Open quantum systems admit two natural definitions of heat: the standard formulation δQ_std = Tr[H dρ] and the entropy-basis formulation δQ_ent = Σ_k dλ_k⟨λ_k|H|λ_k⟩. Their difference W* = δQ_std − δQ_ent is a coherence work term whose thermodynamic status is the subject of this paper. We prove three results. (I) Theorem C (irreducibility): there is no function f: ℝ → ℝ such that δQ_ent + f(W*) satisfies the Clausius equality T dS_vN for all density matrices, proved via a Jacobian rank argument with det J = 3(γ+γ_+)²(15+16 ln 2)/320 ≠ 0 for the complete L₋+L₊ Lindblad system. W* is therefore an irreducible thermodynamic quantity. (II) Theorem N (secular-free second law): the entropy production σ_new = −Tr[D[ρ](log ρ − log ρ_ss)] ≥ 0 requires only complete positivity, not the secular approximation. A concrete example demonstrates σ_std = −0.037 < 0 while σ_new = +0.083 > 0 in the non-secular regime. (III) Theorem H (characterization): W*(D[ρ]) = 0 for all ρ if and only if the Bloch transformation satisfies A_xx=A_yy=A_zz, all cross-axis entries zero, A_xy=−A_yx, and b=0 — equivalent to T₁=T₂ in NMR/quantum-optics language, strictly stronger than the secular condition (T₂ ≤ 2T₁). The three-way identity σ_new = σ_ent − W*/T + Tr[D[ρ](log ρ_ss + βH)], valid for any β, reveals W*/T = σ_ent − σ_std as an experimentally accessible quantity.
Open quantum systems admit two natural definitions of heat: the standard formulation δQ_std = Tr[H dρ] and the entropy-basis formulation δQ_ent = Σ_k dλ_k⟨λ_k|H|λ_k⟩. Their difference W* = δQ_std − δQ_ent is a coherence work term whose thermodynamic status is the subject of this paper. We prove three results. (I) Theorem C (irreducibility): there is no function f: ℝ → ℝ such that δQ_ent + f(W*) satisfies the Clausius equality T dS_vN for all density matrices, proved via a Jacobian rank argument with det J = 3(γ+γ_+)²(15+16 ln 2)/320 ≠ 0 for the complete L₋+L₊ Lindblad system. W* is therefore an irreducible thermodynamic quantity. (II) Theorem N (secular-free second law): the entropy production σ_new = −Tr[D[ρ](log ρ − log ρ_ss)] ≥ 0 requires only complete positivity, not the secular approximation. A concrete example demonstrates σ_std = −0.037 < 0 while σ_new = +0.083 > 0 in the non-secular regime. (III) Theorem H (characterization): W*(D[ρ]) = 0 for all ρ if and only if the Bloch transformation satisfies A_xx=A_yy=A_zz, all cross-axis entries zero, A_xy=−A_yx, and b=0 — equivalent to T₁=T₂ in NMR/quantum-optics language, strictly stronger than the secular condition (T₂ ≤ 2T₁). The three-way identity σ_new = σ_ent − W*/T + Tr[D[ρ](log ρ_ss + βH)], valid for any β, reveals W*/T = σ_ent − σ_std as an experimentally accessible quantity.
The Cartan decomposition su(d) = h ⊕ n encodes a fundamental thermodynamic separation: coherence work W* = δQ_std − δQ_ent (eigenvector-projector rotation) responds exclusively to root-space perturbations (F_α ∈ n), while the entropy-production gap g responds exclusively to Cartan perturbations (F_β ∈ h). This thermodynamic orthogonality is a structural principle from which three results follow non-perturbatively and without secular conditions. Theorem O (thermodynamic orthogonality): The sub-Jacobian of ρ ↦ (W*, g) at ρ₀ = I/d is block-antidiagonal for any d ≥ 2, with det J_sub ≠ 0 iff the dissipator is non-isotropic. Theorem C (irreducibility, from Thm. O): No f: ℝ → ℝ satisfies δQ_ent + f(W*) = T dS_vN for all ρ. Coherence work is an irreducible thermodynamic quantity. Theorem N (second law without secular approximation): σ_new = −Tr[D[ρ](log ρ − log ρ_ss)] ≥ 0 for any GKSL generator; σ_new is the Cartan-orthogonality-invariant measure of irreversibility, with W*/T = σ_ent − σ_std experimentally accessible. Theorem H_d (trivial case of Thm. O): det J_sub = 0, equivalently W* = 0 for all ρ, if and only if A = c I_{d²−1}, b = 0 (T₁ = T₂ for d = 2) — the isotropic dissipator, the unique case in which the Cartan structure imposes no thermodynamic separation. In summary, Theorem O yields three corollaries: (I) W* is irreducible (Theorem C); (II) a non-negative entropy production σ_new ≥ 0 exists within the GKSL framework, without secular conditions (Theorem N); (III) W* = 0 for all ρ iff T₁ = T₂ (Theorem H_d).
