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Proceedings of the Komi Science Centre of the Ural Division of the Russian Academy of Sciences

Lie algebra contractions and the Lindblad equation

Костяков Игорь Владимирович, Куратов Василий Васильевич, N.A. Gromov

Published Jan 1, 2021• 2 citations
Publisher

Abstract

The dynamics of an open quantum system leads to decoherence, which is accompanied by limiting transitions in the Lie algebra of observables and appearance of abelian subalgebras. It is possible to set an inverse problem as well – by a given Lie algebra contraction to find the dynamics of an open quantum system given by the Lindblad equation. The paper proposes examples of finding the Lindblad equation by the known contractions of algebra su(3).

DOI
10.19110/1994-5655-2021-6-36-41

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The dynamics of an open quantum system leads to decoherence, which is accompanied by limiting transitions in the Lie algebra of observables and appearance of abelian subalgebras. It is possible to set an inverse problem as well – by a given Lie algebra contraction to find the dynamics of an open quantum system given by the Lindblad equation. The paper proposes examples of finding the Lindblad equation by the known contractions of algebra su(3).
Open quantum dynamics in the Markovian approximation is described by the Lindblad master equation. The Lindbladian dynamics is closed in the Lie algebra Λ = su(n), i.e. it has su(n) symmetry. We say that the Lindblad equation admits a symmetry reduction if it has an invariant vector subspace Λ0 ⊂Λ with the Lie algebraic structure. Symmetry reductions restrict dynamics to smaller subspaces that additionally are Lie algebras. In these notes, trivial reductions relying onthe reducibility of the Hamiltonian and Lindblad operators are described. Examples of nontrivial reductions in the infinite temperature limit and the parity preserving Majorana reductions are presented. Applicationsto open spin dynamics are discussed.
The evolution of a three-level quantum system interacting with the environment with the symmetry of the Lie algebra su(3) (qutrit) is considered. The evolution of qutrit is obtained when interacting with the environment. The transverse relaxation of the qutrit (due to the diagonal Lindblad operators) has been studied, leading to zero (with the same speedfor the real and imaginary parts) off-diagonal elements of the density matrix. The dynamics of qutrit is obtained in the presence of both transverse and longitudinal relaxation, i.e. when the Lindblad operators are off-diagonal and there are transitions in the system between all energy levels. It is shown that the addition of the longitudinal relaxation ofthe qutrit in addition to the conversion of the offdiagonallements to zero, also affects the diagonal elements of its density matrix. As a result of dissipation, the quantum system becomes incoherent over time. In both cases, initially non-commuting quantum observables in the process of evolution acquire the properties of commuting classical observables,which algebraically manifests itself in the contraction of the algebra su(3).
The quantum damped harmonic oscillator is described by the master equation with usual Lindblad form. The equation has been solved completely by us in arXiv: 0710.2724 [quant-ph]. To construct the general solution a few facts of representation theory based on the Lie algebra su(1,1) were used. In this paper we treat a general model described by a master equation with generalized Lindblad form. Then we examine the algebraic structure related to some Lie algebras and construct the interesting approximate solution. Quantum Computation (Computer) is one of main subjects in Quantum Physics. To realize it we must overcome severe problems arising from Decoherence, so we need to study Quantum Open System to control decoherence (if possible). This paper is a series of [1] and [2], and we study dynamics of a quantum open system. First we explain our purpose in a short manner. See [3] as a general introduction to this subject. ∗E-mail address:
The classical limit of a quantum system with one degree of freedom is examined in terms of the contraction of the underlying non-compact kinematical algebra w 1 (the Weyl-Heisenberg algebra) to the three-dimensional Abelian algebra t 3 . An appropriate definition of the contraction of a Lie algebra and of a sequence of its representations is given. For some quantum systems with simple explicitly integrable dynamics, it is shown how the classical Poisson bracket and classical trajectories are obtained in the limit. Each trajectory is associated with a one-dimensional representation of t 3 within a direct integral of such representations in a Hilbert space. Each of the corresponding generalized one dimensional subspaces is stable under the action of the limiting dynamics, and a superselection rule arises naturally between any two such subspaces.
Abstract In this paper we study the accessibility properties of finite dimensional (N‐level) open quantum systems in the presence of dissipation and relaxation described by the Lindblad master equation. We specifically focus on the unital Lindbladian case where general results can be obtained. The theory of transitive Lie‐group actions is used to classify the system Lie‐algebras of the Lindblad equation for which the reachable sets have nonempty interior. For the special case of n ‐coupled spin‐1/2 systems, we obtain a particularly simple characterization. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)
We investigate the algebraic structure underlying the Lindblad equation for finite-dimensional open quantum systems. By introducing a suitable operator representation of the Liouville superoperator, we show that the dynamics can be formulated in terms of a closed algebra of Hermitian operators that is independent of the particular physical model. This formulation reveals that dissipative dynamics requires a substantially richer algebraic structure than purely unitary evolution, thereby providing a clear characterization of the additional complexity introduced by the Lindbladian. The resulting framework naturally leads to parametrizations of the dynamical map and to differential equations governing its evolution. We further derive recursion relations that enable the efficient construction of the algebra for systems of increasing dimension. Because the algebraic basis is universal, while all model-dependent information enters through a single set of coefficients, the proposed approach significantly reduces the computational cost of constructing the Liouville superoperator compared with direct methods. To facilitate the implementation of the method, we provide a Mathematica notebook containing a one-qubit example that can be systematically extended to an arbitrary number of qubits. The proposed framework therefore provides both a general mathematical description of finite-dimensional Lindblad dynamics and a practical foundation for efficient analytical and numerical implementations.