Dynamical decoupling of unbounded Hamiltonians

Arenz, Christian; Burgarth, Daniel; Facchi, Paolo; Hillier, Robin

doi:10.1063/1.5016495

Cited by 24 publications

(23 citation statements)

References 23 publications

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“…We also did not consider infinite-dimensional systems. Since the adiabatic theorem has itself many generalizations to infinite-dimensional systems and unbounded operators, this connection might pave the way to QZDs with unbounded operators, where there remain many open problems [12,14,38,[50][51][52]. However, our proof via a generalized Baker-Campbell-Hausdorff formula does not easily generalize to infinite dimensions, and finding a more direct bridge between kicked dynamics and the adiabatic theorem would be desirable.…”

Section: Discussionmentioning

confidence: 99%

Quantum Zeno Dynamics from General Quantum Operations

Burgarth

Facchi

Nakazato

et al. 2020

Quantum

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We consider the evolution of an arbitrary quantum dynamical semigroup of a finite-dimensional quantum system under frequent kicks, where each kick is a generic quantum operation. We develop a generalization of the Baker-Campbell-Hausdorff formula allowing to reformulate such pulsed dynamics as a continuous one. This reveals an adiabatic evolution. We obtain a general type of quantum Zeno dynamics, which unifies all known manifestations in the literature as well as describing new types.

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Section: Discussionmentioning

confidence: 99%

Quantum Zeno Dynamics from General Quantum Operations

Burgarth

Facchi

Nakazato

et al. 2020

Quantum

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“…[78], as does any unitary satisfying PFI. The relation of DD to quantum Markovianity has been recently discussed [79][80][81], and was linked to issues in quantum foundations in Ref. [78].…”

Section: Failure Of Dynamical Decouplingmentioning

confidence: 99%

Concepts of quantum non-Markovianity: A hierarchy

2018

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Markovian approximation is a widely-employed idea in descriptions of the dynamics of open quantum systems (OQSs). Although it is usually claimed to be a concept inspired by classical Markovianity, the term quantum Markovianity is used inconsistently and often unrigorously in the literature. In this report we compare the descriptions of classical stochastic processes and quantum stochastic processes (as arising in OQSs), and show that there are inherent differences that lead to the non-trivial problem of characterizing quantum non-Markovianity. Rather than proposing a single definition of quantum Markovianity, we study a host of Markov-related concepts in the quantum regime. Some of these concepts have long been used in quantum theory, such as quantum white noise, factorization approximation, divisibility, GKS-Lindblad master equation, etc.. Others are first proposed in this report, including those we call past-future independence, no (quantum) information backflow, and composability. All of these concepts are defined under a unified framework, which allows us to rigorously build hierarchy relations among them. With various examples, we argue that the current most often used definitions of quantum Markovianity in the literature do not fully capture the memoryless property of OQSs. In fact, quantum non-Markovianity is highly context-dependent. The results in this report, summarized as a hierarchy figure, bring clarity to the nature of quantum non-Markovianity. Conjectures and open questions 60Appendix B FA, PFI and QRF 61Appendix C The AFL model and its relations to FA, QRF and GQRF 61Appendix D Proof of Lemma 1 63Appendix E Hierarchy of classical regression formulas 64References 65ways, including stochastic Heisenberg equations for system operators and evolution equations for the system density operator [7]. However, the generalization of the Markovian assumption from the classical regime to the quantum regime immediately meets some difficulties. The quantum analogues of random variables of the classical system are operators representing system observables. If one restricted to a set or vector of operatorsx that, at any one time t, are mutually commuting, then one could determine a probability distribution P (x) via the quantum state ρ at that time. But even such a restricted set of operators will not be mutually commuting at different times. Thus there is no analogue of the classical conditional distribution appearing in the Markovian condition of Eq. (1). 1 Hence, a different approach is needed to the issue of Markovian and non-Markovian evolution for OQSs. In particular, we will argue, there is no one concept that should be identified with Markovianity in the quantum case, but rather a host of related concepts.In the theory of OQSs, many methods and criteria have been developed that are conceptually related to the general idea of Markovianity. For example, the factorization approximation and quantum white noise limit are often employed in the derivation of the GKS-Lindblad-type master equation from a microscopi...

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“…However, it has recently been recognized that DD can in fact work in the Markovian setting as well, specifically the Davies-Lindblad master equation [65], and a more abstract semigroup framework [66] (see also Refs. [67,68]). Our contribution here is to establish this in the framework of the Redfield master equation and, more importantly, the CGME.…”

Section: Applications and Examplesmentioning

confidence: 99%

Completely positive master equation for arbitrary driving and small level spacing

Mozgunov

Lidar

2020

Quantum

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Markovian master equations are a ubiquitous tool in the study of open quantum systems, but deriving them from first principles involves a series of compromises. On the one hand, the Redfield equation is valid for fast environments (whose correlation function decays much faster than the system relaxation time) regardless of the relative strength of the coupling to the system Hamiltonian, but is notoriously non-completely-positive. On the other hand, the Davies equation preserves complete positivity but is valid only in the ultra-weak coupling limit and for systems with a finite level spacing, which makes it incompatible with arbitrarily fast time-dependent driving. Here we show that a recently derived Markovian coarse-grained master equation (CGME), already known to be completely positive, has a much expanded range of applicability compared to the Davies equation, and moreover, is locally generated and can be generalized to accommodate arbitrarily fast driving. This generalization, which we refer to as the time-dependent CGME, is thus suitable for the analysis of fast operations in gate-model quantum computing, such as quantum error correction and dynamical decoupling. Our derivation proceeds directly from the Redfield equation and allows us to place rigorous error bounds on all three equations: Redfield, Davies, and coarse-grained. Our main result is thus a completely positive Markovian master equation that is a controlled approximation to the true evolution for any time-dependence of the system Hamiltonian, and works for systems with arbitrarily small level spacing. We illustrate this with an analysis showing that dynamical decoupling can extend coherence times even in a strictly Markovian setting.

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Dynamical decoupling of unbounded Hamiltonians

Cited by 24 publications

References 23 publications

Quantum Zeno Dynamics from General Quantum Operations

Quantum Zeno Dynamics from General Quantum Operations

Concepts of quantum non-Markovianity: A hierarchy

Completely positive master equation for arbitrary driving and small level spacing

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