Understanding whether the features of open quantum dynamics are genuinely quantum remains a central challenge in quantum dynamics. Even though the non-Markovian behavior of quantum dynamics has been widely investigated across different settings, there is still no consensus on which properties of a dynamics reflect genuine quantum features and which arise from classical or non-genuine quantum sources. In this review, we provide detailed information on recent developments in characterizing quantum non-Markovianity based on information backflow and the nature of its origin. We also present a survey on how various approaches separate classical and quantum contributions, as well as how they define operational tasks that reveal genuine quantum non-Markovianity. We analyze several frameworks, including state-distinguishability -based, channel-based (``CP-divisibility''), and process-tensor methods. For each framework, we outline the underlying physical motivation, the criteria proposed to distinguish genuine quantum non-Markovianity from practical or apparent memory effects. We further compare different approaches and their strengths and limitations. The review aims to clarify the conceptual and operational aspects of quantum non-Markovian processes based on their nature and to provide a foundation for future research on quantum non-Markovianity and its role in advancing quantum information science and technology.
The problem of defining quantum non-Markovianity has proven elusive, with various inequivalent criteria put forth to address it. We consider the question: what is the weakest notion of quantum non-Markovianity of system dynamics that would account for any aspect of memory? That is, a process indicated to be non-Markovian according to any existing criterion is also non-Markovian according to this criterion, but the converse may not be true. Correspondingly, we identify true memorylessness with the idea of the quantum dynamics being temporally self-similar, whereby the form of the intermediate map that propagates the state between two arbitrary time intervals is independent of the initial time
t0. This is identified with the quantum dynamical semigroup (QDS), renewing advocacy for a familiar candidate. Of practical importance is that our weaker criterion is useful for indicating the non-Markovianity of CP-divisible channels that manifest an effect of counteracting decoherence. This is illustrated with the examples of power-law noise (PLN) and Ornstein-Uhlenbeck Noise (OUN). For physically well motivated channels, we propose a geometric quantification of non-Markovianity according to our definition, which is shown to be applicable to stronger manifestations of quantum non-Markovianity, demonstrated here by application to various channels, including those that are CP-indivisible, P-indivisible and eternally non-Markovian. Finally, we indicate a few open questions that come up as natural consequences of this work.
Quantum non-Markovianity of a quantum noisy channel is typically identified with information backflow or, more generally, with departure of the intermediate map from complete positivity. But here, we also indicate certain non-Markovian channels that cannot be witnessed by the CP-divisibility criterion. In complex systems, non-Markovianity becomes more involved on account of subsystem dynamics. Here we study various facets of non-Markovian evolution, in the context of coined quantum walks, with particular stress on disambiguating the internal vs. environmental contributions to non-Markovian backflow. For the above problem of disambiguation, we present a general power-spectral technique based on a distinguishability measure such as trace-distance or correlation measure such as mutual information. We also study various facets of quantum correlations in the transition from quantum to classical random walks, under the considered non-Markovian noise models. The potential for the application of this analysis to the quantum statistical dynamics of complex systems is indicated.
P-divisibility is a central concept in both classical and quantum non-Markovian processes; in particular, it is strictly related to the notion of information backflow. When restricted to a fixed commutative algebra generated by a complete set of orthogonal projections, any quantum dynamics naturally provides a classical stochastic process. It is indeed well known that a quantum generator gives rise to a
P-divisible quantum dynamics if and only if all its possible classical reductions give rise to divisible classical stochastic processes. However, this property does not hold if one operates a classical reduction of the quantum dynamical maps instead of their generators: As an example, for a unitary dynamics,
P-divisibility of its classical reduction is inevitably lost and the latter thus exhibits information backflow. Instead, for some important classes of purely dissipative qubit evolutions, quantum
P-divisibility always implies classical
P-divisibility and therefore excludes information backflow in both the quantum and classical scenarios. On the contrary, for a wide class of orthogonally covariant qubit dynamics, we show that loss of classical
P-divisibility originates from the classical reduction of a purely dissipative
P-divisible quantum dynamics as in the unitary case. Moreover, such an effect can be interpreted in terms of information backflow due to the coherences developed by the quantumly evolving classical state.
Entropy, and its temporal evolution, play a central role in the foundations of quantum theory and in modern quantum technologies. Here we study, in particular, the relations between the --- in general, non-Markovian --- evolution of an open quantum system, the notions of divisibility of a dynamical map and of distinguishability of quantum states, and the temporal behaviour of various entropy-related quantities such as the Renyi (and sandwiched Renyi) divergences, and the so-called min- and max- conditional entropies. This, in turn, gives rise to an operational meaning of (non-)Markovianity.