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 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.
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 We study Leggett–Garg inequalities (LGIs) for a two level system (TLS) undergoing Markovian dynamics described by unital maps. We first find the analytic expression of LG parameter K 3 (simplest variant of LGIs) in terms of the parameters of two distinct unital maps representing time evolution for intervals: t 1 to t 2 and t 2 to t 3. We then show that the maximum violation of LGI for all possible unital maps can never exceed well known Lüders bound of K 3 L u ¨ d e r s = 3 / 2 over the full parameter space. We further show that if the map for the time interval t 1 to t 2 is non-unitary unital then irrespective of the choice of the map for interval t 2 to t 3 we can never reach Lüders bound. On the other hand, if the measurement operator eigenstates remain pure upon evolution from t 1 to t 2, then depending on the degree of decoherence induced by the unital map for the interval t 2 to t 3 we may or may not obtain Lüders bound. Specifically, we find that if the unital map for interval t 2 to t 3 leads to the shrinking of the Bloch vector beyond half of its unit length, then achieving the bound K 3 L u ¨ d e r s is not possible. Hence our findings not only establish a threshold for decoherence which will allow for K 3 = K 3 L u ¨ d e r s , but also demonstrate the importance of temporal sequencing of the exposure of a TLS to Markovian baths in obtaining Lüders bound.
Macrorealistic description governing the classical realm is known to get uprooted in the quantum domain. The Leggett-Garg inequality [A. J. Leggett and A. Garg, Phys. Rev. Lett. 54, 857 (1985)] proposes to probe the macrorealistic limit emerging from the quantum scenario. It places a bound on the linear combinations of temporal correlations of observables measured sequentially at different time intervals. Violation of the Leggett-Garg inequality, up to the so-called temporal Tsirelson bound (TTB), has been realized in the quantum domain. It is important to understand whether quantum theory permits larger violation beyond the TTB and up to the algebraic maximum value of the Leggett-Garg inequality. To this end, it has been shown [C. Budroni and C. Emary, Phys. Rev. Lett. 113, 050401 (2014)] that a three-term Leggett-Garg inequality can be violated beyond the TTB 3/2 in the time evolution of N>2 level quantum systems by choosing suitable measurement schemes and state-update rules. Furthermore, it is revealed that the algebraic maximum value 3 can be realized in the limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$. Their measurement scheme rules out violation of the Leggett-Garg inequality beyond the TTB by a qubit. Being genuinely quantum in nature, we ask whether such enhanced violation of the Leggett-Garg inequality could be witnessed in a qubit itself. Here, we show that a qubit undergoing time evolution generated by a non-Hermitian PT-symmetric Hamiltonian indeed violates the three-term Leggett-Garg inequality beyond the TTB and up to its algebraic maximum value 3.