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.
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.
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 complete the Hard Gap Criterion of Pillar 6 by deriving, in exact closed form, the dephasing plateau for a soft gap with an arbitrary exponent ss. For 0<s<10<s<1, we prove that the plateau is: Γ∞(s)=J0Δ⋅πssin(πs)Γ∞(s)=ΔJ0⋅sin(πs)πs a universal function of the gap exponent alone, directly related to Euler's reflection formula for the Gamma function. This yields the exact critical coupling: J0(crit)(s)=Δ⋅sin(πs)πsJ0(crit)(s)=Δ⋅πssin(πs) which recovers the hard-gap result J0(crit)(0)=ΔJ0(crit)(0)=Δ of Pillar 6 as a limit and vanishes as s→1−s→1−. For s=1s=1 and s>1s>1 we show that the plateau diverges in the ultraviolet as ln(Λ/Δ)ln(Λ/Δ) and (Λ/Δ)s−1/(s−1)(Λ/Δ)s−1/(s−1), respectively, and we prove that the short-time dephasing rate γpreγpre diverges for any s>0s>0. This establishes that the plateau criterion is the only well-defined criterion for soft gaps. All results are derived algebraically, with no fitting, no adjustable parameters, and no circularity. We provide a complete self-contained Python validation script in the repository.
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.
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.
We study a qubit coupled to a bosonic bath described by a fourth-order scalar field Lagrangian, whose dispersion relation omega^2 = c^2(k^4 + alpha*k^2 + m^2) supports a tunable spectral gap. Below the gap, the local density of states vanishes exactly, suppressing both energy relaxation (T1) and pure dephasing (T2) at leading order. We derive the canonical quantization of the theory from first principles, establish an exact closed-form expression for the local density of states, and map out a phase diagram in (alpha, m^2) parameter space with a sharp gap-closing transition at m^2 = alpha^2/4. At the critical line, the momentum coupling (partial_t phi) uniquely yields an Ohmic bath, placing the system at the spin-boson quantum phase transition boundary, while direct field and gradient couplings produce infrared-divergent spectral functions due to the universal 1/sqrt(omega_k) mode normalization of soft critical modes. The framework provides a concrete design principle for spectral-gap-based qubit protection and connects fourth-order field theory to the spin-boson literature.
Starting from the thermodynamic limit of information recording, this paper derives a complete, universal decoherence formula consisting of three linked equations: 1. A **non-zero lower bound on the decoherence time** of a quantum system: `τ_min = (ℏ / k_B T) · (1 / ln N_eff)`, where `N_eff` is the number of effective environmental degrees of freedom participating in the irreversible recording of the system's information. 2. A **quantitative criterion for the macroscopic–microscopic boundary**: `N_eff^(c) = exp( ℏ / (k_B T · T_sys) )`, where `T_sys` is the characteristic evolution time of the system. When `N_eff > N_eff^(c)`, decoherence completes before a single coherent evolution, and classical behavior emerges. 3. A **thermodynamic lower bound on interference fringe visibility**: `V ≥ exp( − ln N_eff / ln N_eff^(c) )`, derived from the first two equations without additional assumptions. The first equation predicts a *saturation effect*: the decoherence time no longer shortens indefinitely as environmental coupling increases. The second shows that for nearly all laboratory-accessible systems, `N_eff^(c) ≈ 1`, meaning any non-zero environmental coupling suffices to trigger the transition—explaining why macroscopic objects are inevitably classical. The third provides a directly measurable observable for the continuous quantum–classical phase transition. The framework is universal, applies to any quantum system in contact with an environment at temperature `T`, and its three core predictions are independently falsifiable. Based on the boundary criterion, wave–particle duality in double-slit experiments is reinterpreted as a special case of this phase transition occurring under the limiting condition `N_eff^(c) ≈ 1`. Three experimental schemes—a decoherence-saturation ratio test, a temperature-dependence test of the boundary criterion, and a direct observation of the continuous wave–particle transition via ultrafast electron diffraction or circuit QED—are proposed, all feasible on existing quantum information platforms without requiring new equipment.