Quasi-Entropies and Non-Markovianity

Benatti, Fabio; Brancati, Luigi

doi:10.3390/e21101020

Cited by 3 publications

(6 citation statements)

References 32 publications

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“…. ., N − 1 make the unique (N 2 − 1) × (N 2 − 1) Kossakowski matrix K. The interest in this matrix is that, for Markovian dynamics, its PSD is equivalent to the condition of CP of the map [20,21], while for non-Markovian dynamics, it is only a sufficient condition [78]. But here, we are primarily interested in its uniqueness property.…”

Section: Discussionmentioning

confidence: 99%

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Geometric-arithmetic master equation in large and fast open quantum systems

Davidović¹

2022

J. Phys. A: Math. Theor.

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Understanding nonsecular dynamics in open quantum systems is addressed here, with emphasis on systems with large numbers of Bohr frequencies, zero temperature, and fast driving. We employ the master equation, which replaces arithmetic averages of the decay rates in the open system, with their geometric averages, and find that it can improve the second order perturbation theory, known as the Redfield equation, while enforcing complete positivity on quantum dynamics. The characteristic frequency scale that governs the approximation is the minimax frequency: the minimum of the maximum system oscillation frequency and the bath relaxation rate; this needs to be larger than the dissipation rate for it to be valid. The concepts are illustrated on the Heisenberg ferromagnetic spin-chain model. To study the accuracy of the approximation, a Hamiltonian is drawn from the Gaussian unitary ensemble, for which we calculate the fourth order time-convolutionless master equation, in the Ohmic bath at zero temperature. Enforcing the geometric average, decreases the trace distance to the exact solution. Dynamical decoupling of a qubit is examined by applying the Redfield and the geometric-arithmetic master equations, in the interaction picture of the time dependent system Hamiltonian, and the results are compared to the exact path integral solution. The geometric-arithmetic approach is significantly simpler and can be super-exponentially faster compared to the Redfield approach.

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Section: Discussionmentioning

confidence: 99%

“…The inequality (78) relates the accuracy of the geometric-arithmetic approximation of the spectral density mean, to the decay rate of the s-values of the X-matrix, |x n |. Assume that |x n | decays geometrically.…”

Section: The Position-displacement Superoperatorsmentioning

confidence: 99%

Geometric-arithmetic master equation in large and fast open quantum systems

Davidović¹

2022

J. Phys. A: Math. Theor.

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“…The remaining terms, e.g., L ni,mj (t), n = i, m = j; c ni,p (t), n = i, p = 1, ..., N − 1; c p,ni (t), n = i, p = 1, ..., N −1; and c pq (t), p, q = 1, ..., N −1 make the unique (N 2 − 1) × (N 2 − 1) Kossakowski matrix K. The interest in this matrix is that, for Markovian dynamics, its PSD is equivalent to the condition of CP of the map [8,9], while for non-Markovian dynamics, it is only a sufficient condition [36]. But here, we are primarily interested in its uniqueness property.…”

Section: B Superoperator Representationmentioning

confidence: 99%

“…In the pre-Markovian stage where the ME is time-dependent, the answer is that the PSD of the timedependent Kossakowski matrix is sufficient, but not necessary condition for the CP of the map; see Ref. [36] for examples of this and the discussion of the related intertwining maps. The latter are composite maps: the states of the system at time t 1 > 0 are sent to the initial factorized state, by the inverse map (assuming that it exists), which is then sent to the state at time t 2 > t 1 by the time-forward map.…”

Section: Markovian Dynamics As An Intertwining Mapmentioning

confidence: 99%

Systematic Gorini, Kossakowski, Sudarshan and Lindblad Equation for Open Quantum Systems

Davidović¹

2021

Preprint

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In complex, open quantum systems, with many degrees of freedom, it is difficult to start a Markovian dynamics, let alone prepare a factorized system-environment state. Rather, the Markovian dynamics is an intertwining map of a completely positive (CP) non-Markovian dynamics. The Markovian master equation is usually not CP as such, and should be left alone if it works adequately well. Here we study how the intertwining Markovian dynamics can still be well approximated by a CP map. A coherent quantum dynamics can be systematically approximated by the CP, Geometric Arithmetic Master Equation (GAME), but this is not the case for an incoherent dynamics. This dichotomy is responsible for disagreeing claims if a CP Markovian master equation cannot have systematic accuracy. As an example, we show how the dynamical decoupling of a qubit breaks the perturbative, intertwining map, while fixing the accuracy of the GAME to near-exactness.

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“…We emphasize that our procedure can be applied not only to the standard Redfield equation but also to its version with time-dependent coefficients (that can result, for instance, from avoiding the so-called "second Markov" approximation). In this case, our regularization preserves the time dependence of the coefficients but makes the dynamics completely positive divisible [29][30][31]. The residual time dependence is an indicator that our approach could perform better at short times with respect to existing schemes.…”

Section: Introductionmentioning

confidence: 96%

A time-dependent regularization of the Redfield equation

D'Abbruzzo,

Cavina,

Giovannetti

2023

SciPost Phys.

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We introduce a new regularization of the Redfield equation based on a replacement of the Kossakowski matrix with its closest positive semidefinite neighbor. Unlike most of the existing approaches, this procedure is capable of retaining the time dependence of the Kossakowski matrix, leading to a completely positive divisible quantum process. Using the dynamics of an exactly-solvable three-level open system as a reference, we show that our approach performs better during the transient evolution, if compared to other approaches like the partial secular master equation or the universal Lindblad equation. To make the comparison between different regularization schemes independent from the initial state, we introduce a new quantitative approach based on the Choi-Jamiołkowski isomorphism.

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Quasi-Entropies and Non-Markovianity

Cited by 3 publications

References 32 publications

Geometric-arithmetic master equation in large and fast open quantum systems

Geometric-arithmetic master equation in large and fast open quantum systems

Systematic Gorini, Kossakowski, Sudarshan and Lindblad Equation for Open Quantum Systems

A time-dependent regularization of the Redfield equation

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