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.

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 ...

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We study the evolution of a general open quantum system when the system and its environment are initially correlated. We show that the trace distance between two states of the open system can increase above its initial value, and derive tight upper bounds for the growth of the distinguishability of open system states. This represents a generalization of the contraction property of quantum dynamical maps. The obtained inequalities can be interpreted in terms of the exchange of information between the system and the environment, and lead to a witness for system-environment correlations which can be determined through measurements on the open system alone.

We study a recently proposed measure for the quantification of quantum non-Markovianity in the dynamics of open systems which is based on the exchange of information between the open system and its environment. This measure relates the degree of memory effects to certain optimal initial state pairs featuring a maximal flow of information from the environment back to the open system. We rigorously prove that the states of these optimal pairs must lie on the boundary of the space of physical states and that they must be orthogonal. This implies that quantum memory effects are maximal for states which are initially distinguishable with certainty, having a maximal information content. Moreover, we construct an explicit example which demonstrates that optimal quantum states need not be pure states.

One of the most striking consequences of quantum physics is quantum teleportation – the possibility to transfer quantum states over arbitrary distances. Since its theoretical introduction, teleportation has been demonstrated experimentally up to the distance of 143 km. In the original proposal two parties share a maximally entangled quantum state acting as a resource for the teleportation task. If, however, the state is influenced by decoherence, perfect teleportation can no longer be accomplished. Therefore, one of the current major challenges in accomplishing teleportation over long distances is to overcome the limitations imposed by decoherence and the subsequent mixedness of the resource state. Here we show that, in the presence of nonlocal memory effects, perfect quantum teleportation can be achieved even with mixed photon polarisation states. Our results imply that memory effects can be exploited in harnessing noisy quantum systems for quantum communication and that non-Markovianity is a resource for quantum information tasks.