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Structural Theory of Information Backflow in Non-Markovian Relaxation: TC/TCL Formalism and Minimal Phase Diagrams

Koichi Nakagawa

TL;DR

This work addresses how information backflow emerges in minimal non-Markovian relaxation and establishes structural conditions for monotonic relaxation. It fuses TC/TCL projection techniques with non-equilibrium thermo field dynamics (NETFD) to define a universal backflow functional $N_I$ and to map memory effects to observable transients within a doubled Hilbert space. Key contributions include generator-based monotonicity theorems for relative entropy under CP-divisible dynamics, a time-dependent GKSL check, and a NETFD-based decomposition of backflow into classical and intrinsic thermo-field entanglement components, plus a practical phase-diagram algorithm for minimal quantum and classical models (recovering Mittag-Leffler fractional relaxation as a universal envelope). This framework delivers model-independent diagnostics of memory-induced transients and a unified classification of overshoot and revival phenomena, with potential for experimental diagnostics of backflow and entanglement exchange.

Abstract

We develop a structural theory of information backflow in minimal non-Markovian relaxation processes within the framework of nonequilibrium statistical mechanics. The approach is based on the time-convolution (TC) and time-convolutionless (TCL) projection-operator formalisms for reduced dynamics and on the doubling construction of non-equilibrium thermo field dynamics, which provides an embedding representation of dissipative evolution. We introduce a general backflow functional associated with a time-dependent information measure and derive generator-based sufficient conditions for the absence of backflow in terms of divisibility properties and effective relaxation rates. This allows a direct connection between memory kernels in generalized master equations and observable transient phenomena such as entropy overshoot and revival. Furthermore, we propose a decomposition of backflow into classical mixing and intrinsic contributions in the doubled representation, leading to a unified classification of transient regimes. Minimal classical and quantum two-state models are analyzed as analytically tractable examples, yielding explicit phase diagrams and recovering Mittag-Leffler-type fractional relaxation as a universal envelope of non-Markovian damping. The framework provides a constructive TC-to-TCL procedure for extracting effective rates and organizing memory-induced phenomena in a model-independent manner.

Structural Theory of Information Backflow in Non-Markovian Relaxation: TC/TCL Formalism and Minimal Phase Diagrams

TL;DR

This work addresses how information backflow emerges in minimal non-Markovian relaxation and establishes structural conditions for monotonic relaxation. It fuses TC/TCL projection techniques with non-equilibrium thermo field dynamics (NETFD) to define a universal backflow functional and to map memory effects to observable transients within a doubled Hilbert space. Key contributions include generator-based monotonicity theorems for relative entropy under CP-divisible dynamics, a time-dependent GKSL check, and a NETFD-based decomposition of backflow into classical and intrinsic thermo-field entanglement components, plus a practical phase-diagram algorithm for minimal quantum and classical models (recovering Mittag-Leffler fractional relaxation as a universal envelope). This framework delivers model-independent diagnostics of memory-induced transients and a unified classification of overshoot and revival phenomena, with potential for experimental diagnostics of backflow and entanglement exchange.

Abstract

We develop a structural theory of information backflow in minimal non-Markovian relaxation processes within the framework of nonequilibrium statistical mechanics. The approach is based on the time-convolution (TC) and time-convolutionless (TCL) projection-operator formalisms for reduced dynamics and on the doubling construction of non-equilibrium thermo field dynamics, which provides an embedding representation of dissipative evolution. We introduce a general backflow functional associated with a time-dependent information measure and derive generator-based sufficient conditions for the absence of backflow in terms of divisibility properties and effective relaxation rates. This allows a direct connection between memory kernels in generalized master equations and observable transient phenomena such as entropy overshoot and revival. Furthermore, we propose a decomposition of backflow into classical mixing and intrinsic contributions in the doubled representation, leading to a unified classification of transient regimes. Minimal classical and quantum two-state models are analyzed as analytically tractable examples, yielding explicit phase diagrams and recovering Mittag-Leffler-type fractional relaxation as a universal envelope of non-Markovian damping. The framework provides a constructive TC-to-TCL procedure for extracting effective rates and organizing memory-induced phenomena in a model-independent manner.
Paper Structure (29 sections, 4 theorems, 33 equations, 4 figures)

This paper contains 29 sections, 4 theorems, 33 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries: TC and TCL Master Equations
  3. Projection operators and reduced dynamics
  4. Time-convolution (TC) form
  5. Time-convolutionless (TCL) form
  6. Interaction-picture TC/TCL equations in textbook form
  7. NETFD Correspondence and Embedding Interpretation
  8. Doubling construction
  9. Correspondence principle
  10. Information Backflow Functional
  11. General definition
  12. Interpretation
  13. Main Results: Structural Conditions for Monotonicity and Backflow
  14. Setting: TCL dynamics and reference states
  15. Backflow functional
  16. ...and 14 more sections

Key Result

Theorem 1

Assume that the TCL equation eq:TCL_main generates a family of dynamical maps $\Phi(t,0)$ that is completely positive and trace-preserving (CPTP) and CP-divisible: for every $t\ge s\ge 0$ there exists a CPTP map $\Phi(t,s)$ such that Let $\sigma$ be an invariant state: $\Phi(t,0)\sigma=\sigma$ for all $t$. Then the quantum relative entropy is monotone non-increasing: Consequently, for $I(t):=-D(

Figures (4)

  • Figure 1: Information backflow as temporary storage in hidden degrees of freedom and subsequent return to observables. Adapted from NakagawaBackflow2026.
  • Figure 2: Fractional non-Markovian two-state relaxation: intrinsic thermo-field entanglement sector and phase structure. Adapted from NakagawaBackflow2026.
  • Figure 3: Classical three-state non-Markovian relaxation: phase map of entropy/KL overshoot and amplification regimes. Adapted from NakagawaBackflow2026.
  • Figure 4: Markovian two-state relaxation: intrinsic component $b_{\mathrm{qe}}(t)$ in the thermo-field entanglement description. Adapted from NakagawaMarkovTFD2026.

Theorems & Definitions (10)

  • Definition 1: Backflow functional
  • Theorem 1: CP-divisible TCL implies monotonicity of relative entropy
  • proof
  • Theorem 2: Time-dependent Lindblad form implies $N_I=0$
  • proof
  • Proposition 1: Classical divisibility implies monotonic KL relaxation
  • proof
  • Theorem 3: Backflow decomposition via NETFD
  • Definition 2: Decomposed backflow
  • Remark 1