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arXiv:2505.24467 · Reports on Progress in Physics

A universal constraint for relaxation rates for quantum Markov generators: complete positivity and beyond*

Dariusz Chruściński, Frederik vom Ende, Gen Kimura, Paolo Muratore-Ginanneschi

Published Sep 1, 2025• 1 citation
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Abstract

Relaxation rates are key characteristics of quantum processes, as they determine how quickly a quantum system thermalizes, equilibrates, decoheres, and dissipates. While they play a crucial role in theoretical analyses, relaxation rates are also often directly accessible through experimental measurements. Recently, it was shown that for quantum processes governed by Markovian semigroups, the relaxation rates satisfy a universal constraint: the maximal rate is upper-bounded by the sum of all rates divided by the dimension of the Hilbert space. This bound, initially conjectured a few years ago, was only recently proven using classical Lyapunov theory. In this work, we present a new, purely algebraic proof of this constraint. Remarkably, our approach is not only more direct but also allows for a natural generalization beyond completely positive semigroups. We show that complete positivity can be relaxed to two-positivity without affecting the validity of the constraint. This reveals that the bound is more subtle than previously understood: two-positivity is necessary, but even when further relaxed to Schwarz maps, a slightly weaker-yet still non-trivial-universal constraint still holds. Finally, we explore the connection between these bounds and the number of steady states in quantum processes, uncovering a deeper structure underlying their behavior.

DOI
10.1088/1361-6633/ae075f

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Relaxation rates are key characteristics of quantum processes, as they determine how quickly a quantum system thermalizes, equilibrates, decoheres, and dissipates. While they play a crucial role in theoretical analyses, relaxation rates are also often directly accessible through experimental measurements. Recently, it was shown that for quantum processes governed by Markovian semigroups, the relaxation rates satisfy a universal constraint: the maximal rate is upper-bounded by the sum of all rates divided by the dimension of the Hilbert space. This bound, initially conjectured a few years ago, was only recently proven using classical Lyapunov theory. In this work, we present a new, purely algebraic proof of this constraint. Remarkably, our approach is not only more direct but also allows for a natural generalization beyond completely positive semigroups. We show that complete positivity can be relaxed to two-positivity without affecting the validity of the constraint. This reveals that the bound is more subtle than previously understood: two-positivity is necessary, but even when further relaxed to Schwarz maps, a slightly weaker-yet still non-trivial-universal constraint still holds. Finally, we explore the connection between these bounds and the number of steady states in quantum processes, uncovering a deeper structure underlying their behavior.
Relaxation rates are key characteristics of quantum processes, as they determine how quickly a quantum system thermalizes, equilibrates, decoheres, and dissipates. While they play a crucial role in theoretical analyses, relaxation rates are also often directly accessible through experimental measurements. Recently, it was shown that for quantum processes governed by Markovian semigroups, the relaxation rates satisfy a universal constraint: the maximal rate is upper-bounded by the sum of all rates divided by the dimension of the Hilbert space. This bound, initially conjectured a few years ago, was only recently proven using classical Lyapunov theory. In this work, we present a new, purely algebraic proof of this constraint. Remarkably, our approach is not only more direct but also allows for a natural generalization beyond completely positive semigroups. We show that complete positivity can be relaxed to two-positivity without affecting the validity of the constraint. This reveals that the bound is more subtle than previously understood: two-positivity is necessary, but even when further relaxed to Schwarz maps, a slightly weaker—yet still non-trivial—universal constraint still holds. Finally, we explore the connection between these bounds and the number of steady states in quantum processes, uncovering a deeper structure underlying their behavior.
Abstract Relaxation rates define an important characteristics of quantum dynamics. They determine how quickly a quantum system interacting with an environment thermalizes, decoheres, and dissipates. Moreover, relaxation rates are also often directly accessible through experimental measurements. An analysis of relaxation rates of Markovian semigroups covariant w.r.t. maximally commuting subgroup of the unitary group is provided. It is shown by a direct proof that these rates satisfy the recently proved a universal constraint that each single rate is upper-bounded by the sum of all rates divided by the dimension of the Hilbert space. Furthermore, we introduce longitudinal and transversal relaxation rates for a general d -level system and derive nontrivial relations between them in the covariant case. These relations generalize the well-known inequality 2Γ T ≥ Γ L between the longitudinal relaxation rate Γ L and the transversal relaxation rate Γ T for two-level systems. Interestingly, for d ≥ 3, our generalization turns out to be much tighter than those valid for general quantum dynamical semigroups, thereby providing a potential means for the experimental verification of covariant generators.
A general property of relaxation rates in open quantum systems is discussed. We find an interesting constraint for relaxation rates that universally holds in fairly large classes of quantum dynamics, e.g., weak coupling regimes, as well as for entropy nondecreasing evolutions. We conjecture that this constraint is universal, i.e., it is valid for all quantum dynamical semigroups. The conjecture is supported by numerical analysis. Moreover, we show that the conjectured constraint is tight by providing a concrete model that saturates the bound. This universality marks an essential step toward the physical characterization of complete positivity as the constraint is directly verifiable in experiments. It provides, therefore, a physical manifestation of complete positivity. Our conjecture also has two important implications: it provides (i) a universal constraint for the spectra of quantum channels and (ii) a necessary condition to decide whether a given channel is consistent with Markovian evolution.
Abstract Relaxation rates provide important characteristics both for classical and quantum processes. Essentially they control how fast the system thermalizes, equilibrates, decoheres, and/or dissipates. Moreover, very often they are directly accessible to be measured in the laboratory and hence they define key physical properties of the system. Experimentally measured relaxation rates can be used to test validity of a particular theoretical model. Here we analyze a fundamental question: does quantum mechanics provide any nontrivial constraint for relaxation rates? We prove the conjecture formulated a few years ago that any quantum channel implies that a maximal rate is bounded from above by the sum of all the relaxation rates divided by the dimension of the Hilbert space. It should be stressed that this constraint is universal (it is valid for all quantum systems with finite number of energy levels) and it is tight (cannot be improved). In addition, the constraint plays an analogous role to the seminal Bell inequalities and the well known Leggett–Garg inequalities (sometimes called temporal Bell inequalities). Violations of Bell inequalities rule out local hidden variable models, and violations of Leggett–Garg inequalities rule out macrorealism. Similarly, violations of the relaxation bound rule out Markovian (meaning CP-divisible) evolution.
Abstract Unital qubit Schwarz maps interpolate between positive and completely positive maps. It is shown that the relaxation rates of the qubit semigroups of unital maps enjoying the Schwarz property satisfy a universal constraint, which provides a modification of the corresponding constraint known for completely positive semigroups. As an illustration, we consider two paradigmatic qubit semigroups: Pauli dynamical maps and phase-covariant dynamics. This result has two interesting implications: it provides a universal constraint for the spectra of qubit Schwarz maps and gives rise to a necessary condition for a Schwarz qubit map to be Markovian.
In 1976, Gorini, Kossakowski, Sudarshan and Lindblad independently discovered a general form of master equations for an open quantum Markovian dynamics. In honor of all the authors, the equation is nowadays called the GKLS master equation. In this paper, we show universal constraints on the relaxation times valid for any d-level GKLS master equations, which is a generalization of the well-known constraints for 2-level systems. Specifically, we show that any relaxation rate, the inverse-relaxation time, is not greater than half of the sum of all relaxation rates. Since the relaxation times are measurable in experiments, our constraints provide a direct experimental test for the validity of the GKLS master equations, and hence for the conditions of the complete positivity and Markovianity.