Abstract

Buscemi et al. [PRX Quantum 6, 020316 (2025)] introduced the notion of genuine information backflows, namely information revivals in non-Markovian open quantum dynamics that cannot be explained by pre-existing correlations. However, that framework assumes that the system can be initialized in arbitrary quantum states, thereby excluding physically relevant settings in which the accessible state space is restricted, for example by superselection or conservation rules, limited experimental controllability, or classicality constraints. Here we bridge this gap by developing a theory of genuine information backflows for preparable sets of initial states. This extension allows classical stochastic processes, represented by restricted families of mutually commuting states, to be treated as special cases of the same formalism, whereas previous approaches were limited to fully quantum processes. Within this framework, information revivals are classified according to whether the observed increase in mutual information can be explained by inaccessible auxiliary correlations. This leads to distinct notions of system-world, environment-world, and global explainability, which are separated from genuinely dynamical revivals. We establish a hierarchy among these explanatory classes and identify a sufficient condition under which every revival is necessarily non-genuine. Finally, we reveal a structural asymmetry between system-side and environment-side conditional independence.

Introduction

The exchange and redistribution of information between a system and its environment is a defining feature of open quantum dynamics [5]. In particular, temporary increases in correlations between an open system and an isolated reference are commonly regarded as signatures of information revivals and play a central role in the characterization of memory effects and non-Markovian behavior [1]. Operationally, such revivals are typically identified through an increase in mutual information following a preceding degradation [7, 2].

Recent developments have refined this perspective by distinguishing between genuine and extrinsic information revivals [3]. Genuine revivals correspond to information backflow arising from memory effects in the system–environment interaction itself, whereas extrinsic revivals may instead be explained through pre-existing correlations independent of the system [3]. Consequently, the observation of a revival alone is insufficient to establish the presence of genuine memory effects.

The analysis of [3], however, is restricted to pure maximally entangled reference–system states. Such an assumption excludes a broad class of physically relevant scenarios involving arbitrary mixed states, including classical, quantum, and hybrid correlations commonly encountered in realistic noisy settings. More fundamentally, it relies on the implicit assumption that the entire state space of the system is operationally accessible. In practice, experimental, structural, or fundamental constraints often restrict the set of states that can be prepared, so that the dynamics is only defined on a limited family of inputs. In such situations, reduced-channel descriptions may fail to uniquely characterize the evolution [2].

In this work, we investigate information revivals within a minimal three-time scenario comprising initial, intermediate, and final observation times. To distinguish genuinely dynamical revivals from those that can be attributed to extrinsic correlations (i.e., correlations that do not encode information about the system and therefore cannot represent information backflow), we introduce a suitable local interaction model allowing also for "inert" degrees of freedom, as in [3].

The key difference here is that, in order to accommodate general sets of initial states, the reference–system state is no longer assumed to be pure. As a result, inert degrees of freedom need not be confined to the environment: they may reside on the system side, or be distributed across both system and environment.

Within this setting, revivals admit a classification in terms of their explainability, distinguishing system-world, environment-world, and global explanations from genuine revivals. We establish a hierarchy among these explanatory classes and provide a sufficient condition under which any revival is necessarily non-genuine. Finally, we identify a structural asymmetry between system-side and environment-side conditional independence: while system-side decoupling enforces global explainability, the corresponding environment-side condition does not, in general, imply the same conclusion.

Formalization

To investigate information revivals in open quantum dynamics, we consider the minimal temporal scenario consisting of three observation times, "snapshots," $t_{0} < t_{1} < t_{2}$. The corresponding system states may be regarded as three-time snapshots of the evolution: an initial time $t_{0}$, an intermediate time $t_{1}$, and a final time $t_{2}$. Since revivals concern a temporary loss and a subsequent recovery of information, at least three times are necessary to characterize the phenomenon [3].The evolution of an open system is fundamentally a process and should therefore be represented by an object capable of describing the evolution of all operationally accessible states simultaneously [2]. Let $\mathcal{S}$ denote the set of physically preparable states of the system. In general, $\mathcal{S}$ is a convex subset of $\mathcal{L}_{+,1}(\mathcal{H})$, the set of positive semidefinite unit-trace operators acting on the finite-dimensional Hilbert space $\mathcal{H}$ associated with the system.