This work introduces a unified dynamical law of irreversibility that links three fundamental quantities in open quantum systems: coherence change, information flux, and irreversible entropy production.For finite-dimensional systems evolving under completely positive trace-preserving (CPTP) Markovian semigroups, we prove the exact identity κ˙(t)=Φinfo(t)−ΔS(t),\dot{\kappa}(t) = \Phi_{\text{info}}(t) - \Delta S(t),κ˙(t)=Φinfo(t)−ΔS(t), where κ(t)=S(ρdiag)−S(ρ)\kappa(t) = S(\rho_{\text{diag}}) - S(\rho)κ(t)=S(ρdiag)−S(ρ) is the entropy-valued coherence, Φinfo(t)=−ddtS(ρt∥ν)\Phi_{\text{info}}(t) = -\frac{d}{dt} S(\rho_t \Vert \nu)Φinfo(t)=−dtdS(ρt∥ν) is the information flux relative to a canonical maximum-entropy reference state ν\nuν, and ΔS(t)=Tr[L(ρt)(logρt−logσ)]≥0\Delta S(t) = \text{Tr}[ \mathcal{L}(\rho_t)(\log \rho_t - \log \sigma)] \ge 0ΔS(t)=Tr[L(ρt)(logρt−logσ)]≥0 is the Spohn entropy production rate. The result establishes an exact entropy-balance identity connecting quantum coherence, thermodynamic irreversibility, and information flow — all expressed in consistent entropy-per-time units.Operational protocols are provided for experimental validation using modern quantum platforms (e.g., superconducting qubits, trapped ions, or cavity QED).A residual test, R(t)=κ˙−Φinfo+ΔSR(t) = \dot{\kappa} - \Phi_{\text{info}} + \Delta SR(t)=κ˙−Φinfo+ΔS, offers a falsifiable criterion for detecting non-Markovianity or hidden memory flux. The framework unifies information geometry, resource-theoretic coherence, and nonequilibrium thermodynamics into a single experimentally testable law, illuminating how quantum structure can be sustained only when information inflow compensates entropy production. Keywords:Quantum thermodynamics, information theory, open quantum systems, coherence, entropy production, relative entropy, Lindblad dynamics, irreversibility, information geometry.
We present a systematic thermodynamic analysis of nonsecular GKLS master equations. When some energy transitions in the system are comparable with the dissipative rate, the secular approximation is not valid, and the positivity of the dynamics must be ensured via other approximations. The dynamics then presents features reminiscent from strong coupling to the environment: e.g. the equilibrium state with an environment at thermal equilibrium is not a Gibbs state of the local system Hamiltonian and the dynamics couples energy coherences and populations. Nonetheless, we build a unified, consistent thermodynamic framework for those dynamics. We focus on two approximations beyond the full secular approximation (the local and the more accurate geometric-arithmetic approximations), and then employ systematic perturbation theory on microscopic expressions for the first and second law of thermodynamics which relate irreversibility to the increase of system-environment correlations. We show that, in spite of the weak system-bath coupling, the system-bath interaction energy participates to the energy balance, and is directly linked to the Lamb-shift Hamiltonian. We compare this microscopic entropy production with the definition based on the contractivity of the reduced system dynamics (Spohn inequality). We show that, unlike for secular master equations, the two entropy production rates differ because of the presence of non-vanishing stationary coherences in the energy eigenbasis. However, in the case of a single thermal bath, the difference is purely transient, and no work can be cyclically extracted from the steady-state despite its non-Gibbs form. Finally, we illustrate our results with simple examples, clarifying and completing the thermodynamic picture of Markovian dynamics in the quantum regime.
A consistent theory of non-equilibrium thermodynamics for Markovian open quantum systems has been developed in the late seventies in analogy with Classical Irreversible Thermodynamics. The time-evolution of these open systems is usually described by means of effective master equations in Lindblad form, that turn out to be reliable when there is a separation of time-scales between system and environment, such that memory effects are negligible and the so-called Markovian approximation is justified. In this framework, the variations of energy and entropy in the system are consistently described, distinguishing between heat and work contributions and providing a statement of the second law of thermodynamics as positivity of the entropy production.\nHowever, there is empirical evidence that many physical systems like photosynthetic complexes,\nopto-mechanical resonators and superconducting qubits, just to mention a few, experience more general non-Markovian dynamics. The formulation of the laws of thermodynamics in a non-Markovian setting is matter of current research and represents the main topic of the present work. As a first step, we show that the entropy production defined as in the Markovian case can be negative for a class of non-Markovian dynamics. We argue that this outcome should not be interpreted as a violation of the second law of thermodynamics because the environment must be explicitly taken into account in the balance of entropy in the non-Markovian setting. In order to justify this claim we adopt a more general point of view, studying a closed bipartite quantum system, such that each of the two subsystems plays the role of a finite environment out of equilibrium for the other one. We concentrate on the balance of energy first and construct an effective Hamiltonian for each subsystem using physically reasonable requirements; then we define heat and work as in the standard Markovian treatment, with the effective Hamiltonian replacing the free Hamiltonian. It turns out that, in our framework, the work power is perfectly balanced between subsystems, while the correlations can store a part of energy locally inaccessible and exchange it with both subsystems in the form of heat.\nConcerning the balance of entropy, a quite general formulation of the second law of thermodynamics can be given as follows: under the assumption of a factorized initial state for the compound system, the sum of the total variations of the entropies in the two subsystems is always nonnegative. We show with an explicit example that this general formulation does not correspond to the statement presented in the framework of Markovian master equations, which should not be considered a priori the second law of thermodynamics.\nIn the last part of the thesis we concentrate of the so-called fluctuation relations, that are results extending the thermodynamic formalism beyond the behavior of average quantities. After reviewing the main theoretical outcomes, such as the Jarzynski equality and the Crooks fluctuation theorem, we describe a proposal to access experimentally the work performed on an ensemble of diatomic molecules by a time-dependent electric field coupled with their vibrational degree of freedom. This procedure could then be used to test the quantum Jarzynski equality. With respect to the results so far appeared in the literature, in which the left-hand side of the equality is inferred from an experiment and the right-hand side is computed according to a model, in our proposed setting we should be able to estimate from the experiment both the left-hand side and the right-hand side of the equality, independently.
