Abstract
We discuss the conceptually different definitions used for the non-Markovianity of classical and quantum processes. The well-established definition of non-Markovianity of a classical stochastic process represents a condition on the Kolmogorov hierarchy of the n-point joint probability distributions. Since this definition cannot be transferred to the quantum regime, quantum non-Markovianity has recently been defined and quantified in terms of the underlying quantum dynamical map, using either its divisibility properties or the behavior of the trace distance between pairs of initial states. Here, we investigate and compare these definitions and their relations to the classical notion of non-Markovianity by employing a large class of non-Markovian processes, known as semi-Markov processes, which admit a natural extension to the quantum case. A number of specific physical examples are constructed that allow us to study the basic features of the classical and the quantum definitions and to evaluate explicitly the measures of quantum non-Markovianity. Our results clearly demonstrate several fundamental differences between the classical and the quantum notion of non-Markovianity, as well as between the various quantum measures of non-Markovianity. In particular, we show that the divisibility property in the classical case does not coincide with Markovianity and that the non-Markovianity measure based on divisibility assigns equal infinite values to different dynamics, which can be distinguished by exploiting the trace distance measure. A simple exact expression for the latter is also obtained in a special case.
Highlights
We discuss the conceptually different definitions used for the non-Markovianity of classical and quantum processes
Starting within the classical framework, we will analyze how the non-Markovianity of a process reflects itself in the behavior of its one-point probability density, which naturally leads to criteria for the characterization of non-Markovian behavior in the dynamics
While knowledge of a non-Markovian classical process requires information on all the conditional probability densities, when studying the dynamics of an open system one only considers the evolution of the state, expressed by a probability vector in the classical case and a statistical operator in the quantum case
Summary
We will state that the map (t, 0) is P-divisible if (t, s) sends states into states but is only positive, and that it is indivisible if neither P-divisibility nor CP-divisibility holds Its matrix elements satisfy ( (t, s))i j 0 and k ( (t, s))kj = 1, which provide the necessary and sufficient conditions ensuring that probability vectors are sent into probability vectors [20] Once again this need not generally be true even if the map (s, 0) is invertible as a linear operator. Note that here we are only considering the one-point probabilities p(t), which are certainly not sufficient to assess the Markovianity or non-Markovianity of a process according to the mathematically precise definition used in classical probability theory
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
