Information Backflow Diagrams Unify Entanglement Revivals and Entropy Overshoots in Models

Researchers are increasingly focused on understanding memory effects within systems exhibiting non-Markovian dynamics, typically identifying them through either entanglement revivals or entropy overshoots. Koichi Nakagawa from Hoshi University, Tokyo, Japan, alongside colleagues, present a novel information-backflow diagram approach that elegantly unifies these previously disparate phenomena within minimal models. This work establishes a common functional to analyse both entanglement and classical information, utilising fractional extensions of dissipative models and semi-Markov processes to delineate a critical boundary governing information flow. By demonstrating an analogous backflow emerges from the mathematical structure of the memory kernel itself, rather than the system’s parameters, Nakagawa et al. provide a compact and model-agnostic framework for classifying non-Markovianity and interpreting the return of ‘memory’ as non-monotonic information.

The research, published on January 28, 2026, introduces a functional, NI, that quantifies the return of information to an observed sector, revealing a shared embedding narrative where memory stored in hidden degrees of freedom manifests as non-monotonic information flow. Researchers achieved this by meticulously comparing two analytically tractable minimal models, a fractional two-state dissipative model embedded within thermo-field dynamics (TFD) on the quantum side, and a three-state classical model with memory introduced via generalized master equations or semi-Markov processes. This work establishes a compact, model-agnostic route to classify non-Markovianity, moving beyond discipline-specific diagnostics and towards a unified understanding of memory effects.

The study unveils a backflow functional, NI, defined as the integral of information-like observables I(t) over time intervals where I(t) is positive, effectively measuring the total return of information. This functional is expressed mathematically as NI ≡ Z ∞ 0 Θ( I(t)) I(t) dt, where Θ is the Heaviside step function, ensuring integration only occurs when information increases. The team proposes that non-Markovianity arises from an embedding principle, where memory is explicitly encoded in hidden degrees of freedom and subsequently returns to the observed system, creating the observed non-monotonic behaviour. By embedding both quantum and classical dynamics into larger Markovian frameworks, TFD for the quantum case and auxiliary variables for the classical case, the researchers demonstrate that information backflow corresponds to information temporarily stored in these hidden degrees of freedom.
On the quantum side, scientists employed a fractional (Caputo) extension of a two-state relaxation model, yielding a closed-form intrinsic entanglement component, b(α)qe(t) = 1 4[Eα(−λαtα)]2 sin2(ωt), where α governs the fractional derivative, λ defines the dissipative scale, and ω controls coherent oscillations. This expression clearly shows how non-Markovian memory enters through the slowly decaying Mittag, Leffler function, Eα, while oscillations provide opportunities for repeated revivals. An integrated revival measurement could therefore quantify the degree of non-Markovianity.

Information Backflow and Non-Markovianity Quantification via TFD reveal

Scientists investigated memory effects in non-Markovian dynamics by proposing a novel information-backflow diagram approach, unifying entanglement revivals and classical entropy overshoots through a common backflow functional. The study pioneered a method to quantify non-Markovianity by integrating time intervals where information-like observables increase, denoted as NI, effectively measuring the return of information to the observed system. Researchers engineered a fractional (Caputo) extension of a two-state dissipative model, embedding it within thermo-field dynamics (TFD) to yield a closed-form intrinsic entanglement component and a revival measure, delineating a sharp boundary near α = 1/2 in the parameter plane. The team employed a three-state classical model, promoting its Markov generator to either an exponential-kernel generalized master equation, allowing for exact Markov embedding, or a semi-Markov process utilising Erlang-2 waiting times.
To quantify non-monotonicity, scientists calculated the entropy overshoot and employed Kullback-Leibler (KL) divergence-based diagnostics on the probability simplex. Experiments harnessed a fractional Mittag, Leffler memory kernel within the classical dynamics, revealing an analogous backflow emergence around α = 1/2, suggesting the boundary’s origin lies within the kernel’s mathematical structure rather than the parameter α itself. This work developed the backflow functional, expressed as NI ≡ ∫₀ ∞ Θ(I(t)) I(t) dt, where I(t) represents an information-like observable, such as quantum correlation, Shannon entropy, or KL divergence, derived from the observed state. The approach enables a precise quantification of information return by summing the increases in I(t) over time, effectively capturing the extent of memory effects.

