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Journal of Physics A Mathematical and Theoretical

Corralling spectra of generators of time-continuous Markov evolution (both classical and quantum)

Dariusz Chruściński, Sergey Denisov, Wojciech Tarnowski, Karol Życzkowski

Published Mar 30, 2026• 0 citations
Publisher

Abstract

Abstract The spectral properties of operators generating Markov evolution provide essential information about processes induced by these operators. The celebrated theorem of Perron and Frobenius implies that spectra of classical discrete-time Markov operators, represented by stochastic matrices, are restricted to the unit disk. This property holds also for spectra of quantum stochastic maps (quantum channels), which describe quantum Markovian evolution in discrete time. Moreover, the spectra of stochastic N × N matrices are additionally restricted to a subset of the unit disk, called Karpelevi c ˘ region, the shape of which depends onN. We pose the question of whether the spectra of generators that induce continuous-time Markovian evolution can be corralled in a similar way. We propose renormalization procedures that allow us to answer the question affirmatively. The eigenvalues of the renormalized classical Markov generators are confined to modified Karpelevi c ˘ regions, whereas the eigenvalues of the renormalized quantum Markov generators are confined to the unit disk. We discuss spectral corralling for different symmetry classes of the generators and also make a few steps beyond the conventional assumptions of complete positivity and strict Markovianity. Our approach enables direct comparisons of the spectra of Markov generators arising from diverse fields, spanning multiple timescales, parameter ranges, and dimensionalities.

