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Complete positivity of non-Markovian quantum dynamics

Björn Witt

Published Aug 1, 2016• 0 citations
Publisher

Abstract

The hierarchy equations of motion provide an elegant formalism for the description of non-Markovian quantum dynamics, which has successfully been applied to a broad variety of physical problems ranging from light-harvesting complexes to resonant tunnelling junctions. When these equations are derived from microscopic models of system and environment, then they typically expand to an infinite series of coupled differential equations. The truncation of this series is accompanied by an approximation of the dynamics and can give rise to a violation of complete positivity. The goal of this thesis is to investigate under what conditions a set of hierarchy equations of motion induces completely positive dynamics. Such conditions are in particular crucial for a phenomenological understanding of the equations of motion. Several approaches to this problem are proposed and compared for classical systems before a generalisation to quantum mechanical problems is carried out. The main result is a concise set of operator inequalities that are sufficient for complete positivity and that can be implemented efficiently in terms of a semi-definite program. The applicability of this technique is demonstrated by various explicit examples. For truncated hierarchy equations for which a violation of complete positivity is inevitable the framework is extended such that the degree of this violation can be estimated and the validity of the truncation can be gauged. In particular in the presence of initial system-environment correlations a dynamical process can also be considered physically reasonable, when the time evolution is not completely positive and it is shown how hierarchy equations that describe such scenarios can be identified. Eventually, the framework is altered such that it gives rise to conditions on hierarchy equations that are sufficient not only for completely positive but even Markovian dynamics.

DOI
10.25560/45311

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The hierarchy equations of motion provide an elegant formalism for the description of non-Markovian quantum dynamics, which has successfully been applied to a broad variety of physical problems ranging from light-harvesting complexes to resonant tunnelling junctions. When these equations are derived from microscopic models of system and environment, then they typically expand to an infinite series of coupled differential equations. The truncation of this series is accompanied by an approximation of the dynamics and can give rise to a violation of complete positivity. The goal of this thesis is to investigate under what conditions a set of hierarchy equations of motion induces completely positive dynamics. Such conditions are in particular crucial for a phenomenological understanding of the equations of motion. Several approaches to this problem are proposed and compared for classical systems before a generalisation to quantum mechanical problems is carried out. The main result is a concise set of operator inequalities that are sufficient for complete positivity and that can be implemented efficiently in terms of a semi-definite program. The applicability of this technique is demonstrated by various explicit examples. For truncated hierarchy equations for which a violation of complete positivity is inevitable the framework is extended such that the degree of this violation can be estimated and the validity of the truncation can be gauged. In particular in the presence of initial system-environment correlations a dynamical process can also be considered physically reasonable, when the time evolution is not completely positive and it is shown how hierarchy equations that describe such scenarios can be identified. Eventually, the framework is altered such that it gives rise to conditions on hierarchy equations that are sufficient not only for completely positive but even Markovian dynamics.
We show that complete positivity is not only sufficient but also necessary for the validity of the quantum data-processing inequality. As a consequence, the reduced dynamics of a quantum system are completely positive, even in the presence of initial correlations with its surrounding environment, if and only if such correlations do not allow any anomalous backward flow of information from the environment to the system. Our approach provides an intuitive information-theoretic framework to unify and extend a number of previous results.
We study the role of correlations with the environment as the source of non-Markovian quantum evolutions. We first focus on the impact that correlations with the environment can have on the dynamical map that evolve the system. We expand the set of initial states of a system and its environment that are known to guarantee completely positive reduced dynamics for the system when the combined state evolves unitarily. We characterize the correlations in the initial state in terms of its quantum discord. The induced maps can be not completely positive when quantum correlations including, but not limited to, entanglement are present. We discuss the implications and limitations of the Markov approximation necessary to derive the Kossakowski-Lindblad master equation. A generalized non-Markovian master equation is derived from the dynamical map of systems correlated with their environment. The physical meaning of not completely positive maps is studied to obtain a consistent theory of non-Markovian quantum dynamics. These are associated to inverse maps necessary to establish correlations and they give rise to a canonical embedding map that is local in time. This master equation goes beyond the Kossakowski-Lindblad master equation. Non-equilibrium quantum thermodynamics can be be studied within this theory. Through out this discussion, the general dynamics of two interacting qubits is used as an example for illustrations.
We construct a large class of non-Markovian master equations that describe the dynamics of open quantum systems featuring strong memory effects, which relies on a quantum generalization of the concept of classical semi-Markov processes. General conditions for the complete positivity of the corresponding quantum dynamical maps are formulated. The resulting non-Markovian quantum processes allow the treatment of a variety of physical systems, as is illustrated by means of various examples and applications, including quantum optical systems and models of quantum transport.
We define non-Markovian quantum dynamics as evolution in which the current state depends on all past states, and completely characterize its structure under the assumptions of complete positivity and non-signalling. The resulting continuous-time dynamics is an integro-differential equation that augments the Gorini-Kossakowski-Sudarshan-Lindblad equation with a memory integral, and is capable of describing the quantum state of systems exposed to noise with any integrable power spectral density with no further approximations. We then establish a formalism to evaluate multi-time correlations of measurement outcomes in this general setting, obviating the need for a regression theorem. As an application, we derive the emission spectrum of a driven two-level system coupled to a non-Markovian bath: the familiar Mollow triplet acquires a frequency-dependent linewidth that encodes the memory of the bath. Our work provides a rigorous yet transparent description of the quantum state of non-Markovian systems, opening the door for state estimation and state-based quantum control beyond the Markovian regime.
Maps that are not completely positive (CP) are often useful to describe the dynamics of open systems. An apparent violation of complete positivity can occur because there are prior correlations of the principal system with the environment, or if the applied transformation is correlated with the state of the system. We provide a physically motivated definition of accessible non-CP maps and derive two necessary conditions for a map to be accessible. We also show that entanglement between the system and the environment is not necessary to generate a non-CP dynamics. We describe two simple approximations that may be sufficient for some problems in process tomography, and then outline what these methods may be able to tell us in other situations where non-CP dynamics naturally arise.