Researchers embedded the observed non-Markovian dynamics into higher-dimensional Markovian dynamics, utilising both exponential-kernel generalized master equations and thermo-field dynamics to explicitly represent memory storage in auxiliary degrees of freedom. Specifically, the quantum model’s intrinsic entanglement component, b(α) qe (t) = 1/4 [E α (−λαt α )] 2 sin 2 (ωt), demonstrates how the Mittag, Leffler function E α encodes long-tailed memory, sustaining repeated revivals governed by the parameter α. The integrated positive-slope measurement reveals sustained quantum coherence.

Information Backflow Unifies Quantum and Classical Revivals

Scientists have developed a novel framework for classifying non-Markovianity, revealing a shared mechanism underlying both quantum and classical systems. The research, published recently, introduces an information-backflow diagram approach that unifies seemingly disparate phenomena, entanglement revivals in quantum mechanics and classical entropy overshoots, through a common functional. Experiments utilising a fractional (Caputo) extension of a two-state dissipative model, embedded within thermo-field dynamics (TFD), yielded a closed-form intrinsic entanglement component, demonstrating persistent revivals governed by a Mittag, Leffler memory envelope. Measurements confirm a sharp boundary appears near α = 1/2 in the parameter plane, delineating a critical point in the system’s behaviour.

The team measured information flow using a newly defined functional, NI, quantifying the total return of information to the observed sector by integrating only time intervals where the information-like observable, I(t), increases. Results demonstrate that NI = 0 when I(t) is monotone non-increasing, and NI 0 signifies the return of information, effectively capturing the essence of non-Markovian memory. On the classical side, a three-state model was investigated, with memory introduced via either an exponential-kernel generalized master equation or a semi-Markov process with Erlang-2 waiting times. Scientists recorded the entropy overshoot, ∆H, and employed Kullback-Leibler (KL) divergence-based diagnostics on the probability simplex to quantify non-monotonicity.

To strengthen the quantum, classical symmetry, a fractional Mittag, Leffler memory kernel was introduced into the classical dynamics, and remarkably, an analogous backflow transition emerged around α ≃ 1/2. This finding indicates the boundary isn’t solely a quantum phenomenon, but rather originates from the mathematical structure of the memory kernel itself. The study’s core achievement lies in providing a compact, model-agnostic route to classify non-Markovianity via phase diagrams of information backflow, interpreting these diagrams through a shared embedding narrative. Specifically, the intrinsic entanglement component, b(α)qe(t), was calculated to be 1/4 [Eα(−λαtα)]2 sin2(ωt), where λ governs the dissipative scale and ω controls coherent oscillations. The integrated positive-slope measure quantifies the degree of non-Markovianity.

Information Backflow Unifies Quantum and Classical Dynamics

Scientists have developed a novel information-backflow phase-diagram approach that unifies the understanding of quantum entanglement revivals and classical entropy overshoots within non-Markovian models. This work establishes a common ground for interpreting these seemingly disparate phenomena through a shared embedding mechanism, where information is stored in hidden degrees of freedom and subsequently returns to the observed sector. The research introduces a single operational quantity, the backflow functional, to quantify this information return, offering a compact and model-agnostic method for classifying non-Markovianity. By employing both a fractional two-state dissipative model in quantum mechanics and a three-state model in classical dynamics, the authors demonstrate that the observed boundary between Markovian and non-Markovian behaviour originates from the mathematical structure of the memory kernel, specifically the Mittag, Leffler kernel, rather than being an inherent property of the system itself. The authors acknowledge that their analysis relies on specific model choices and parameter ranges, potentially limiting the direct applicability to all non-Markovian systems. Future research could explore the extension of this framework to more complex models and investigate the role of different memory kernels in shaping the information backflow dynamics.