Measures Of non-Markovianity Research Articles

Non-Markovianity measure using two-time correlation functions

We investigate non-Markovianity measure using two-time correlation functions for open quantum systems. We define non-Markovianity measure as the difference between the exact two-time correlation function and the one obtained in the Markov limit. Such non-Markovianity measure can easily be measured in experiments. We found that the non-Markovianity dynamics in different time scale crucially depends on the system-environment coupling strength and other physical parameters such as the initial temperature of the environment and the initial state of the system. In particular, we obtain the short-time and long-time behaviors of non-Markovianity for different spectral densities. We also find that the thermal fluctuation always reduce the non-Markovian memory effect. Also, the non-Markovianity measure shows non-trivial initial state dependence in different time scales.

Gaussian interferometric power as a measure of continuous-variable non-Markovianity

We investigate the non-Markovianity of continuous variable Gaussian quantum channels through the evolution of an operational metrological quantifier, namely the Gaussian interferometric power, which captures the minimal precision that can be achieved using bipartite Gaussian probes in a black-box phase estimation setup, where the phase shift generator is \emph{a priori} unknown. We observe that the monotonicity of the Gaussian interferometric power under the action of local Gaussian quantum channels on the ancillary arm of the bipartite probes is a natural indicator of Markovian dynamics; consequently, its breakdown for specific maps can be used to construct a witness and an effective quantifier of non-Markovianity. In our work, we consider two paradigmatic Gaussian models, the damping master equation and the quantum Brownian motion, and identify analytically and numerically the parameter regimes that give rise to non-Markovian dynamics. We then quantify the degree of non-Markovianity of the channels in terms of Gaussian interferometric power, showing in particular that even nonentangled probes can be useful to witness non-Markovianity. This establishes an interesting link between the dynamics of bipartite continuous variable open systems and their potential for optical interferometry. The results are an important supplement to the recent research on characterization of non-Markovianity in continuous variable systems.

Generalized trace-distance measure connecting quantum and classical non-Markovianity

We establish a direct connection of quantum Markovianity of an open quantum system to its classical counterpart by generalizing the criterion based on the information flow. Here, the flow is characterized by the time evolution of Helstrom matrices, given by the weighted difference of statistical operators, under the action of the quantum dynamical evolution. It turns out that the introduced criterion is equivalent to P-divisibility of a quantum process, namely divisibility in terms of positive maps, which provides a direct connection to classical Markovian stochastic processes. Moreover, it is shown that similar mathematical representations as those found for the original trace distance based measure hold true for the associated, generalized measure for quantum non-Markovianity. That is, we prove orthogonality of optimal states showing a maximal information backflow and establish a local and universal representation of the measure. We illustrate some properties of the generalized criterion by means of examples.

Hierarchy of non-Markovianity andk-divisibility phase diagram of quantum processes in open systems

In recent years, many efforts have been devoted to the construction of a proper measure of quantum non-Markovianity. However, those proposed measures are shown to be at variance in different situations. In this work, we utilize the theory of $k$-positive maps to generalize a hierarchy of $k$ divisibility and develop a powerful tool, called $k$-divisibility phase diagram, to show the landscape of non-Markovianity. This allows one to study different measures in a unified framework and provides a deeper insight into the nature of quantum non-Markovianity. By exploring the phase diagram with several paradigms, we can explain the origin of the discrepancy between three frequently used measures and find the condition under which these measures coincide with each other.

Dynamics of non-Markovianity in the presence of a driving field

We investigate a two-level system in a cavity QED by considering the effects of amplitude damping, phase damping and driving field. We have studied the non-Markovianity in resonance and non-resonance limits in the presence of these effects using Breuer–Laine–Piilo (BLP) non-Markovianity measure (N BLP). The evolution of the system is derived using the time convolutionless (TCL) master equation. In some conditions, it is shown that in the presence of a driving field, the N BLP increases in the resonance and non-resonance limits. We have also found the exact solution of the master equation in order to investigate the effect of temperature- and environment-excited states. We have shown that the behaviour of non-Markovianity is very different from what one can see from the TCL approach. We have also presented some explanation about the behaviour of non-Markovianity in the exact solution using quantum discord (QD).

Using non-Markovian measures to evaluate quantum master equations for photosynthesis.

When dealing with system-reservoir interactions in an open quantum system, such as a photosynthetic light-harvesting complex, approximations are usually made to obtain the dynamics of the system. One question immediately arises: how good are these approximations, and in what ways can we evaluate them? Here, we propose to use entanglement and a measure of non-Markovianity as benchmarks for the deviation of approximate methods from exact results. We apply two frequently-used perturbative but non-Markovian approximations to a photosynthetic dimer model and compare their results with that of the numerically-exact hierarchy equation of motion (HEOM). This enables us to explore both entanglement and non-Markovianity measures as means to reveal how the approximations either overestimate or underestimate memory effects and quantum coherence. In addition, we show that both the approximate and exact results suggest that non-Markonivity can, counter-intuitively, increase with temperature, and with the coupling to the environment.