Open quantum systems — those coupled to external baths, reservoirs, or measurement apparatuses — sit at the intersection of thermodynamics, information theory, and non-equilibrium dynamics. A candidate structural pattern emerges from recent preprints across quant-ph, math-ph, and cond-mat.stat-mech: the thermodynamic behavior of open quantum systems is more richly constrained than secular (Lindblad) approximations suggest, and this additional structure manifests simultaneously in (i) the steady-state coherences and entropy production of nonsecular master equations, (ii) the noise signatures accessible near Lindbladian exceptional points, (iii) multi-time quantum correlations that survive thermalization and carry pre-collapse information, (iv) the engineering of bosonic mode symmetries that trade error-protection for logical controllability, and (v) the conserved-charge counting that demarcates quantum integrability from chaos. This synthesis is a *heuristic reading*, not a derivation: the five threads share vocabulary (coherence, entropy, symmetry, information) but do not yet share a unified formal structure. The unifying hypothesis is that *steady-state thermodynamic observables systematically undercount the information retained in open quantum dynamics*, and that the deficit is recoverable through higher-order temporal and spectral probes. Falsification path: if current-noise spectra near a Lindbladian exceptional point show no anomalous time-delay structure beyond what secular theory predicts, and if multi-time correlators in evaporating black-hole analogs reduce exactly to single-time thermal distributions, the hypothesis collapses. Sources span quant-ph and cond-mat.stat-mech preprints from June 2026; all are v1 preprints, not peer-reviewed. --- Authorship: Saluca Agentic AI Research Team (Saluca LLC). AI-drafted from arXiv preprint corpus on the date in the filename. Cited arXiv preprints: 2606.09790v1, 2606.09805v1, 2606.10141v1, 2606.10744v1, 2606.11010v1, 2606.11392v1, 2606.11991v1, 2606.12161v1, 2606.12173v1, 2606.12310v1, 2606.12313v1, 2606.13251v1, 2606.13377v1, 2606.13383v1, 2606.13404v1, 2606.13453v1, 2606.13504v1
In a series of papers, we intend to take the perspective of open quantum systems and examine from their nonequilibrium dynamics the conditions when the physical quantities, their relations, and the laws of thermodynamics become well defined and viable for quantum many-body systems. We first describe how an open-system nonequilibrium dynamics (ONEq) approach is different from the closed combined system + environment in a global thermal state (CGTs) setup. Only after the open system equilibrates will it be amenable to conventional thermodynamics descriptions, thus quantum thermodynamics (QTD) comes at the end rather than assumed in the beginning. The linkage between the two comes from the reduced density matrix of ONEq in that stage having the same form as that of the system in the CGTs. We see the open-system approach having the advantage of dealing with nonequilibrium processes as many experiments in the near future will call for. Because it spells out the conditions of QTD's existence, it can also aid us in addressing the basic issues in quantum thermodynamics from first principles in a systematic way. We then study one broad class of open quantum systems where the full nonequilibrium dynamics can be solved exactly, that of the quantum Brownian motion of N strongly coupled harmonic oscillators, interacting strongly with a scalar-field environment. In this paper, we focus on the internal energy, heat capacity, and the third law. We show for this class of physical models, amongst other findings, the extensive property of the internal energy, the positivity of the heat capacity, and the validity of the third law from the perspective of the behavior of the heat capacity toward zero temperature. These conclusions obtained from exact solutions and quantitative analysis clearly disprove claims of negative specific heat in such systems and dispel allegations that in such systems the validity of the third law of thermodynamics relies on quantum entanglement. They are conceptually and factually unrelated issues. Entropy and entanglement will be the main theme of our second paper on this subject matter.