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arXiv:1408.3056

Surprising Interactions of Markovian noise and Coherent Driving

Shai Machnes, Martin Bodo Plenio

Published Aug 13, 2014• 7 citations
arXivHTMLPDF

Abstract

We derive the explicit commutation relations for the generators of quantum dynamical semigroup - Markovian superoperator evolution, allowing the extension of Baker-Campbell-Hausdorff-type relations to general Lindblad-type evolutions. This provides a novel tool for exploring the interaction of time-dependent coherent and Markovian evolutions - a surprisingly rich set of behaviors which include deformation by coherent driving of Markovian terms, generation of new Lindblad terms from time-dependent noise and even a coherent driving term generated by the interaction of time-separated Markovian noises. Finally, we consider the Lindblad superoperators as vectors in a higher space, allowing us to extract the infinitely divisible subspace of a given channel and to recover its Lindblad form.

DOI
10.48550/arxiv.1408.3056

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We derive the explicit commutation relations for the generators of quantum dynamical semigroup - Markovian superoperator evolution, allowing the extension of Baker-Campbell-Hausdorff-type relations to general Lindblad-type evolutions. This provides a novel tool for exploring the interaction of time-dependent coherent and Markovian evolutions - a surprisingly rich set of behaviors which include deformation by coherent driving of Markovian terms, generation of new Lindblad terms from time-dependent noise and even a coherent driving term generated by the interaction of time-separated Markovian noises. Finally, we consider the Lindblad superoperators as vectors in a higher space, allowing us to extract the infinitely divisible subspace of a given channel and to recover its Lindblad form.
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Abstract We develop a notion of dephasing under the action of a quantum Markov semigroup in terms of convergence of operators to a block-diagonal form determined by irreducible invariant subspaces. If the latter are all one-dimensional, we say the dephasing is maximal . With this definition, we show that a key necessary requirement on the Lindblad generator is bistochasticity, and focus on characterizing whether a maximally dephasing evolution may be described in terms of a unitary dilation with only classical noise, as opposed to a genuine non-commutative Hudson–Parthasarathy dilation. To this end, we make use of a seminal result of Kümmerer and Maassen on the class of commutative dilations of quantum Markov semigroups. In particular, we introduce an intrinsic quantity constructed from the generator, the Hamiltonian obstruction , which vanishes if and only if the latter admits a self-adjoint representation and quantifies the hindrance to having a classical diffusive noise model.
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In recent years, growing attention has been devoted to the possibility that theories with deformed symmetries, associated with certain models of non-commutative spacetime, may encode a fundamental form of decoherence. This effect should be described by a Lindblad-like evolution governed by the non-trivial Hopf algebra structure of the time-evolution generators. In this work we provide a detailed analysis of such possibility for similar Hopf algebra deformations of the Hamiltonian of a qubit. Starting from a critical examination of the very definition of time evolution through the generalized adjoint action, we explore whether a coherent and physically viable framework can be established. In particular, our analysis shows that a more general combination of adjoint actions always guarantees a von Neumann dynamics and, also in the case of deformed spacetime symmetries considered in the literature, a physically viable Lindblad evolution cannot be established.
Semigroups describing the time evolution of open quantum systems in finite-dimensional spaces have generators of a special form, known as Lindblad generators. These generators and the corresponding processes of time evolution are analyzed and characterized as decay, dissipation and dephasing. In relation to these processes the Hilbert space of the system is equipped with a special structure, a decomposition into a sum of mutually orthogonal subspaces. The complete set of all the stationary states and the asymptotic behavior of the evolutions are presented in detail. Some unusual special facts about invariant operators and symmetries are studied and examples are demonstrated. Perturbation theory for the structure and for the stationary states is discussed and performed in case studies.