A measure of non-Markovianity for unital quantum dynamical maps

One of the most important topics in the study of the dynamics of open quantum systems is the information exchange between system and environment. Based on the features of back-flow information from an environment to a system, an approach is provided to detect non-Markovianity for unital dynamical maps. The method takes advantage of non-contraction property of the von Neumann entropy under completely positive and trace-preserving unital maps. Accordingly, for the dynamics of a single qubit as an open quantum system, the sign of the time derivative of the density matrix eigenvalues of the system determines the non-Markovianity of unital quantum dynamical maps. The main characteristics of the measure are to make the corresponding calculations and optimization procedure simpler.

Admissible memory kernels for random unitary qubit evolution

We analyze the random unitary evolution of a qubit within the memory kernel approach. We provide sufficient conditions which guarantee that the corresponding memory kernel generates physically legitimate quantum evolution. Interestingly, we are able to recover several well-known examples and to generate new classes of nontrivial qubit evolution. Surprisingly, it turns out that a class of quantum evolutions with a memory kernel generated by our approach gives rise to the vanishing of a non-Markovianity measure based on the distinguishability of quantum states.

Bounds for the divisibility-based and distinguishability-based non-Markovianity measures

We derive an upper bound for the distinguishability-based non-Markovianity measure of a two-level system and prove that for certain master equations the exact value of the measure achieves this bound. Furthermore, we obtain an easily calculable lower bound for the divisibility-based non-Markovianity measure of an $n$-level system. We illustrate the calculation of these bounds through examples, considering in detail the spin-boson model. We show that the differences between the two measures in the spin-boson model are caused by the drift vector that is also responsible for the nonunitality of the dynamical map.

Non-Markovianity through flow of information between a system and an environment

Exchange of information between a quantum system and its surrounding environment plays a fundamental role in the study of the dynamics of open quantum systems. Here we discuss the role of the information exchange in the non-Markovian behavior of dynamical quantum processes following the decoherence approach, where we consider a quantum system that is initially correlated with its measurement apparatus, which in turn interacts with the environment. We introduce a new way of looking at the information exchange between the system and environment using the quantum loss, which is shown to be closely related to the measure of non-Markovianity based on the quantum mutual information. We also extend the results of [Phys. Rev. Lett. 112, 210402 (2014)] by Fanchini et al. in several directions, providing a more detailed investigation of the use of the accessible information for quantifying the backflow of information from the environment to the system. Moreover, we reveal a clear conceptual relation between the entanglement and mutual information based measures of non-Markovianity in terms of the quantum loss and accessible information. We compare different ways of studying the information flow in two theoretical examples. We also present experimental results on the investigation of the quantum loss and accessible information for a two-level system undergoing a zero temperature amplitude damping process. We use an optical approach that allows full access to the state of the environment.

Comparative study of non-Markovianity measures in exactly solvable one- and two-qubit models

In this paper we present a detailed critical study of several recently proposed non-Markovianity measures. We analyse their properties for single qubit and two-qubit systems in both pure-dephasing and dissipative scenarios. More specifically we investigate and compare their computability, their physical meaning, their Markovian to non-Markovian crossover, and their additivity properties with respect to the number of qubits. The bottom-up approach that we pursue is aimed at identifying similarities and differences in the behavior of non-Markovianity indicators in several paradigmatic open system models. This in turn allows us to infer the leading traits of the variegated phenomenon known as non-Markovian dynamics and, possibly, to grasp its physical essence.

Coherence, correlation and non-Markovianity in qubit systems

By means of recently presented two typical measures of non-Markovianity, we study the non-Markovian dynamics of a qubit system coupled to a single qubit environment. Special attention is paid to the problem of environmental coherence on the effect of non-Markovianity of system dynamics and on the effect of entanglement evolution. We also study the relation between non-Markovianity and system-environment correlations and the time evolution of coherence of the environment itself.

Locality and universality of quantum memory effects.

The modeling and analysis of the dynamics of complex systems often requires to employ non-Markovian stochastic processes. While there is a clear and well-established mathematical definition for non-Markovianity in the case of classical systems, the extension to the quantum regime recently caused a vivid debate, leading to many different proposals for the characterization and quantification of memory effects in the dynamics of open quantum systems. Here, we derive a mathematical representation for the non-Markovianity measure based on the exchange of information between the open system and its environment, which reveals the locality and universality of non-Markovianity in the quantum state space and substantially simplifies its numerical and experimental determination. We further illustrate the application of this representation by means of an all-optical experiment which allows the measurement of the degree of memory effects in a photonic quantum process with high accuracy.

