We study the spectral excitation functional D(C ∥χ) := −R ∞0 C(ω)log Fβ(ω) dω,where Fβ(ω) = (1 − e−βω)/(βω) is the Bogoliubov–Kubo–Mori (BKM) thermalweight and C is a positive-frequency spectral density. Three new results, allproved rigorously and verified numerically, are:(i) Exact closed-form evolution. Under KMS Lindblad dynamics, the fulltime trajectory of D is D(Ct ∥ χt) = Deq +Pk(nk(0) − n¯k) w(ωk) e−Γkt,where w(ω) = − log Fβ(ω) and Γk = γk(2n¯k + 1). This is not merely amonotonicity statement; it gives the trajectory in closed form.(ii) Bernstein characterization. The function t 7→ D(Ct ∥ χt) − Deq iscompletely monotone if and only if every mode is initially over-occupiedrelative to thermal equilibrium. This links the conditional monotonicityof D to the Bernstein–Widder theorem for Laplace transforms of positivemeasures.(iii) Spectral H-theorem. The weighted L1Pimbalance functional F(t) :=k|nk(t) − n¯k| w(ωk) is universally non-increasing under KMS Lindbladdynamics, decays exponentially with rate Γmin, and satisfies |D(Ct ∥χt) −Deq| ≤ F(t) for all t ≥ 0. This resolves the open question of whethera universally monotone spectral functional exists and characterises theprecise sense in which D can violate monotonicity.We also establish self-containedly: the Todd identification F−1β = td(βω), theBose–Einstein decomposition of D, the Laplace harmonic series, the completePetz information-loss chain with energy-independent bounds, the conditionalmonotonicity theorem in sharp form with an explicit two-mode counterexample,and the large-N limiting density (the folded semicircle on [0, 2√2] with meanenergy 8√2/(3π)). All results are verified numerically to machine precision.
We study the spectral excitation functional D(C ∥χ) := −R ∞0 C(ω)log Fβ(ω) dω,where Fβ(ω) = (1 − e−βω)/(βω) is the Bogoliubov–Kubo–Mori (BKM) thermalweight and C is a positive-frequency spectral density. Three new results, allproved rigorously and verified numerically, are:(i) Exact closed-form evolution. Under KMS Lindblad dynamics, the fulltime trajectory of D is D(Ct ∥ χt) = Deq +Pk(nk(0) − n¯k) w(ωk) e−Γkt,where w(ω) = − log Fβ(ω) and Γk = γk(2n¯k + 1). This is not merely amonotonicity statement; it gives the trajectory in closed form.(ii) Bernstein characterization. The function t 7→ D(Ct ∥ χt) − Deq iscompletely monotone if and only if every mode is initially over-occupiedrelative to thermal equilibrium. This links the conditional monotonicityof D to the Bernstein–Widder theorem for Laplace transforms of positivemeasures.(iii) Spectral H-theorem. The weighted L1Pimbalance functional F(t) :=k|nk(t) − n¯k| w(ωk) is universally non-increasing under KMS Lindbladdynamics, decays exponentially with rate Γmin, and satisfies |D(Ct ∥χt) −Deq| ≤ F(t) for all t ≥ 0. This resolves the open question of whethera universally monotone spectral functional exists and characterises theprecise sense in which D can violate monotonicity.We also establish self-containedly: the Todd identification F−1β = td(βω), theBose–Einstein decomposition of D, the Laplace harmonic series, the completePetz information-loss chain with energy-independent bounds, the conditionalmonotonicity theorem in sharp form with an explicit two-mode counterexample,and the large-N limiting density (the folded semicircle on [0, 2√2] with meanenergy 8√2/(3π)). All results are verified numerically to machine precision.
An exact spectral representation of entropy response in thermal quantum systems is es-tablished using the Bogoliubov–Kubo–Mori (BKM) bilinear form. The response kernel isshown to define a universal spectral filter acting on modular frequencies, inducing a sharpnotion of spectral locality at scale 1/β. Under standard Lieb–Robinson locality assumptions,this structure implies explicit exponential decay bounds for correlations and mutual informa-tion with a thermal length scale proportional to β. The results provide a unified derivation ofthermal locality from modular spectral structure without introducing additional dynamicalassumptions.
We investigate the extension of the Kubo--Martin--Schwinger (KMS) thermal equilibrium condition to bounded non-Hermitian Hamiltonians with real spectra and biorthogonal eigensystems, providing a unified framework through three complementary constructions: a complete KMS theorem under quasi-Hermiticity, a biorthogonal KMS-type identity whose positivity characterises quasi-Hermiticity, and a quantum-detailed-balance condition for the associated open-system dynamics. Our main result is a thermodynamic characterisation of quasi-Hermiticity: for any diagonalisable H∈Md(C) with real spectrum, the biorthogonal Gibbs functional ωbi(A)=Zbi−1∑ne−βEn⟨φn∣A∣ψn⟩ satisfies ωbi(A†A)≥0 for all A if and only if H is quasi-Hermitian. The proof reconstructs the metric η directly from the eigenprojectors of ωbi via the Riesz representation theorem, yielding a metric-free criterion for quasi-Hermiticity. Under the quasi-Hermitian hypothesis, we prove that the η-Gibbs state ωη(A)=Zη−1Tr[ηe−βHA] satisfies the full analytic KMS condition using the Hadamard three-line theorem and Bari's theorem on Riesz bases. The transported state generally differs from the Gibbs state of the isospectral Hermitian partner whenever [η,h]=0, so the KMS property cannot be obtained by similarity transformation alone. Finally, within the Haag--Hugenholtz--Winnink programme, we establish the Tomita--Takesaki modular structure of the η-Gibbs state in finite dimensions, while the construction of a compatible C∗-norm and the proof of σ-weak continuity remain open.
