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arXiv:1311.4213 · Physical Review Letters

Degree of Non-Markovianity of Quantum Evolution

Dariusz Chruściński, Sabrina Maniscalco

Published Mar 26, 2014• 351 citations
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Abstract

We propose a new characterization of non-Markovian quantum evolution based on the concept of non-Markovianity degree. It provides an analog of a Schmidt number in the entanglement theory and reveals the formal analogy between quantum evolution and the entanglement theory: Markovian evolution corresponds to a separable state and the non-Markovian one is further characterized by its degree. It enables one to introduce a non-Markovianity witness-an analog of an entanglement witness, and a family of measures-an analog of Schmidt coefficients, and finally to characterize maximally non-Markovian evolution being an analog of the maximally entangled state. Our approach allows us to classify the non-Markovianity measures introduced so far in a unified rigorous mathematical framework.

DOI
10.1103/physrevlett.112.120404

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We propose a new characterization of non-Markovian quantum evolution based on the concept of non-Markovianity degree. It provides an analog of a Schmidt number in the entanglement theory and reveals the formal analogy between quantum evolution and the entanglement theory: Markovian evolution corresponds to a separable state and the non-Markovian one is further characterized by its degree. It enables one to introduce a non-Markovianity witness-an analog of an entanglement witness, and a family of measures-an analog of Schmidt coefficients, and finally to characterize maximally non-Markovian evolution being an analog of the maximally entangled state. Our approach allows us to classify the non-Markovianity measures introduced so far in a unified rigorous mathematical framework.
The degree of non-Markovianity allows one to characterize quantum evolutions that depart from a Markovian regime in a similar way as the Schmidt number measures the degree of entanglement of pure states. Maximally non-Markovian dynamics are analogous to maximally entangled states [D. Chru\ifmmode \acute{s}\else \'{s}\fi{}ci\ifmmode \acute{n}\else \'{n}\fi{}ski and S. Maniscalco, Phys. Rev. Lett. 112, 120404 (2014)]. Here we demonstrate that there exists a class of maximally non-Markovian quantum evolutions where the associated environment (degrees of freedom not belonging to the system) obeys a Markovian (memoryless) dynamics, which in turn is unperturbed by the system state or dynamics. These properties imply the absence of any ``physical environment-to-system backflow of information.'' Non-Markovian features (as usual in quantum systems coupled to dissipative classical degrees of freedom) arise from a unidirectional dependence of the system dynamics on the reservoir states.
Calculating the degree of non-Markovianity of a quantum process, for a high-dimensional system, is a difficult task given complex maximization problems. Focusing on the entanglement-based measure of non-Markovianity we propose a numerically feasible quantifier for finite-dimensional systems. We define the non-Markovianity measure in terms of a class of entanglement quantifiers named witnessed entanglement which allow us to write several entanglement based measures of non-Markovianity in a unique formalism. In this formalism, we show that the non-Markovianity, in a given time interval, can be witnessed by calculating the expectation value of an observable, making it attractive for experimental investigations. Following this property we introduce a quantifier base on the entanglement witness in an interval of time; we show that measure is a bonafide measure of non-Markovianity. In our example, we use the generalized robustness of entanglement, an entanglement measure that can be readily calculated by a semidefinite programming method, to study impurity atoms coupled to a Bose-Einstein condensate.
We present a comprehensive and up-to-date review of the concept of quantum non-Markovianity, a central theme in the theory of open quantum systems. We introduce the concept of a quantum Markovian process as a generalization of the classical definition of Markovianity via the so-called divisibility property and relate this notion to the intuitive idea that links non-Markovianity with the persistence of memory effects. A detailed comparison with other definitions presented in the literature is provided. We then discuss several existing proposals to quantify the degree of non-Markovianity of quantum dynamics and to witness non-Markovian behavior, the latter providing sufficient conditions to detect deviations from strict Markovianity. Finally, we conclude by enumerating some timely open problems in the field and provide an outlook on possible research directions.
A profound understanding of decoherence and dissipation in quantum dynamics is crucial for the realistic modeling of the evolution of quantum systems. In open quantum dynamics one distinguishes between a memoryless, so-called Markovian evolution and dynamics incorporating memory effects, termed non-Markovian. In this work we study how prevalent memory effects are in the set of all such dynamics. We thus investigate how often a Markovian description is applicable. This question is approached by investigating randomly generated two-step qubit dynamics with respect to different concepts and witnesses of non-Markovianity. We observe that almost all dynamics are non-Markovian, and only a small (yet finite) fraction is Markovian. Furthermore, we study how this proportion changes when considering certain subclasses such as lower rank or mixed-unitary dynamics. Importantly, our results shed light on the relative ratios of -- and interrelations between -- the sets of dynamics that are non-Markovian with respect to different criteria. Finally, we investigate the fraction of dynamics in which the memory effects are necessarily of quantum nature and establish a connection between two recently developed concepts that characterize the quantumness of memory in non-Markovian dynamics.