This preprint (Pillar 11 of the Gapped Quantum Systems Framework) presents an exact information-theoretic analysis of pure-dephasing channels coupled to a bath with a hard spectral gap. Core Findings:Using the exact dephasing function derived in our previous work, we rigorously prove that the spectral gap induces strictly positive non-Markovianity (information backflow) and causes the exact quantum capacity to exhibit damped oscillations around its asymptotic plateau. We also derive the exact quantum Fisher information and a novel identity linking the Rényi-2 entropy to the concurrence. Methodology & Rigor:The derivation relies entirely on exact analytical methods and established quantum information theorems (e.g., BDS quantum capacity theorem), without any fitting, phenomenological parameters, or circular reasoning. Reproducibility:The upload includes the PDF, LaTeX source, high-resolution figures, and self-contained Python scripts (test_pillar11.py, plot_pillar11.py) that validate every analytical claim to machine precision and generate the figures.
This preprint (Pillar 11 of the Gapped Quantum Systems Framework) presents an exact information-theoretic analysis of pure-dephasing channels coupled to a bath with a hard spectral gap. Core Findings:Using the exact dephasing function derived in our previous work, we rigorously prove that the spectral gap induces strictly positive non-Markovianity (information backflow) and causes the exact quantum capacity to exhibit damped oscillations around its asymptotic plateau. We also derive the exact quantum Fisher information and a novel identity linking the Rényi-2 entropy to the concurrence. Methodology & Rigor:The derivation relies entirely on exact analytical methods and established quantum information theorems (e.g., BDS quantum capacity theorem), without any fitting, phenomenological parameters, or circular reasoning. Reproducibility:The upload includes the PDF, LaTeX source, high-resolution figures, and self-contained Python scripts (test_pillar11.py, plot_pillar11.py) that validate every analytical claim to machine precision and generate the figures.
This manuscript (Pillar 13 of the unified gapped-bath framework) presents an exact derivation of the quantum thermodynamic properties of a central spin undergoing pure dephasing in a hard-gap bosonic bath. By strictly avoiding the Lindblad master equation and any phenomenological approximations, we prove that the spectral gap acts as a thermodynamic resource. The gap protects a finite residual ergotropy at long times, E∞=(ℏω0/2)exp(−J0/Δ)E∞=(ℏω0/2)exp(−J0/Δ), and caps entropy production strictly below the classical maximum. At the exact critical coupling J0=4Δ/π2J0=4Δ/π2, the residual ergotropy ratio is the pure mathematical number e−4/π2≈0.666787e−4/π2≈0.666787. We also demonstrate that pure dephasing induces zero heat exchange (Q=0Q=0), with the entropy saturating at exactly 0.9000460.900046 bits for J0=ΔJ0=Δ. The paper includes rigorous proofs, five falsifiable predictions, a comparison with standard frameworks (Lindblad, Caldeira-Leggett), and a fully reproducible Python validation script.
We establish a unified theoretical framework for the conservation of quantum information in gapped dephasing baths. Building on the exact tripartite complementarity relation V^2 + C^2 = 1 and the non-circular crossover criterion \Gamma = \gamma_{pre} t_{ML} = 1, we classify the dynamics into distinct universality classes determined by the infrared (IR) structure of the bath. We prove the "Hard Gap Theorem": an abrupt hard gap (s=0) strictly preserves local coherence at zero temperature, while any soft gap (s>0) generates an IR divergence that forces complete local decoherence, even at infinitesimal coupling. Furthermore, we derive the exact finite-temperature phase diagram. We show that the critical coupling scales as J_0^{crit} \propto \Delta^2 / (k_B T) at high temperatures, a result that corrects previous conjectures and unifies the thermodynamic behavior across all gap geometries. This theorem is shown to be mathematically equivalent to the generation of pure topological constants in other gapped sectors, including the Casimir effect (c_0 = 0.8343), massive Unruh radiation (e^{-2\Delta/a}), and the Bekenstein entropy bound (C_d k_B). This preprint serves as the unifying Pillar 6 of a five-part series on gapped quantum systems.
