Phase Diagrams of Information Backflow: Unifying Entanglement Revivals and Entropy Overshoots in Minimal Non-Markovian Models

TL;DR

This work addresses the lack of a unified framework for non-Markovian information backflow by introducing a single backflow functional $N_I$ that measures the total information returned during positive-slope intervals of an information observable. It develops two minimal, analytically tractable models—quantum fractional two-state relaxation embedded via thermo-field dynamics and classical three-state memory dynamics with memory extensions (GME and Erlang-2)—and demonstrates that memory effects manifest as backflow in both cases, linked through explicit embedding narratives. A key finding is a phase boundary near $α\approx 1/2$ in both quantum and classical settings, indicating that the boundary is driven by the memory kernel structure (Mittag-Leffler) rather than quantum specifics. The paper thus provides a compact, model-agnostic phase-diagram framework for non-Markovianity, with kernel-level universality and practical implications for analyzing memory effects across quantum and classical domains.

Abstract

Memory effects in non-Markovian dynamics are often diagnosed either via quantum-correlation revivals or via non-monotonic classical information measures, yet a unified minimal framework comparing their ``backflow phases'' is still lacking. Here we propose an information-backflow phase-diagram approach that places \emph{quantum entanglement revivals} and \emph{classical entropy overshoots} on the same footing through a common backflow functional $N_I=\int_{\dot I>0}\dot I\,dt$. On the quantum side, we employ a fractional (Caputo) extension of a two-state dissipative model embedded by thermo-field dynamics (TFD), yielding a closed-form intrinsic entanglement component $b^{(α)}_{qe}(t)=\frac14[E_α(-λ^αt^α)]^2\sin^2(ωt)$ and an integrated revival measure $N_{qe}$ that delineates a sharp boundary near $α\simeq 1/2$ in the $(α,ω/λ)$ plane. On the classical side, we consider a three-state model whose Markov generator is promoted either to an exponential-kernel generalized master equation (with exact Markov embedding) or to a semi-Markov process with Erlang-2 waiting times. We quantify non-monotonicity by the entropy overshoot $ΔH$ and KL-based diagnostics on the probability simplex. To strengthen the quantum--classical symmetry, we further introduce a \emph{fractional Mittag--Leffler memory kernel} in the classical dynamics and show that an analogous backflow transition emerges around $α\simeq 1/2$, indicating that the boundary originates from the kernel's mathematical structure rather than from quantumness per se. Overall, our results provide a compact, model-agnostic route to classify non-Markovianity by phase diagrams of information backflow and to interpret them via a shared embedding narrative: memory stored in hidden degrees of freedom returns to the observed sector as non-monotonic information flow.

Figures (4)

• Figure 1: (Color online) Unified concept of information backflow. Left: embedding picture where memory is stored in hidden degrees of freedom and can return to the observed sector. Center: schematic time series of an information observable $I(t)$ highlighting intervals with $\dot I(t)>0$. Right: definition of the backflow functional $N_I=\int_{\dot I>0}\dot I\,dt$.
• Figure 2: (Color online) Quantum fractional two-state model. (a) Intrinsic entanglement component $b^{(\alpha)}_{qe}(t)$ for different fractional orders $\alpha$ at fixed $\omega/\lambda$, showing persistent revivals for smaller $\alpha$. (b) Phase diagram in the $(\alpha,\omega/\lambda)$ plane using $N_{qe}=\int_{\dot b_{qe}>0}\dot b_{qe}\,dt$, indicating a sharp boundary near $\alpha=1/2$ separating weak and strong revival regimes.
• Figure 3: (Color online) Classical three-state memory model. (a) Schematic three-state network and memory construction (exponential-kernel GME and Erlang-2 semi-Markov). (b) Phase map of entropy overshoot $\Delta H$ on the probability simplex for fixed memory parameters. (c) Comparison between exponential-kernel and Erlang-2 semi-Markov constructions, showing possible amplification of non-monotonicity and robustness via KL-based diagnostics.
• Figure 4: (Color online) Unified comparison of backflow phases. (a) Quantum parameter-space phase diagram (Fig. \ref{['fig:quantum']}b) placed alongside the classical simplex phase map (Fig. \ref{['fig:classical']}b). (b) Backflow functional $N_I$ computed with matched protocol choices ($I=b_{qe}$ for quantum, $I=H$ or $I=-D_{\rm KL}$ for classical), demonstrating unified classification by the total information-return measure.