The Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) equation, a cornerstone of open-system theory, describes Markovian semigroup dynamics in which information about the initial preparation cannot revive. Beyond this regime, information revivals are widely interpreted as backflow from a non-Markovian environment. However, a decrease followed by an increase in information retrievable from the system need not imply that information has returned from the environment [PRX Quantum 6, 020316 (2025)]. It may instead arise from noise correlations unrelated to the system's initial preparation, carried by inert degrees of freedom that never interact with the system and therefore cannot mediate backflow. Such revivals are called extrinsic; genuine backflow is reserved for revivals that cannot be explained this way. The framework developed in that work, however, assumes access to arbitrary initial system states, an assumption often incompatible with physical constraints or limited experimental control. This leaves open which properties of the available preparations are necessary for an information revival to certify genuine backflow. Here we extend the framework to allow the description of restricted families of initial states and show that genuine backflow is possible only if the family contains noncommuting states. If all available states commute, as in a classical stochastic process, every revival can be explained by a suitable inert extension and is therefore extrinsic. Noncommutativity of the available system states is thus necessary for information revivals to provide an operational signature of environmental memory.
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Information revivals in open-system dynamics need not represent information returning from an interacting environment: correlations with inert auxiliary registers can sometimes remove the revival. This paper asks when restricted initial preparations permit a revival to certify “genuine” backflow, extending a framework previously formulated for unrestricted channel inputs.
The setting is a finite-dimensional, three-time process with an initially independent environment and two system–environment unitaries. A fixed reference–system probe $\rho_{RQ}$ encodes the available preparations through steering; information revivals mean $I(R;Q'')>I(R;Q')$. A revival is extrinsic if some compatible model admits inert system-side and/or environment-side registers whose inclusion makes this mutual information nonincreasing.
The main no-go theorem shows that every revival from a separable probe is extrinsic. Its proof stores a separable-decomposition label in an inert classical register $S$, giving $I(R;Q|S)=0$ and screening the reference correlations throughout evolution. A second result proves that the steered preparations commute exactly when the probe is quantum–classical on the system. Consequently, genuine backflow requires an entangled probe and noncommuting preparations. System-side and environment-side explainability are also shown to be incomparable, and conditional-mutual-information identities establish asymmetric stability of exact screening under additional conditioning.
These necessary conditions are not sufficient. The no-go conclusion depends on allowing system-side extensions; excluding them changes the classification. For noncommuting preparations, results depend on the chosen probe, not solely its steered set. Numerical illustrations test selected extensions and do not certify genuine backflow by exhaustive exclusion.
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21 pages, 4 figures. Submitted to Open Systems & Information Dynamics
The Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) equation, a cornerstone of open-system theory, describes Markovian semigroup dynamics in which information about the initial preparation cannot revive. Beyond this regime, information revivals are widely interpreted as backflow from a non-Markovian environment. However, a decrease followed by an increase in information retrievable from the system need not imply that information has returned from the environment [PRX Quantum 6, 020316 (2025)]. It may instead arise from noise correlations unrelated to the system's initial preparation, carried by inert degrees of freedom that never interact with the system and therefore cannot mediate backflow. Such revivals are called extrinsic; genuine backflow is reserved for revivals that cannot be explained this way. The framework developed in that work, however, assumes access to arbitrary initial system states, an assumption often incompatible with physical constraints or limited experimental control. This leaves open which properties of the available preparations are necessary for an information revival to certify genuine backflow. Here we extend the framework to allow the description of restricted families of initial states and show that genuine backflow is possible only if the family contains noncommuting states. If all available states commute, as in a classical stochastic process, every revival can be explained by a suitable inert extension and is therefore extrinsic. Noncommutativity of the available system states is thus necessary for information revivals to provide an operational signature of environmental memory.
Information revivals in open-system dynamics need not represent information returning from an interacting environment: correlations with inert auxiliary registers can sometimes remove the revival. This paper asks when restricted initial preparations permit a revival to certify “genuine” backflow, extending a framework previously formulated for unrestricted channel inputs.
The setting is a finite-dimensional, three-time process with an initially independent environment and two system–environment unitaries. A fixed reference–system probe ρRQ encodes the available preparations through steering; information revivals mean I(R;Q′′)>I(R;Q′). A revival is extrinsic if some compatible model admits inert system-side and/or environment-side registers whose inclusion makes this mutual information nonincreasing.
The main no-go theorem shows that every revival from a separable probe is extrinsic. Its proof stores a separable-decomposition label in an inert classical register S, giving I(R;Q∣S)=0 and screening the reference correlations throughout evolution. A second result proves that the steered preparations commute exactly when the probe is quantum–classical on the system. Consequently, genuine backflow requires an entangled probe and noncommuting preparations. System-side and environment-side explainability are also shown to be incomparable, and conditional-mutual-information identities establish asymmetric stability of exact screening under additional conditioning.
These necessary conditions are not sufficient. The no-go conclusion depends on allowing system-side extensions; excluding them changes the classification. For noncommuting preparations, results depend on the chosen probe, not solely its steered set. Numerical illustrations test selected extensions and do not certify genuine backflow by exhaustive exclusion.
