Irreducibility, Second-Law Justification, and Characterization of Coherence Work in Open Quantum Systems

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.