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Exponential Stability of Dynamic Petz Recovery in Primitive Quantum Markov Semigroups

Roberto Isai Crotone

Published Apr 30, 2026• 0 citations
Publisher

Abstract

This document establishes a non-asymptotic exponential bound for the reconstruction error induced by a time-dependent Petz recovery map in primitive quantum Markov semigroups (QMS). Specifically, the decay rate is governed by the spectral gap of the Lindbladian, providing a rigorous quantitative relation between dissipative mixing and the recoverability of quantum information.

DOI
10.5281/zenodo.19915217

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This document establishes a non-asymptotic exponential bound for the reconstruction error induced by a time-dependent Petz recovery map in primitive quantum Markov semigroups (QMS). Specifically, the decay rate is governed by the spectral gap of the Lindbladian, providing a rigorous quantitative relation between dissipative mixing and the recoverability of quantum information.
We investigate the mixing properties of primitive Markovian Lindblad dynamics (i.e., quantum Markov semigroups), where the detailed balance is disrupted by a coherent drift term. It is known that the sharp L2L^2L2-exponential convergence rate of Lindblad dynamics is determined by the spectral gap of the generator. We show that incorporating a Hamiltonian component into a detailed balanced Lindbladian can generically enhance its spectral gap, thereby accelerating the mixing. In addition, we analyze the asymptotic behavior of the spectral gap for Lindblad dynamics with a large coherent contribution. However, estimating the spectral gap, particularly for a non-detailed balanced Lindbladian, presents a significant challenge. In the case of hypocoercive Lindblad dynamics, we extend the variational framework originally developed for underdamped Langevin dynamics to derive fully explicit and constructive exponential decay estimates for convergence in the noncommutative L2L^2L2-norm. This analysis relies on establishing a quantum analog of space-time Poincaré inequality. Furthermore, we provide several examples with connections to quantum noise and quantum Gibbs samplers as applications of our theoretical results.
We establish a relation between the exponential decay rates of quantum Markov semigroups with respect to different inner products. More precisely, it was conjectured by Fagnola, Poletti, Sasso and Umanità that for a Gaussian quantum Markov semigroup, the exponential decay rate with respect to the KMS inner product is bounded below by the exponential decay rate for the GNS inner product. We show that this is indeed the case and not limited to Gaussian quantum Markov semigroups, but holds for quantum Markov semigroups with a faithful normal invariant state on arbitrary von Neumann algebras. Additionally, the KMS inner product can be replaced by a whole class of inner products induced by operator monotone functions.
An important issue in developing quantum technology is that quantum states are so sensitive to noise. We propose a protocol that introduces reverse dynamics, in order to precisely control quantum systems against noise described by the Lindblad master equation. The reverse dynamics can be obtained by constructing the Petz recovery map in continuous time. By providing the exact form of the Hamiltonian and jump operators for the reverse dynamics, we explore the potential of utilizing the near-optimal recovery of the Petz map in controlling noisy quantum dynamics. While time-dependent dissipation engineering enables us to fully recover a single quantum trajectory, we also design a time-independent recovery protocol to protect encoded quantum information against decoherence. Our protocol can efficiently suppress only the noise part of dynamics thereby providing an effective unitary evolution of the quantum system.
We introduce the notion of the BKM coercivity constant and show that the positivity of this constant governs the exponential relative entropy decay of the semigroup. Indeed, we prove that the modified log-Sobolev constant is equivalent to the BKM coercivity constant up to a constant depending on the asymptotic conditional expectation. We also discover that the BKM coercivity constants for matrix amplifications are already attained with a qubit auxiliary system, which also coincides with the GNS spectral gap of the generating Lindbladian. As a corollary, we establish a criterion that the complete modified log-Sobolev inequality holds if and only if the GNS gap of the semigroup is strictly positive. All the discussion above applies to quantum Markov semigroups that admit a faithful asymptotic conditional expectation as long-time equilibration, with no detailed balance condition assumed. As an application, we show the finite-dimensional Chen-Kastoryano-Gilyén Gibbs samplers always satisfy the complete modified log-Sobolev inequality.
Open quantum systems weakly coupled to the environment are modeled by completely positive, trace preserving semigroups of linear maps. The generators of such evolutions are called Lindbladians. In the setting of quantum many-body systems on a lattice it is natural to consider Lindbladians that decompose into a sum of local interactions with decreasing strength with respect to the size of their support. For both practical and theoretical reasons, it is crucial to estimate the impact that perturbations in the generating Lindbladian, arising as noise or errors, can have on the evolution. These local perturbations are potentially unbounded, but constrained to respect the underlying lattice structure. We show that even for polynomially decaying errors in the Lindbladian, local observables and correlation functions are stable if the unperturbed Lindbladian has a unique fixed point and a mixing time which scales logarithmically with the system size. The proof relies on Lieb-Robinson bounds, which describe a finite group velocity for propagation of information in local systems. As a main example, we prove that classical Glauber dynamics is stable under local perturbations, including perturbations in the transition rates which may not preserve detailed balance.