The study of information revivals, witnessing the violation of certain data-processing inequalities, has provided an important paradigm in the study of non-Markovian quantum stochastic processes. Although often used interchangeably, we argue here that the notions of “revivals” and “backflows,” i.e., flows of information from the environment back into the system, are distinct: an information revival can occur without any backflow ever taking place. In this paper, we examine in detail the phenomenon of noncausal revivals and relate them to the theory of short Markov chains and squashed non-Markovianity. We also provide an operational condition, in terms of system-only degrees of freedom, to witness the presence of genuine backflow that cannot be explained by noncausal revivals. As a byproduct, we demonstrate that focusing on processes with genuine backflows, while excluding those with only noncausal revivals, resolves the issue of nonconvexity of Markovianity, thus enabling the construction of a convex resource theory of genuine quantum non-Markovianity.
We address the dynamics of quantum correlations for a bipartite continuous-variable quantum system interacting with its fluctuating environment. In particular, we consider two independent quantum oscillators initially prepared in a Gaussian state, e.g., a squeezed thermal state, and compare the dynamics resulting from local noise, i.e., oscillators coupled to two independent external fields, to that originating from common noise, i.e., oscillators interacting with a single common field. We prove non-Markovianity (nondivisibility) of the dynamics in both regimes and analyze the connections between nondivisibility, backflow of information, and revivals of quantum correlations. Our main results may be summarized as follows: (i) revivals of quantumness are present in both scenarios, however, the interaction with a common environment better preserves the quantum features of the system; (ii) the dynamics is always nondivisible but revivals of quantum correlations are present only when backflow of information is present as well. We conclude that nondivisibility in its own is not a resource to preserve quantum correlations in our system, i.e., it is not sufficient to observe recoherence phenomena. Rather, it represents a necessary prerequisite to obtain backflow of information, which is the true ingredient to obtain revivals of quantumness.
A well established approach to describe open quantum system dynamics is the celebrated Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation. In this article we aim to shed light on semigroup dynamics that are not of this form. Such dynamics may appear in real physical systems describing asymptotic long-time dynamics. Thus, they correspond to semigroup dynamics applicable to a restricted set of initial states only. As a way of motivating our studies, first we review known results about the single-mode Gaussian Quantum Brownian motion channel and corresponding master equations. Next we discuss a conceptually simpler family of qubit semigroup generators and identify parameter regions of completely positive, merely positive, and non-positive dynamics. We then focus on the latter case to which we will refer to as restricted-positive, as the semigroup generator may well be physically meaningful when applied to a restricted set of admissible initial states.
A Markovian process of a system is defined classically as a process in which the future state of the system is fully determined by only its present state, not by its previous history. There have been several measures of non-Markovianity to quantify the degrees of non-Markovian effect in a process of an open quantum system based on information backflow from the environment to the system. However, the condition for the witness of the system information backflow does not coincide with the classical definition of a Markovian process. Recently, a new measure with a condition that coincides with the classical definition in the relevant limit has been proposed. Here, we focus on the new definition (measure) for quantum non-Markovian processes, and characterize the Markovian condition as a quantum process that has no information backflow through the reduced environment state (IBTRES) and no system-environment correlation effect (SECE). The action of IBTRES produces non-Markovian effects by flowing the information of quantum operations performed by an experimenter at earlier times back to the system through the environment, while the SECE can produce non-Markovian effect without carrying any earlier quantum operation information. We give the necessary and sufficient conditions for no IBTRES and no SECE, respectively, and show that a process is Markovian if and only if it has no IBTRES and no SECE. The quantitative measures and algorithms for calculating non-Markovianity, IBTRES, and solely SECE are explicitly presented.
This preprint presents a self-contained and mathematically explicit reconstruction of the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation starting from exact microscopic unitary dynamics through the Nakajima-Zwanzig projection operator formalism. Beginning with the exact non-local generalized master equation, we systematically analyze the hierarchy of approximations required for the emergence of a Markovian quantum dynamical semigroup. The Born (weak-coupling), Markov (time-localization), and secular approximations are derived and interpreted as successive information-reduction steps, each associated with clearly identifiable mathematical assumptions and physical consequences. Particular attention is devoted to the role of secularization in transforming the Redfield generator into a completely positive GKSL structure. To provide a concrete worked example, the formalism is applied to a damped two-level system coupled to a bosonic thermal reservoir, where the Redfield and GKSL equations are derived explicitly and compared at the microscopic level. Beyond the derivation itself, the work develops a systematic map of validity limits and failure regimes. The breakdown of the Born, Markov, secular, factorization, and stationarity assumptions is analyzed in terms of environmental correlation times, spectral-density structure, non-Markovian memory effects, information backflow, and loss of semigroup behavior. The resulting framework clarifies the precise operational domain in which GKSL dynamics can be regarded as a physically reliable effective description and provides a rigorous foundation for applications involving entropy-production theorems, quantum thermodynamics, and open-system information theory.
We analyze the exact dynamics of a generalized quantum Brownian motion model, employing Gaussian master equation methods. We demonstrate that, by modulating the relative weights of specific interaction channels, we can control the degree of non-Markovianity of the system, and induce a transition from non-Markovian to Markovian regimes. The non-Markovianity of the evolution is formally characterized by leveraging the Gorini--Kossakowski--Sudarshan--Lindblad theorem and by employing quantitative measures of information backflow. Finally, we clarify the physical mechanism behind the phenomenology of this model, thereby providing a systematic platform for environment engineering through the strategic tuning of dissipation channels.