Currently, there is no systematic way to describe a quantum process with memory solely in terms of experimentally accessible quantities. However, recent technological advances mean we have control over systems at scales where memory effects are non-negligible. The lack of such an operational description has hindered advances in understanding physical, chemical and biological processes, where often unjustified theoretical assumptions are made to render a dynamical description tractable. This has led to theories plagued with unphysical results and no consensus on what a quantum Markov (memoryless) process is. Here, we develop a universal framework to characterise arbitrary non-Markovian quantum processes. We show how a multi-time non-Markovian process can be reconstructed experimentally, and that it has a natural representation as a many body quantum state, where temporal correlations are mapped to spatial ones. Moreover, this state is expected to have an efficient matrix product operator form in many cases. Our framework constitutes a systematic tool for the effective description of memory-bearing open-system evolutions.

We derive a necessary and sufficient condition for a quantum process to be Markovian which coincides with the classical one in the relevant limit. Our condition unifies all previously known definitions for quantum Markov processes by accounting for all potentially detectable memory effects. We then derive a family of measures of non-Markovianity with clear operational interpretations, such as the size of the memory required to simulate a process, or the experimental falsifiability of a Markovian hypothesis.In classical probability theory, a stochastic process is the collection of joint probability distributions of a system's state (described by random variable X) at different times, {P (X k , t k ; X k−1 , t k−1 ; . . . ; X 1 , t 1 ; X 0 , t 0 ) ∀k ∈ N}; to be a valid process, these distributions must additionally satisfy the Kolmogorov consistency conditions [1]. A Markov process is one where the state X k of the system at any time t k only depends conditionally on the state of the system at the previous time step, and not on the remaining history. That is, the conditional probability distributions satisfy P (X k , t k |X k−1 , t k−1 ;. . .; X 0 , t 0 ) = P (X k , t k |X k−1 , t k−1 ) (1) for all k. This simple looking condition has profound implications, leading to a massively simplified description of the stochastic process. The study of such processes forms an entire branch of mathematics, and the evolution of physical systems is frequently approximated to be Markov (when it is not exactly so). This is in part due to the fact that the properties of Markov processes make them easier to manipulate analytically and computationally [2].Implicit in this description of a classical process is the assumption that the value of X j at a given time can be observed without affecting the subsequent evolution. This assumption cannot be valid for quantum processes. In quantum theory, a measurement must be performed to infer the state of system. And the measurement process, in general, must disturb that state. Therefore, unlike its classical counterpart, a generic quantum stochastic process cannot be described without interfering with it [3]. These complications make it challenging to define the process independently of the control operations of the experimenter. From a technical perspective, a serious consequence of this is that joint probability distributions of quantum observables at different times do not satisfy the Kolmogorov conditions [1], and do not constitute stochastic processes in the classical sense.Nevertheless, temporal correlations between observables do play an important role in the dynamics of many open quantum systems, e.g. in the emission spectra of quantum dots [4] and in the vibrational motion of interacting molecular fluids [5]. Quantifying memory effects, and clearly defining * felix.pollock@monash.edu † kavan.modi@monash.edu the boundary between Markovian and non-Markovian quantum processes, represents an important challenge in describing such systems. Attempts at solving this problem tend to take a...

We expand the set of initial states of a system and its environment that are known to guarantee completely positive reduced dynamics for the system when the combined state evolves unitarily. We characterize the correlations in the initial state in terms of its quantum discord [H. Ollivier and W. H. Zurek, Phys. Rev. Lett. 88, 017901 (2001)]. We prove that initial states that have only classical correlations lead to completely positive reduced dynamics. The induced maps can be not completely positive when quantum correlations including, but not limited to, entanglement are present. We outline the implications of our results to quantum process tomography experiments.

We introduce the quantum stochastic walk (QSW), which determines the evolution of generalized quantum mechanical walk on a graph that obeys a quantum stochastic equation of motion. Using an axiomatic approach, we specify the rules for all possible quantum, classical and quantum-stochastic transitions from a vertex as defined by its connectivity. We show how the family of possible QSWs encompasses both the classical random walk (CRW) and the quantum walk (QW) as special cases, but also includes more general probability distributions. As an example, we study the QSW on a line, the QW to CRW transition and transitions to genearlized QSWs that go beyond the CRW and QW. QSWs provide a new framework to the study of quantum algorithms as well as of quantum walks with environmental effects.Many classical algorithms, such as most Markov-chain Monte Carlo algorithms, are based on classical random walks (CRW), a probabilistic motion through the vertices of a graph. The quantum walk (QW) model is a unitary analogue of the CRW that is generally used to study and develop quantum algorithms [1,2,3]. The quantum mechanical nature of the QW yields different distributions for the position of the walker, as a QW allows for superposition and interference effects [4]. Algorithms based on QWs exhibit an exponential speedup over their classical counterparts have been developed [5,6,7]. QWs have inspired the development of an intuitive approach to quantum algorithm design [8], some based on scattering theory [9]. They have recently been shown to be capable of performing universal quantum computation [10].The transition from the QW into the classical regime has been studied by introducing decoherence to specific models of the discrete-time QW [11,12,13,14]. Decoherence has also been been studied as non-unitary effects on continuous-time QW in the context of quantum transport, such as environmentally-assisted energy transfer in photosynthetic complexes [15,16,17,18,19] and state transfer in superconducting qubits [20,21]. For the purposes of experimental implementation, the vertices of the graph in a walk can be implemented using a qubit per vertex (an inefficient or unary mapping) or by employing a quantum state per vertex (the binary or efficient mapping). The choice of mapping impacts the simulation efficiency and their robustness under decoherence [22,23,24]. The previous proposed approaches for exploring decoherence in quantum walks have added environmental-effects to a QW based on computational or physical models such as pure dephasing [17] but have not considered walks where the environmental effects are constructed axiomatically from the underlying graph.In this work, we define the quantum stochastic walk (QSW) using a set of axioms that incorporate unitary and non-unitary effects. A CRW is a type of classical stochastic processes. From the point of view of the theory of open quantum systems, the generalization of a classical stochastic process to the quantum regime is known to be a quantum stochastic process [16,25,26,27,28,29] ...

We extend the Runge-Gross theorem for a very general class of open quantum systems under weak assumptions about the nature of the bath and its coupling to the system. We show that for Kohn-Sham (KS) time-dependent density functional theory, it is possible to rigorously include the effects of the environment within a bath functional in the KS potential. A Markovian bath functional inspired by the theory of nonlinear Schrödinger equations is suggested, which can be readily implemented in currently existing real-time codes. Finally, calculations on a helium model system are presented.