The flow of quantum information in local dephasing channels is analyzed over short and long times in case the structured reservoir of frequency modes exhibits a spectral gap in the density of modes over low frequencies. The presence of the low-frequency gap with upper cut-off frequency [Formula: see text] produces an infinite sequence of long-time intervals over which information backflow appears. Though generally irregular, the time intervals exhibit, under certain conditions, upper bounds: the [Formula: see text]th episode of information backflow has already started at the instant [Formula: see text], and already ended at the instant [Formula: see text], for every [Formula: see text]. The intervals become regular over long times, tend to the asymptotic length [Formula: see text], as the supremum value, and are described analytically in terms of the structure of the spectral density near the upper cut-off frequency of the spectral gap. Consequently, engineering structured reservoirs of frequency modes with a spectral gap over low frequencies produces in local dephasing channels regular sequences of information backflow and recoherence over long times, along with non-Markovian evolution.
The energy of the bosonic bath and the flow of quantum information in local dephasing channels are studied over short and long times in case the distribution of frequency modes of the bosonic bath exhibits a low-frequency gap. The initial conditions consist in special correlations between the qubit and the bosonic bath or are factorized, and involve thermal states of the whole system or of the bath. The low-frequency gap generates damped oscillations of the bath energy around the asymptotic value, for the correlated initial conditions, and induces the open system to alternately loose and gain information, for the factorized initial configurations. The long-time oscillations of the bath energy become regular and the frequency of the oscillations coincides with the upper cut-off frequency of the spectral gap. Regular long-time sequences of intervals are found over which the bath energy increases (decreases), for the correlated initial conditions, and information is lost (gained) by the open system, for the factorized initial configurations, even at different temperatures. This relation is reversed, if compared to the one obtained without the low-frequency gap, and can fail if the spectral density is tailored near the spectral gap according to power laws with odd natural powers.
We demonstrate that the topological entanglement entropy (γ = log D) is a pure universal number that strictly requires a hard gap (s=0) to exist. Building on the Hard Gap Criterion established in "The Hard Gap Criterion: Universality Classes and Thermodynamic Scaling in Gapped Dephasing Baths", we prove that a soft gap (s ≥ 1) induces anyon proliferation, destroys the topological order, and drives γ → 0. This establishes that the conservation of many-body quantum information is not just a property of the Hamiltonian, but a direct consequence of the infrared spectral structure of the environment. We show that γ is the many-body analogue of the entanglement concurrence C in the conservation law V² + C² = 1 established in "Conservation of Quantum Information in Gapped Baths: The Duality-Depleted Regime". The area law term represents local decoherence, while the topological constant represents globally protected information. The derivation uses only established theorems (the area law of entanglement entropy, the Kitaev-Preskill topological correction, and the algebraic theory of anyons) and contains no fitting, no adjustment, and no circularity. We provide exact numerical validation via the stabilizer formalism, computing the exact pure numbers for the Toric code (log 2 ≈ 0.693147), the Ising model (log 2), and the Fibonacci model (≈ 0.642965). The exact diagonalization confirms that the topological ground state is exactly pure (total stabilizer rank equals the number of qubits), validating the perfect delocalization of quantum information. We state four falsifiable predictions and provide a complete, reproducible Python script.
We explore signatures of the non-Markovianity in the time-resolved energy transfer processes for quantum open systems. Focusing on typical systems such as the exact solvable damped Jaynes-Cummings model and the general spin-boson model, we establish quantitative links between the time-resolved energy current and the symmetric logarithmic derivative quantum Fisher information (SLD-QFI) flow, one of measures quantifying the non-Markovianity, within the framework of non-Markovian master equations in time-local forms. From the relationships, we find in the damped Jaynes-Cummings model that the SLD-QFI backflow from the reservoir to the system always correlates with an energy backflow, thus we can directly witness the non-Markovianity from the dynamics of the energy current. In the spin-boson model, the relation is built on the rotating-wave approximation, calibrated against exact numerical results, and proven reliable in the weak coupling regime. We demonstrate that whether the non-Markovianity guarantees the occurrence of an energy backflow depends on the bath spectral function. For the Ohmic and sub-Ohmic cases, we show that no energy backflow occurs and the energy current always flow out of the system even in the non-Markovian regime. While in the super-Ohmic case, we observe that the non-Markovian dynamics can induce an energy backflow.