We construct a general measure for the degree of non-Markovian behavior in open quantum systems. This measure is based on the trace distance which quantifies the distinguishability of quantum states. It represents a functional of the dynamical map describing the time evolution of physical states, and can be interpreted in terms of the information flow between the open system and its environment. The measure takes on nonzero values whenever there is a flow of information from the environment back to the open system, which is the key feature of non-Markovian dynamics.

The dynamical behavior of open quantum systems plays a key role in many applications of quantum mechanics, examples ranging from fundamental problems, such as the environment-induced decay of quantum coherence and relaxation in many-body systems, to applications in condensed matter theory, quantum transport, quantum chemistry and quantum information. In close analogy to a classical Markovian stochastic process, the interaction of an open quantum system with a noisy environment is often modeled phenomenologically by means of a dynamical semigroup with a corresponding time-independent generator in Lindblad form, which describes a memoryless dynamics of the open system typically leading to an irreversible loss of characteristic quantum features. However, in many applications open systems exhibit pronounced memory effects and a revival of genuine quantum properties such as quantum coherence, correlations and entanglement. Here, recent theoretical results on the rich non-Markovian quantum dynamics of open systems are discussed, paying particular attention to the rigorous mathematical definition, to the physical interpretation and classification, as well as to the quantification of quantum memory effects. The general theory is illustrated by a series of physical examples. The analysis reveals that memory effects of the open system dynamics reflect characteristic features of the environment which opens a new perspective for applications, namely to exploit a small open system as a quantum probe signifying nontrivial features of the environment it is interacting with. This article further explores the various physical sources of non-Markovian quantum dynamics, such as structured environmental spectral densities, nonlocal correlations between environmental degrees of freedom and correlations in the initial system-environment state, in addition to developing schemes for their local detection. Recent experiments addressing the detection, quantification and control of non-Markovian quantum dynamics are also briefly discussed.

We study the dynamics of quantum and classical correlations in the presence of nondissipative decoherence. We discover a class of initial states for which the quantum correlations, quantified by the quantum discord, are not destroyed by decoherence for times t t, on the other hand, classical correlations do not change in time and only quantum correlations are lost due to the interaction with the environment. Therefore, at the transition timet the open system dynamics exhibits a sudden transition from classical to quantum decoherence regime.PACS numbers: 03.65. Ta,03.65.Yz,03.67.Mn The interaction of a quantum system with its environment causes the rapid destruction of crucial quantum properties, such as the existence of quantum superpositions and of quantum correlations in composite systems [1,2]. Contrarily to the exponential decay characterizing the transition from a quantum superposition to the corresponding statistical mixture, entanglement may disappear completely after a finite time, an effect known as entanglement sudden death [3]. There exists, however, quantum correlations more general and more fundamental than entanglement. Several measures of these quantum correlations have been investigated in the literature [4][5][6][7][8][9], and among them the quantum discord [4,5], has recently received a great deal of attention [10][11][12][13][14][15][16][17][18][19][20][21][22][23]. The total correlations (quantum and classical) in a bipartite quantum system are measured by the quantum mutual information I(ρ AB ) defined aswhere ρ A(B) and ρ AB are the reduced density matrix of subsystem A(B) and the density matrix of the total system, respectively, and S(ρ) = −T r{ρ log 2 ρ} is the von Neumann entropy. The quantum discord is then defined as The major obstacle to the development of quantum technologies has been, until now, the destruction of all quantum properties caused by the inevitable interaction of quantum systems with their environment. The fact that, under certain conditions, quantum correlations useful for quantum algorithms are completely unaffected by the environment, for long time intervals, may constitute a new breakthrough to quantum technologies such as, e.g., quantum computers. A crucial aspect of the dynamics is that, while the total quantum correlations measured by the discord remain constant, classical correlations are lost. Interestingly, in this dynamical region, entanglement decays exponentially in time but at the same time quantum-correlationsother-than-entanglement, measured by dissonance [9], increase monotonically until t =t. We prove analytically that, for certain initial Belldiagonal states, when discord starts to decay, i.e., for t >t, the classical correlations become constant in time. Therefore, there exists an instant of timet at which the system stops losing classical correlation and starts losing quantum correlations. The timet depends on a single parameter characterizing the initial state. The class of i...