Quantum regression theorem and non-Markovianity of quantum dynamics

We explore the connection between two recently introduced notions of non-Markovian quantum dynamics and the validity of the so-called quantum regression theorem. While non-Markovianity of a quantum dynamics has been defined looking at the behaviour in time of the statistical operator, which determines the evolution of mean values, the quantum regression theorem makes statements about the behaviour of system correlation functions of order two and higher. The comparison relies on an estimate of the validity of the quantum regression hypothesis, which can be obtained exactly evaluating two points correlation functions. To this aim we consider a qubit undergoing dephasing due to interaction with a bosonic bath, comparing the exact evaluation of the non-Markovianity measures with the violation of the quantum regression theorem for a class of spectral densities. We further study a photonic dephasing model, recently exploited for the experimental measurement of non-Markovianity. It appears that while a non-Markovian dynamics according to either definition brings with itself violation of the regression hypothesis, even Markovian dynamics can lead to a failure of the regression relation.

Quantum energy and coherence exchange with discrete baths

Coherence and quantum average energy exchange are studied for a system particle as a function of the number N of constituents of a discrete bath model. The time evolution of the energy and coherence, determined via the system purity (proportional to the linear entropy of the quantum statistical ensemble), are obtained solving numerically the Schrödinger equation. A new simplified stochastic Schrödinger equation is derived which takes into account the discreteness of the bath. The environment (bath) is composed of a finite number N of uncoupled harmonic oscillators (HOs), characterizing a structured bath, for which a non-Markovian behavior is expected. Two distinct physical situations are assumed for the system particle: the HO and the Morse potential. In the limit N→∞ the bath is assumed to have an ohmic, sub-ohmic or super-ohmic spectral density. In the case of the HO, for very low values of N (≲10) the mean energy and purity oscillate between HO and bath indefinitely in time, while for intermediate and larger values (N∼10→500) they start to decay with two distinct time regimes: exponential for relatively short times and power-law for larger times. In the case of the Morse potential we only observe an exponential decay for large values of N while for small N’s, due to the anharmonicity of the potential, no recurrences of the mean energy and coherences are observed. Wave packet dynamics is used to determine the evolution of the particle inside the system potentials. For both systems the time behavior of a non-Markovianity measure is analyzed as a function of N and is shown to be directly related to the time behavior of the purity.

Competition between memory-keeping and memory-erasing decoherence channels

We study the competing effects of simultaneous Markovian and non-Markovian decoherence mechanisms acting on a single spin. We show the existence of a threshold in the relative strength of such mechanisms above which the spin dynamics becomes fully Markovian, as revealed by the use of several non-Markovianity measures. We identify a measure-dependent nested structure of such thresholds, hinting at a causality relationship amongst the various non-Markovianity witnesses used in our analysis. Our considerations are then used to argue the unavoidably non-Markovian evolution of a single-electron quantum dot exposed to both intrinsic and Markovian technical noise, the latter of arbitrary strength.

Non-Markovianity through Accessible Information

The degree of non-Markovianity of quantum processes has been characterized in several different ways in the recent literature. However, the relationship between the non-Markovian behavior and the flow of information between the system and the environment through an entropic measure has not been yet established. We propose an entanglement-based measure of non-Markovianity by employing the concept of assisted knowledge, where the environment E, acquires information about a system S, by means of its measurement apparatus A. The assisted knowledge, based on the accessible information in terms of von-Neumann entropy, monotonically increases in time for all Markovian quantum processes. We demonstrate that the signatures of non-Markovianity can be captured by the nonmonotonic behaviour of the assisted knowledge. We explore this scenario for a two-level system undergoing a relaxation process, through an experimental implementation using an optical approach that allows full access to the state of the environment.

Degree of non-Markovianity of quantum evolution.

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.

Non-Markovian Quantum Probes

We review the most recent developments in the theory of open quantum systems focusing on situations in which the reservoir memory effects, due to long-lasting and non-negligible correlations between system and environment, play a crucial role. These systems are often referred to as non-Markovian systems. After a brief summary of different measures of non-Markovianity that have been introduced over the last few years we restrict our analysis to the investigation of information flow between system and environment. Within this framework we introduce an important application of non-Markovianity, namely its use as a quantum probe of complex quantum systems. To illustrate this point we consider quantum probes of ultracold gases, spin chains, and trapped ion crystals and show how properties of these systems can be extracted by means of non-Markovianity measures.

Quantum non-Markovian behavior at the chaos border

In this work we study the non-Markovian behavior of a qubit coupled to an environment in which the corresponding classical dynamics change from integrable to chaotic. We show that in the transition region, where the dynamics has both regular islands and chaotic areas, the average non-Markovian behavior is enhanced to values even larger than those in the regular regime. This effect can be related to the non-Markovian behavior as a function of the initial state of the environment, where maxima are attained at the regions dividing separate areas in classical phase space, particularly at the borders between chaotic and regular regions. Moreover, we show that the fluctuations of the fidelity of the environment—which determine the non-Markovianity measure—give a precise image of the classical phase portrait.