We prove that a single abstract theorem governs information loss in quantum systems under Lindbladdynamics, in neural networks under domain shift, in gravitational decoherence, and in non-Markovian open systems. The theorem requires three objects: a metrized configuration space (X,d),a positive semidefinite operator ρ on X decomposed into shells by d, and a distance-monotone contraction—any completely positive map that suppresses off-diagonal elements monotonically in d.From these alone, we derive the spectral bottleneck D∗, the peeling order, the entropy–shell connectionvia the Kubo-Mori kernel, and the parameter-free collapse time ratio. Lindblad dephasing,gravitational self-energy channels, spectral projection in transfer learning, and time-dependentnon-Markovian dynamics are all shown to be instantiations of the same abstract structure, with D∗emerging from the same theorem in every case.
An arbitrary function in the eigenfunction space of some quantum-mechanical Hamiltonian may be thought to represent the initial configuration ψ(q,0) of a nonstationary state. The system develops in time according to ψ(q,t)=exp(−itH)ψ(q,0)= ∑ k=0∞(−it)kk!Hkψ(q,0).Defining F(t)≡〈ψ(q,0), ψ(q,t) 〉, and hk≡〈ψ(q,0), Hkψ(q,0) 〉, and taking the Fourier transformG(ω)= ∫ −∞∞dtexp(−iωt)F(t),we obtain G(ω)=2π ∑ k=0∞hkk!(ω).In terms of the formal expansion in the energy eingenfunctions ψ(q,0)= ∑ n=0∞cnφn(q)+ ∫ 0∞dωc(ω)φ(ω,q),the Fourier transform represents G(ω)=2π ∑ n=0∞|cn|2δ(ω+ωn)+2π|c(−ω)|2,which exhibits, in principle, the entire eigenvalue spectrum. In this paper, a direct method of calculating eigenvalue spectra, based on the foregoing principle, is proposed. Two modifications are required for computational practicability: (i) use of a finite representation for the delta function and truncation of the summation (a); (ii) replacement of the integrals hk by hk(q′)≡Hkψ(q,0)]q=q′. The modified spectral function is taken to be GN(τ,ω,q′)=2π ∑ k=0Nhk(q′)k!χτ(k)(ω),with χτ(ω)≡sinτω/πω. The sequence GN(τ,ω,q′) is shown to converge as N→∞ if in the Expansion (b) the coefficients cn and c(ω) decrease with ω as exp(—ω/λ) or faster. Assuming convergence, the spectral function represents a broadened eigenvalue spectrum.
For the damped quantum harmonic oscillator at finite temperature, we study the ratio Δ_q⋆/g_F between a Lindblad-resolvent-induced quadratic form W_q⋆(A) = ⟨A · K_β⋆(A)⟩{ρ_β} (with K_β⋆ = (−ℒ⋆)^{−1} − I, ℒ⋆ the Heisenberg-picture Lindblad generator that admits the thermal state ρ_β as a stationary state) and the exponential-family Fisher variance g_F(A) = Var{ρ_β}(A), evaluated on polynomial perturbations A_p = (a + a†)^p. We establish five exact results. Theorem A (hypergeometric decomposition): the projector weight R_l onto the l-th Hermite eigenmode of −ℒ⋆ admits a closed hypergeometric form. Theorem B (closed-form factorial moments): all factorial moments admit closed product formulas. Theorem 1 (DC ratio): the resolvent-to-Fisher ratio equals −2 for every polynomial order p. Theorem 2 (off-diagonal closed form): gives the off-diagonal deviation matrix. Corollary 3: establishes negative semidefiniteness of the full deviation matrix. Proofs use only Hermite orthogonality, OU spectral decomposition, and the Bény–Osborne Wasserstein gradient flow formulation.
Given a family of von Neumann subalgebras of a fixed factor, we construct an intrinsic distance d(α, β) := -log(Δ(α, β)/2), defined purely from the supremum of commutator norms between unit-ball elements, with no reference to operator-norm proximity, Lip-norms, or spectral triples. We first determine the metric structure of this invariant: the quasi-triangle inequality d(α, γ) ≤ d(α, β) + d(β, γ) + log 2 holds unconditionally on totally ordered subfamilies A_α ⊆ A_β ⊆ A_γ and fails for non-nested triples, with an explicit M2(C) counterexample using the tracial state, giving a sharp chain-versus-general dichotomy. Our main result is an exponential clustering theorem: if the dynamics is implemented by a positive Hamiltonian with spectral gap γ and unique invariant vector, and if d(θ_s(A_α), A_β) = +∞ for all |s| ≤ τ, then the full connected correlator obeys |ω(AB) - ω(A)ω(B)| ≤ (1 + 2/(πγτ)) e^(-γτ/2) for unit-ball A, B: cluster decomposition emerges from commutator data plus a spectral gap, with no metric input. This upgrades the elementary antisymmetric bound |ω(AB) - ω(BA)| ≤ 2e^(-d(α,β)), which cannot be extended to the symmetric part by commutator methods alone; the pair (d, τ) defines an emergent causal cone on the net. As an application to algebraic quantum field theory, within a Haag–Kastler net satisfying Einstein causality and a causal-faithfulness hypothesis, d(O1, O2) = +∞ if and only if O1 and O2 are spacelike separated. The distance is invariant under every automorphism that permutes the net, in particular under modular flow, and the construction applies to factors of all Murray–von Neumann types, including the Type III1 factors of relativistic quantum field theory.