Recently, a measure for the non-Markovian behavior of quantum processes in open systems has been developed which is based on the quantification of the flow of information between the open system and its environment [Phys. Rev. Lett. 103, 210401 (2009)]. The information flow is connected to the rate of change of the trace distance between quantum states which can be interpreted in terms of the distinguishability of these states. Here, we elaborate the mathematical details of this theory, present applications to specific physical models, and discuss further theoretical and experimental implications, as well as relations to alternative approaches proposed recently.

Realistic quantum mechanical systems are always exposed to an external environment. This often induces Markovian processes in which the system loses information to its surroundings. However, many quantum systems exhibit nonMarkovian behaviour with a flow of information from the environment back to the system 1-5 . The environment usually consists of large number of degrees of freedom which are difficult to control, but some sophisticated schemes for reservoir engineering have been developed 6 . The control of open systems plays a decisive role, for example, in proposals for entanglement generation 7-9 and dissipative quantum computation 10 , for the design of quantum memories 11 and in quantum metrology 12 . Here we report an all-optical experiment which allows one to drive the open system from the Markovian to the non-Markovian regime, to control the information flow between the system and the environment, and to determine the degree of non-Markovianity by measurements on the open system.The standard approach to the dynamics of open quantum systems employs the concept of a quantum Markov process which is given by a semigroup of completely positive dynamical maps and a corresponding quantum master equation with a generator in Lindblad form 13,14 . Very recently, a toolbox for the engineering of such quantum Markov processes in a multi-qubit system of trapped ions has been realized experimentally 15 and technological developments have also allowed experimental studies of quantum correlations in open systems 16,17 . Within a microscopic approach, quantum Markovian master equations are usually obtained by means of the Born-Markov approximation, which presupposes a weak system-environment coupling and several further, mostly rather drastic approximations. However, in many processes occurring in nature these approximations are not applicable, a situation which occurs, in particular, in the cases of strong systemenvironment couplings, structured and finite reservoirs, and low temperatures, as well as in the presence of large initial systemenvironment correlations. In the case of substantial quantitative and qualitative deviations from the dynamics of a quantum Markov process one often speaks of a non-Markovian process, implying that the dynamics is governed by significant memory effects. Quite recently important steps towards the development of a general consistent theory of non-Markovian quantum dynamics have been made which try to rigorously define the border between Markovian The measure for quantum non-Markovianity constructed in ref.19 is based on the idea that memory effects in the open system dynamics can be characterized in terms of the flow of information between the open system and its environment. It has been used recently, for example, to describe this information flow in the energy transfer dynamics of photosynthetic complexes 2,4 , and to characterize memory effects of the dynamics of qubits in spin baths 5 . Here, we present the results of an experiment which enables one, through a careful preparation ...

Open quantum systems that interact with structured reservoirs exhibit non-Markovian dynamics. We present a quantum jump method for treating the dynamics of such systems. This approach is a generalization of the standard Monte Carlo wave function (MCWF) method for Markovian dynamics. The MCWF method identifies decay rates with jump probabilities and fails for non-Markovian systems where the time-dependent rates become temporarily negative. Our non-Markovian quantum jump approach circumvents this problem and provides an efficient unraveling of the ensemble dynamics.

We study the exact entanglement dynamics of two qubits in a common structured reservoir. We demonstrate that, for certain classes of entangled states, entanglement sudden death occurs, while for certain initially factorized states, entanglement sudden birth takes place. The backaction of the non-Markovian reservoir is responsible for revivals of entanglement after sudden death has occurred, and also for periods of disentanglement following entanglement sudden birth.

Many complex systems present an intrinsic bipartite structure where elements of one set link to elements of the second set. In these complex systems, such as the system of actors and movies, elements of one set are qualitatively different than elements of the other set. The properties of these complex systems are typically investigated by constructing and analyzing a projected network on one of the two sets (for example the actor network or the movie network). Complex systems are often very heterogeneous in the number of relationships that the elements of one set establish with the elements of the other set, and this heterogeneity makes it very difficult to discriminate links of the projected network that are just reflecting system's heterogeneity from links relevant to unveil the properties of the system. Here we introduce an unsupervised method to statistically validate each link of a projected network against a null hypothesis that takes into account system heterogeneity. We apply the method to a biological, an economic and a social complex system. The method we propose is able to detect network structures which are very informative about the organization and specialization of the investigated systems, and identifies those relationships between elements of the projected network that cannot be explained simply by system heterogeneity. We also show that our method applies to bipartite systems in which different relationships might have different qualitative nature, generating statistically validated networks in which such difference is preserved.