Complete positivity, finite-temperature effects, and additivity of noise for time-local qubit dynamics

Lankinen, Juho; Lyyra, Henri; Sokolov, Boris; Teittinen, Jose; Ziaei, Babak; Maniscalco, Sabrina

doi:10.1103/physreva.93.052103

Cited by 25 publications

(26 citation statements)

References 57 publications

(74 reference statements)

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“…Figure 5 shows the initial state dependence of QSL for the eternal non-Markovianity model. Since the master equation is in the Lindblad form and γ 3 (t) can have negative values, we know that this dynamics is not CP-divisible, but is still CPTP according to the results of [32]. The condition for optimal evolution from equation (22) for this system is where e 2 (t) is a non-zero time-dependent, but not a dependent function.…”

Section: Eternal Non-markovianitymentioning

confidence: 93%

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There is no general connection between the quantum speed limit and non-Markovianity

2019

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The quantum speed limit (QSL) sets a bound on the minimum time required for a quantum system to evolve between two states. For open quantum systems this quantity depends on the dynamical map describing the time evolution in presence of the environment, on the evolution time τ, and on the initial state of the system. We consider a general single qubit open dynamics and show that there is no simple relationship between memory effects and the tightness of the QSL bound. We prove that only for specific classes of dynamical evolutions and initial states, there exists a link between non-Markovianity and the QSL. Our results shed light on the connection between information back-flow between system and environment and the speed of quantum evolution. than 1 when it is non-Markovian. Their result suggests that in the Markovian case the dynamics saturates the bound, giving the most efficient evolution, whereas in the non-Markovian case the actual limit can still be lower than the evolution time. The explicitly derived dependency between QSL and non-Markovianity has proven useful in several applications [10,[15][16][17][18][19][20][21][22][23][24][25][26][27][28].Our main goal is to tackle the question of the connection between non-Markovianity and the QSL not starting from a specific model but in full generality, looking in detail at the role played by the dynamical map, the evolution time τ, and the initial state, in the achievement of the QSL bound. We show that, for the most general cases, there is no simple connection between the Markovian to non-Markovian crossover and the QSL. Under certain more restrictive assumptions, however, we can characterize families of one-qubit dynamical maps for which the QSL speed-up coincides with the onset of non-Markovianity, as indicated by the Breuer-Laine-Piilo (BLP) non-Markovianity measure [13]. For these families we derive analytical formulas for the QSL as a function of the BLP measure. Our results also show that, for a given open quantum system model, both the evolution time τ and the initial state play a key role and cannot be overlooked when making claims on the QSL. As an example, we generalize results in [4] to a broader set of pure initial states, and show that the QSL bound is saturated only for very few initial states even in the fully Markovian case.The paper is structured as follows. In section 2 we briefly present the formalism of open quantum systems and recall the common mathematical definitions of QSL. In section 3 we present the Jaynes-Cummings (JC) model used in [4] and discuss briefly their results concerning non-Markovianity and quantum speed-up. In section 4 we study how the actual evolution time affects the QSL for the same JC system. In section 5, we calculate the general conditions for the QSL optimal dynamics, and study the connection between BLP non-Markovianity and QSL. In section 6 we study the initial state dependence of the QSL for the Markovian dynamics arising from Pauli and phase-covariant master equations. In section 7 we study the effects of Markov...

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Section: Eternal Non-markovianitymentioning

confidence: 93%

“…The example dynamics considered in this paper arise from two very general families of master equations, namely the phase-covariant master equation [32][33][34][35][36]: 3 1 2 3 3 3 and the Pauli master equation [37,38]:…”

Section: Qsl Non-markovianity and Open Quantum Systemsmentioning

confidence: 99%

There is no general connection between the quantum speed limit and non-Markovianity

2019

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“…, zz and ( ) H t 11 LS (see, e.g., [44,82]). Crucially, we see how the secular master equation in equation (38) precisely corresponds to the most general form of a master equation associated with a PC qubit dynamics recalled in section 3, see equation (21).…”

Section: Secular Approximationmentioning

confidence: 99%

Fundamental limits to frequency estimation: a comprehensive microscopic perspective

Haase¹,

Smirne²,

Kołodyński³

et al. 2018

New J. Phys.

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We consider a metrology scenario in which qubit-like probes are used to sense an external field that affects their energy splitting in a linear fashion. Following the frequency estimation approach in which one optimizes the state and sensing time of the probes to maximize the sensitivity, we provide a systematic study of the attainable precision under the impact of noise originating from independent bosonic baths. Specifically, we invoke an explicit microscopic derivation of the probe dynamics using the spin-boson model with weak coupling of arbitrary geometry. We clarify how the secular approximation leads to a phase-covariant (PC) dynamics, where the noise terms commute with the field Hamiltonian, while the inclusion of non-secular contributions breaks the PC. Moreover, unless one restricts to a particular (i.e., Ohmic) spectral density of the bath modes, the noise terms may contain relevant information about the frequency to be estimated. Thus, by considering general evolutions of a single probe, we study regimes in which these two effects have a non-negligible impact on the achievable precision. We then consider baths of Ohmic spectral density yet fully accounting for the lack of PC, in order to characterize the ultimate attainable scaling of precision when N probes are used in parallel. Crucially, we show that beyond the semigroup (Lindbladian) regime the Zeno limit imposing the 1/N 3/2 scaling of the mean squared error, recently derived assuming PC, generalises to any dynamics of the probes, unless the latter are coupled to the baths in the direction perfectly transversal to the frequency encoding-when a novel scaling of 1/N 7/4 arises. As our microscopic approach covers all classes of dissipative dynamics, from semigroup to non-Markovian ones (each of them potentially non-phase-covariant), it provides an exhaustive picture, in which all the different asymptotic scalings of precision naturally emerge.

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“…This constitutes our definition of additivity, as studied previously in [54][55][56]. This is distinct from the concept of additive decoherence rates explored, for example, in [60,61]. The additivity assumption permits one to unambiguously identify the particle current J t P a ( ) and energy current J t E a ( ) entering the system from B α as .…”

Section: Preliminariesmentioning

confidence: 97%

Non-additive dissipation in open quantum networks out of equilibrium

2018

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We theoretically study a simple non-equilibrium quantum network whose dynamics can be expressed and exactly solved in terms of a time-local master equation. Specifically, we consider a pair of coupled fermionic modes, each one locally exchanging energy and particles with an independent, macroscopic thermal reservoir. We show that the generator of the asymptotic master equation is not additive, i.e. it cannot be expressed as a sum of contributions describing the action of each reservoir alone. Instead, we identify an additional interference term that generates coherences in the energy eigenbasis, associated with the current of conserved particles flowing in the steady state. Notably, non-additivity arises even for wide-band reservoirs coupled arbitrarily weakly to the system. Our results shed light on the non-trivial interplay between multiple thermal noise sources in modular open quantum systems.

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Complete positivity, finite-temperature effects, and additivity of noise for time-local qubit dynamics

Cited by 25 publications

References 57 publications

There is no general connection between the quantum speed limit and non-Markovianity

There is no general connection between the quantum speed limit and non-Markovianity

Fundamental limits to frequency estimation: a comprehensive microscopic perspective

Non-additive dissipation in open quantum networks out of equilibrium

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