Non-Markovianity and memory effects in quantum open systems

Abstract

Although a number of measures for quantum non-Markovianity have been proposed recently, it is still an open question whether these measures directly characterize the memory effect of the environment, i.e., the dependence of a quantum state on its past in a time evolution. In this paper, we present a criterion and propose a measure for non-Markovianity with clear physical interpretations of the memory effect. The non-Markovianity is defined by the inequality T (t2, t0) = T (t2, t1)T (t1, t0) in terms of memoryless dynamical map T introduced in this paper. This definition is conceptually distinct from that based on divisibility used by Rivas et al (Phys. Rev. Lett 105, 050403 (2010)), whose violation is manifested by non-complete positivity of the dynamical map. We demonstrate via a typical quantum process that without Markovian approximation, nonzero memory effects (non-Markovianity) always exist even if the non-Markovianity is zero by the other non-Marovianity measures.

Non-Markovianity in the collision model with environmental block

“…whereâ is the annihilation operator, r ( ) t is the density operator of the field, g=1 is the coupling strength and γ(t) is the damping rate. We note that the evolution of a generic Gaussian state, governed by equation (25), can be described in terms of the covariance matrix, as shown in equation (15), with the matrices…”

“…For instance, the CPT condition is violated if a quantum system is strongly coupled to the environment, so that the dynamics becomes non-Markovian [12]. In recent years a number of criteria characterizing non-Markovianity have been proposed, from different perspectives, basing on the dynamical divisibility [13][14][15][16], back-flow of information characterized by trace distance [4,17], Fisher information [18], mutual information [19], relative entropy [20,21], accessible information [22], Gaussian interferometric power [23] and response functions [24]. For recent reviews, see [25,26].…”

Maximally non-Markovian quantum dynamics without environment-to-system backflow of information

“…The previous point of view was introduced in the seminal contribution of Breuer, Laine, and Piilo [6], where the memory indicator is given by the (non-monotonous) behavior of the distinguishability between two initial states. Since then, many other indicators and measures were introduced, such as for example based on the divisibility of the propagator [7][8][9][10], geometry of the set of accessible states [11], negativity of the dissipative rates in a canonical form of the quantum master equation [12], non-Markovianity degree [13,14], spectra of the dynamical map [15], quantum regression theorem [16], power spectrum [17], Fisher information flow [18], mutual information [19], fidelity [20], and accessible information [21] just to name a few [4,5]. In addition, many aspects were analyzed such as for example the relation with the definition of non-Markovian classical stochastic processes [22,23], and the correspondence between the different (inequivalent) non-Markovian indicators and measures [24][25][26][27][28][29].…”

A Note on Markovian Quantum Dynamics

“…Mathematically, a quantum Markovian process can be described by a master equation in the Lindblad form or equivalently by completely positive divisible maps [17]. Besides the above master equation or divisibility approach [18], there are many other ways to describe non-Markovianity, such as the breakdown of the semigroup property [22], the increasing of distinguishability between two evolving states [3], the nonmonotonic behaviours of mutual information items [13], the negative decay rate [10] or the inequality of the memoryless dynamical maps [11] (for more details, see the articles [1,2,4,7,12,15,16,20]).…”