

Abstract

We prove that the Gibbs states of classical, and commuting-Pauli, Hamiltonians are stable under weak local decoherence: i.e., we show that the effect of the decoherence can be locally reversed. In particular, our conclusions apply to finite-temperature equilibrium critical points and ordered low-temperature phases. In these systems the unconditional spatio-temporal correlations are long-range, and local (e.g., Metropolis) dynamics exhibits critical slowing down. Nevertheless, our results imply the existence of local "decoders" that undo the decoherence, when the decoherence strength is below a critical value. An implication of these results is that thermally stable quantum memories have a threshold against decoherence that remains nonzero as one approaches the critical temperature. Analogously, in diffusion models, stability of data distributions implies the existence of computationally-efficent local denoisers in the late-time generation dynamics.

Figures (3)

• Figure 1: (a) A phase diagram of a low-temperature or critical Gibbs state $e^{-\beta H}$ under local perturbations $\mathcal{E}$ with strength $\epsilon$. $\mathcal{R}$ is the recovery map we construct and $\mathcal{F}$ is the Gibbs sampler which could exhibit slowdown behavior. (b) An annulus-shaped tripartition $\textcolor{Cerulean}{A}\textcolor{Green}{B}\textcolor{Goldenrod}{C}$ used to define the Markov length. $\textcolor{Cerulean}{A}$ is a single qubit, $\textcolor{Green}{B}$ surrounds $\textcolor{Cerulean}{A}$ with a radius $d_{AC}$, and $\textcolor{Goldenrod}{C}$ is the rest of the system. (c) a local recovery channel $\mathcal{R}_{s,t}$ acting on $\textcolor{Cerulean}{A}\textcolor{Green}{B}$ that reverses the effect of $\mathcal{E}_{s,t}$ supported on $\textcolor{Cerulean}{A}$.
• Figure 2: (a) Schematic phase diagram for thermally stable quantum memories, as a function of temperature and noise strength. A mixed state with parameters $(T,p)$ is prepared by starting from a Gibbs state of temperature $T$ and applying noise of strength $p$ to every qudit. For noise below the information-theoretic threshold, quantum information can be recovered by optimal decoding. At the information-theoretic threshold, the Markov length diverges. Passive decoders such as heat-bath dynamics are suboptimal, so their thresholds have no information-theoretic significance. (b) Schematics of the diffusion model and its relation to mixed-state phases. The generation process has to violate locality at some point ($\mathcal{R}_{\rm{global}}$) but near the data distribution the generation dynamics can be local ($\mathcal{R}_{\rm{local}}$).
• Figure 3: (a) A one-dimensional Gibbs distribution $P(A_1 B_1 B_2 C_1)$ of four bits in a line and nearest-neighbor interactions. We partition the four bits into $\textcolor{Cerulean}{A_1}$, $\textcolor{Green}{B_1B_2}$, and $\textcolor{Goldenrod}{C_1}$. (b) After applying a product of local stochastic processes on each bit, we obtain a joint distribution $P(A_1' B_1' B_2' C_1' A_1 B_1 B_2 C_1)$. (c) Conditioned on post-selecting $b_1' b_2'$ on $B$, the conditional distribution $P(A_1 B_1 B_2 C_1 | b_1' b_2')$. (d) The conditional distribution $P(A_1 B_1 B_2 C_1 | b_1' b_2')$ is a Gibbs distribution with pinning terms. (e) Identifying the post-selected transition matrices as pinning terms. (f) blocking spins on the same site but in different time slices to form super-spins.

Theorems & Definitions (54)

• Definition 1
• Theorem 2
• Definition 3
• Definition 4
• Theorem 5
• Theorem 6
• Proposition 7
• Proposition 8
• Lemma 9: Pinning Lemma
• proof