DOI
10.1088/1751-8121/ae58d2

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Abstract The spectral properties of operators generating Markov evolution provide essential information about processes induced by these operators. The celebrated theorem of Perron and Frobenius implies that spectra of classical discrete-time Markov operators, represented by stochastic matrices, are restricted to the unit disk. This property holds also for spectra of quantum stochastic maps (quantum channels), which describe quantum Markovian evolution in discrete time. Moreover, the spectra of stochastic N × N matrices are additionally restricted to a subset of the unit disk, called Karpelevi c ˘ region, the shape of which depends onN. We pose the question of whether the spectra of generators that induce continuous-time Markovian evolution can be corralled in a similar way. We propose renormalization procedures that allow us to answer the question affirmatively. The eigenvalues of the renormalized classical Markov generators are confined to modified Karpelevi c ˘ regions, whereas the eigenvalues of the renormalized quantum Markov generators are confined to the unit disk. We discuss spectral corralling for different symmetry classes of the generators and also make a few steps beyond the conventional assumptions of complete positivity and strict Markovianity. Our approach enables direct comparisons of the spectra of Markov generators arising from diverse fields, spanning multiple timescales, parameter ranges, and dimensionalities.
Continuous-time Markovian evolution appears to be manifestly different in classical and quantum worlds. We consider ensembles of random generators of N-dimensional Markovian evolution, quantum and classical ones, and evaluate their universal spectral properties. We then show how the two types of generators can be related by superdecoherence. In analogy with the mechanism of decoherence, which transforms a quantum state into a classical one, superdecoherence can be used to transform a Lindblad operator (generator of quantum evolution) into a Kolmogorov operator (generator of classical evolution). We inspect spectra of random Lindblad operators undergoing superdecoherence and demonstrate that, in the limit of complete superdecoherence, the resulting operators exhibit spectral density typical to random Kolmogorov operators. By gradually increasing strength of superdecoherence, we observe a sharp quantum-to-classical transition. Furthermore, we define an inverse procedure of supercoherification that is a generalization of the scheme used to construct a quantum state out of a classical one. Finally, we study microscopic correlation between neighboring eigenvalues through the complex spacing ratios and observe the horseshoe distribution, emblematic of the Ginibre universality class, for both types of random generators. Remarkably, it survives both superdecoherence and supercoherification.
The classical embeddability problem asks whether a given stochastic matrix T , describing transition probabilities of a d -level system, can arise from the underlying homogeneous continuous-time Markov process. Here, we investigate the quantum version of this problem, asking of the existence of a Markovian quantum channel generating state transitions described by a given T . More precisely, we aim at characterising the set of quantum-embeddable stochastic matrices that arise from memoryless continuous-time quantum evolution. To this end, we derive both upper and lower bounds on that set, providing new families of stochastic matrices that are quantum-embeddable but not classically-embeddable, as well as families of stochastic matrices that are not quantum-embeddable. As a result, we demonstrate that a larger set of transition matrices can be explained by memoryless models if the dynamics is allowed to be quantum, but we also identify a non-zero measure set of random processes that cannot be explained by either classical or quantum memoryless dynamics. Finally, we fully characterise extreme stochastic matrices (with entries given only by zeros and ones) that are quantum-embeddable.
Similar to its classical version, quantum Markovian evolution can be either time-discrete or time-continuous. Discrete quantum Markovian evolution is usually modeled with completely positive trace-preserving maps, while time-continuous evolution is often specified with superoperators referred to as "Lindbladians." Here, we address the following question: Being given a quantum map, can we find a Lindbladian that generates an evolution identical-when monitored at discrete instances of time-to the one induced by the map? It was demonstrated that the problem of getting the answer to this question can be reduced to an NP-complete (in the dimension N of the Hilbert space, the evolution takes place in) problem. We approach this question from a different perspective by considering a variety of machine learning (ML) methods and trying to estimate their potential ability to give the correct answer. Complimentarily, we use the performance of different ML methods as a tool to validate a hypothesis that the answer to the question is encoded in spectral properties of the so-called Choi matrix, which can be constructed from the given quantum map. As a test bed, we use two single-qubit models for which the answer can be obtained using the reduction procedure. The outcome of our experiment is that, for a given map, the property of being generated by a time-independent Lindbladian is encoded both in the eigenvalues and the eigenstates of the corresponding Choi matrix.
Motivated by a model presented by S. Gudder, we study a quantum generalization of Markov chains and discuss the relation between these maps and open quantum random walks, a class of quantum channels described by S. Attal et al. We consider processes which are nonhomogeneous in time, i.e., at each time step, a possibly distinct evolution kernel. Inspired by a spectral technique described by L. Saloff-Coste and J. Zúñiga, we define a notion of ergodicity for nonhomogeneous quantum Markov chains and describe a criterion for ergodicity of such objects in terms of singular values. As a consequence we obtain a quantum version of the classical probability result concerning the behavior of the columns (or rows) of the iterates of a stochastic matrix induced by a finite, irreducible, aperiodic Markov chain. We are also able to relate the ergodic property presented here with the notions of weak and uniform ergodicity known in the literature of noncommutative L1L^1L1-spaces.
Abstract We reformulate relaxation constraints for open quantum dynamics in a discrete interaction-based framework, where evolution is indexed by successive system–environment transactions rather than by continuous time. We distinguish two levels of characterisation: mode contraction factors of the one-step propagator and the channel displacement from identity. In the homogeneous semigroup-embeddable completely positive-divisible regime, the Markovian relaxation-rate bound is recovered exactly from the one-step propagator eigenvalues, which makes explicit the structural assumptions behind its validity. Beyond this regime, the appropriate constraints become path-dependent and memory-sensitive, so that a single spectral speed limit no longer characterises the overall relaxation structure. We illustrate these findings with minimal qubit models.
Gaussian quantum Markov semigroups are the natural non-commutative extension of classical Ornstein-Uhlenbeck semigroups. They arise in open quantum systems of bosons where canonical non-commuting random variables of positions and momenta come into play. If there exists a faithful invariant density we explicitly compute the optimal exponential convergence rate, namely the spectral gap of the generator, in non-commutative L 2 spaces determined by the invariant density showing that the exact value is the lowest eigenvalue of a certain matrix determined by the diffusion and drift matrices. The spectral gap turns out to depend on the non-commutative L 2 space considered, whether the one determined by the so-called GNS or KMS multiplication by the square root of the invariant density. In the first case, it is strictly positive if and only if there is the maximum number of linearly independent noises. While, we exhibit explicit examples in which it is strictly positive only with KMS multiplication. We do not assume any symmetry or quantum detailed balance condition with respect to the invariant density.