Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Exact Quantum Thermodynamics of Hard-Gap Dephasing: Residual Ergotropy and Entropy Production without Lindblad

Ali Bourizk

Published Oct 30, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.5281/zenodo.23788027

Related papers

Gap-Induced Non-Markovianity and Oscillatory Quantum Capacity in Dephasing Channels

Ali Bourizk

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.

•Oct 30, 2026•0 citations

The Hard Gap Criterion: Universality Classes and Thermodynamic Scaling in Gapped Dephasing Baths

Ali Bourizk

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.

•Oct 23, 2026•0 citations

Exact Dephasing Entropy of Hawking Radiation in a Gapped Bath: An Open Quantum System Derivation of the Rising Page Curve

Ali Bourizk

We study the entanglement entropy of Hawking radiation using the open quantum systems framework. For a vacuum with a spectral gap ΔΔ, the radiation entropy saturates at a sub-maximal value, leaving a residual visibility (η(0)≈0.9000) that protects quantum information. This provides an analytical description of the rising Page curve.

•Oct 28, 2026•0 citations

Thermodynamics of a qubit undergoing dephasing

Stefano Marcantoni

The thermodynamics of a qubit undergoing dephasing due to the coupling with the external environment is discussed. First of all, we assume the dynamics of the system to be described by a master equation in Lindblad form. In this framework, we review a standard formulation of the first and second law of thermodynamics that has been known in literature for a long time. After that, we explicitly model the environment with a set of quantum harmonic oscillators choosing the interaction such that the global dynamics of system and bath is analytically solvable and the Lindblad master equation is recovered in the weak-coupling limit. In this generalized setting, we can show that the correlations between system and bath play a fundamental role in the heat exchange. Moreover, the internal entropy production of the qubit is proven to be positive for arbitrary coupling strength.

Journal of Physics Conference Series•May 1, 2017•4 citations

Soft Gaps Destroy Quantum Information: Topological Protection as a Hard Gap Phenomenon

Ali Bourizk

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.

•Oct 27, 2026•0 citations

Finite-Bath Open Quantum Systems: Exact Dynamics

Devvrat Tiwari, Subhashish Banerjee

In this work, we introduce a method for deriving exact master equations from the dynamical map for finite open quantum systems coupled to (in)finite reservoirs, using the principle of minimal dissipation. The exact dynamics of the central spin model, which models a finite-bath open quantum system, is developed for two interaction types: Heisenberg and stochastic pure-dephasing interactions. The Heisenberg interaction yields a novel phase-covariant quantum channel in the strong-coupling regime, offering a new platform for studying a range of quantum information protocols. The stochastic pure-dephasing interaction provides the microscopic derivation of the paradigmatic non-Markovian random telegraph noise (RTN) channel, establishing its quantum foundation and offering insight into stochastic couplings. We derive the closed-form master equations for both models. As a demonstration, we explore the thermodynamic performance of these systems as quantum batteries. A direct relationship between quantum heat current and charging power is revealed, and RTN quantum batteries are shown to have advantages in charge storage.

quant-pharXiv:2511.07180•The Journal of Chemical Physics•Nov 11, 2025•0 citations
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 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 thermodynamics of a qubit undergoing dephasing due to the coupling with the external environment is discussed. First of all, we assume the dynamics of the system to be described by a master equation in Lindblad form. In this framework, we review a standard formulation of the first and second law of thermodynamics that has been known in literature for a long time. After that, we explicitly model the environment with a set of quantum harmonic oscillators choosing the interaction such that the global dynamics of system and bath is analytically solvable and the Lindblad master equation is recovered in the weak-coupling limit. In this generalized setting, we can show that the correlations between system and bath play a fundamental role in the heat exchange. Moreover, the internal entropy production of the qubit is proven to be positive for arbitrary coupling strength.
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.
In this work, we introduce a method for deriving exact master equations from the dynamical map for finite open quantum systems coupled to (in)finite reservoirs, using the principle of minimal dissipation. The exact dynamics of the central spin model, which models a finite-bath open quantum system, is developed for two interaction types: Heisenberg and stochastic pure-dephasing interactions. The Heisenberg interaction yields a novel phase-covariant quantum channel in the strong-coupling regime, offering a new platform for studying a range of quantum information protocols. The stochastic pure-dephasing interaction provides the microscopic derivation of the paradigmatic non-Markovian random telegraph noise (RTN) channel, establishing its quantum foundation and offering insight into stochastic couplings. We derive the closed-form master equations for both models. As a demonstration, we explore the thermodynamic performance of these systems as quantum batteries. A direct relationship between quantum heat current and charging power is revealed, and RTN quantum batteries are shown to have advantages in charge storage.