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

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 $b^{(α)}{qe}(t)=\frac14[Eα(-λ^αt^α)]^2\sin^2(ωt)$ 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.