In the idealized case where every quantum state is operationally accessible, namely $\mathcal{S}=\mathcal{L}_{+,1}(\mathcal{H})$, the evolution of the open-system dynamics from an initial time $t_{0}$ to later times $t_{1}$ and $t_{2}$ is naturally represented by completely positive trace-preserving (CPTP) maps,
$$ \mathcal{N}{t{0}\to t_{1}} \qquad \mathcal{N}{t{0}\to t_{2}} $$ or equivalently by their corresponding Choi operators [4, 6]. In this setting, the evolution of any possible state is fully specified by the channels. However, we note that, in general, the intermediate step $t_{1}\to t_{2}$ need not admit a CPTP description, reflecting the possible non-divisibility of the dynamics [3]. For the purposes of the present work, such a description is overly restrictive. In realistic scenarios, not every state allowed by quantum theory is operationally accessible. Experimental constraints, preparation procedures, or structural restrictions may limit the possible state space to a proper subset $$ \mathcal{S} \subsetneq \mathcal{L}_{+,1}({\mathcal{H}}) $$ This limitation becomes particularly relevant when seeking a unified framework capable of treating both quantum and classical systems. For example, classical systems may be represented as subsets of mutually commuting states, in which case different CPTP maps may give rise to the same evolution [2]. Consequently, a channel description alone no longer uniquely characterizes the operational process. We thus need to introduce a formalism that allows us to treat such more general situations. Such a formalism is provided by the idea of preparable systems, namely, subsets $\mathcal{S}$ such that there exists a finite dimensional Hilbert space $\mathcal{K}$ and a completely positive map $\mathcal{P}:\mathcal{L}(\mathcal{K})\to\mathcal{L}(\mathcal{H})$ such that for all $\rho \in \mathcal{S}$ there exists $R\geqslant 0$ in $\mathcal{L}(\mathcal{K})$ with$$ \rho = \frac{\mathcal{P}(R)}{\operatorname{Tr}\left[\mathcal{P}(R)\right]} $$ [2]This means that every accessible state in $\mathcal{S}$ can be generated through a common physical preparation procedure applied to an underlying "raw" system, possibly followed by post-selection. A key consequence of this construction is that all possible preparations may be encoded in a single bipartite state $\rho_{RQ}$, where the auxiliary reference system $R$ captures the structure of the accessible preparation space [2]. Importantly, $\rho_{RQ}$ need not be pure and may contain arbitrary mixtures of classical and quantum correlations. This stands in contrast to Ref. [3], where the reference-system state is taken to be maximally entangled, consistently with the fact that therein the object of interest is the reduced channel. In this work, rather than focusing on a single initial state, we adopt a description in terms of triples of states $$ \rho_{RQ},\sigma_{RQ'},\tau_{RQ''} $$ which are compatible. That is, they represent the evolution of accessible reference–system correlations across three ordered times, $$ t_{0} \rightarrow t_{1} \rightarrow t_{2}. $$ Throughout, the reference system $R$ remains non-interacting. Consequently, every compatible triple necessarily satisfies $$ \rho_{R}=\sigma_{R}=\tau_{R}. $$ A revival is defined as an increase in mutual information between the reference and the system at the final time $t=t_{2}$ relative to the intermediate time $t=t_{1}$ $$ I(R; Q''){2} > I(R; Q'){1} $$ under the local evolution constraint (i.e., the data-processing inequality) [2] $$ I(R; Q){0} \geqslant I(R; Q'){1} \quad\text{and}\quad I(R; Q){0} \geqslant I(R; Q''){2}. $$ We define the dynamics through a **Local Interaction Model** (LIM). A compatible triple of states is said to admit a LIM if there exists an initially uncorrelated environment $E$, prepared in some state $\omega_{E}$, with local unitary interactions $$ U_{QE \rightarrow Q'E'}, \qquad V_{Q'E' \rightarrow Q''E''}, $$ such that $$ \sigma_{RQ'} = \operatorname{Tr}{E'}\left[ U \left(\rho{RQ} \otimes \omega_{E}\right) U^\dagger \right], $$ $$ \tau_{RQ''} = \operatorname{Tr}{E''}\left[ V U \left(\rho{RQ} \otimes \omega_{E}\right) U^\dagger V^\dagger \right]. $$ Although the dimension of their product remains conserved, for any given LIM, the local dimensions of $Q$ and $E$ can change.The observation of a revival alone does not determine whether it reflects genuine information backflow from the environment or can instead be accounted for through inaccessible auxiliary correlations [3]. To distinguish between these possibilities, we introduce an **extended Local Interaction Model** (ext-LIM), obtained by augmenting the LIM with inert auxiliary systems $S$ and $F$ on the system and environment sides, respectively. These auxiliary systems do not participate in the interaction dynamics, but may carry correlations capable of explaining an observed revival. A revival is said to be *explainable* (i.e., non-genuine) if there exists an ext-LIM satisfying the following explanatory conditions. A revival is called *environment-world explainable* if $$ I(R;Q'F){1} \geqslant I(R;Q''F){2}, $$ or equivalently using quantum conditional mutual information (QCMI), $$ I(R;E''|Q''F){2} \geqslant I(R;E'|Q'F){1}, $$ *system-world explainable* if $$ I(R;Q'S){1} \geqslant I(R;Q''S){2}, $$ equivalently, $$ I(R;E''|Q''S){2} \geqslant I(R;E'|Q'S){1}, $$ or *globally explainable* if $$ I(R;Q'SF){1} \geqslant I(R;Q''SF){2}, $$ equivalently, $$ I(R;E''|Q''SF){2} \geqslant I(R;E'|Q'SF){1}. $$ That is, a revival is explainable whenever the apparent increase in mutual information disappears once suitable inert degrees of freedom are taken into account. If no such explanatory extension exists, the revival is classified as *genuine*. #The Results The presented framework induces a hierarchy for the different classes of explanations for information revivals. Within this framework, system-world and environment-world explainability each imply global explainability: $$ \text{System} \implies \text{Global}, \qquad \text{Environment} \implies \text{Global}. $$ This follows from the fact that any system-world (respectively environment-world) explanation may always be extended by adjoining an inert auxiliary system on the environment side (respectively system side), thereby yielding a valid global explanation. However, the converse implications fail in general: \begin{equation} \text{Global} \;\not\Rightarrow\; \text{System}, \qquad \text{Global} \;\not\Rightarrow\; \text{Environment}. \end{equation} Thus, the observation of a global explanation does not guarantee the existence of explanatory correlations localized exclusively to either the system or environment side. **Theorem 1 (Sufficient Condition)** *Consider a compatible triple $$ (\rho_{RQ},\sigma_{RQ'},\tau_{RQ''}) $$ admitting an ext-LIM. If there exists a system-side inert extension $S$ such that $$ I(R;Q|S){0} = 0, $$ then any information revival occurring after $t{1}$ is necessarily non-genuine. In particular, all such revivals are system-world explainable, and therefore globally explainable.* **Theorem 2 (Structural Asymmetry I)** *Consider a compatible triple $$ (\rho_{RQ},\sigma_{RQ'},\tau_{RQ''}) $$ admitting an ext-LIM. If $$ I(R;E'|Q'S){1} = 0, $$ then $$ I(R; E'|Q'SF){1} = 0 $$ is enforced. Consequently, the global explanatory condition is thus automatically satisfied.* **Remark 2** *The condition $$ I(R;Q|S){0} = 0 $$ implies that, conditioned on the inert system $S$, the reference and system are initially independent. Operationally, all reference–system correlations are already mediated through auxiliary degrees of freedom on the system-side at $t{0}$. Consequently, any subsequent revival cannot reflect genuine dynamical backflow and instead is always system-world and, thus, globally explained.* **Proposition 1 (Structural Asymmetry II)** *Consider a compatible triple $$ (\rho_{RQ},\sigma_{RQ'},\tau_{RQ''}) $$ admitting an ext-LIM. The condition $$ I(R;E'|Q'F)_{1} = 0 $$ does not, in general, guarantee global explainability.* **Remark 3** *In contrast to the system-side case, environment-side conditional independence alone is insufficient to enforce global explainability. This asymmetry originates from the placement of the reference system $R$ within the system-side sector of the total composite system.*

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