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.
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 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 develop a structural framework in which physical reality is derived from the spectral properties of quantum states. Within the approach of Universal Modular Dynamics (UMD), the density operator ρ is taken as the fundamental object, and its modular generator K =−log ρ provides a representation of informational content. We show that the distribution of modular energies, defined by the spectrum of K, serves as a primary structure from which higher-level physical organization emerges. In particular, we demonstrate how spectral distributions give rise to stable structures, induce locality through optimal partitioning, and generate geometric relations via correlation-based distances. Dynamics is introduced as an intrinsic ordering of states under modular evolution, while renormalization appears as a spectral phenomenon governed by statistical properties of the modular spectrum. This establishes a connection between informational structure and physical scale. The resulting construction defines a sequential chain of emergence, ρ → p(k) → structure → locality → geometry → dynamics → scale, in which each level arises from properties of the previous one without introducing external assumptions. This work provides a unified perspective in which physical structure is understood as an emergent consequence of spectral information encoded in quantum states.
This thesis analyses several locality properties of extended quantum lattice systems, considering both spins and fermions. We begin by assuming locality of the Hamiltonian and then discuss how this translates into locality of the time evolution and the thermal states. Starting from a local Hamiltonian, it is well known that the Heisenberg time evolution satisfies Lieb-Robinson bounds, which quantify its locality. We strengthen these bounds for long-range interacting fermions and for commuting interactions. Furthermore, we establish the first Lieb-Robinson bounds for Hamiltonians whose local terms can grow linearly away from the origin. We also prove stability of Lieb-Robinson bounds with respect to local perturbations in the Hamiltonian. For the thermal states we focus on three types of locality known as decay of correlations, the local perturbations perturb locally (LPPL) principle and local indistinguishability. As a key tool we use spectral filters, which were introduced by Hastings in the 2000s for various applications. It enable us to use Lieb-Robinson bounds to obtain information about locality properties of states. In particular, we use quantum belief propagation to prove an equivalence of decay of correlations, LPPL and local indistinguishability for Gibbs states at any positive temperature. Using the improved Lieb-Robinson bounds mentioned before, we obtain improvements for decay of correlations in the ground state, the locality of the spectral flow and LPPL for long-range interacting systems and commuting Hamiltonians. Furthermore, we adjust the filter functions to achieve exponential decay for certain applications that were previously limited to stretched exponential decay. Spectral filters are also used to rigorously justify response theory, which usually requires a gapped system to begin with. To relax this assumption, we introduce various notions of local gaps, discuss their relations and use one to justify response theory and Kubo’s formula in the absence of a global gap. In another result we use local indistinguishability as an input to prove bulk-edge correspondence at finite temperature, in the sense that the bulk magnetisation is equal to the edge current. This result has mostly been studied for non-interacting models or at zero temperature before. Finally, we present a class of weakly interacting spin systems with an on-site gap and prove that the ground state satisfies a strong form of LPPL. Moreover, we show that these systems satisfy decay of correlations and local indistinguishability uniformly in temperature.
Random states play a foundational role in different branches of modern quantum science. In this work, we study a recently proposed random state ensemble induced from von Neumann entropy through the Bogoliubov-Kubo-Mori (BKM) metric. In particular, we derive an exact yet explicit formula of average entanglement entropy over BKM ensemble. In obtaining the formula, we only make use of properties of normalization constant of the ensemble in the absence of its correlation kernel, contrary to average entropy computation of other ensembles. This new framework paves the way for calculating higher-order cumulants of BKM ensemble beyond the average.
We investigate the information-theoretic structure underlying causal horizons through the formalism of modular quantum channels. By analyzing the singular value decomposition of the channel induced by the partial trace over inaccessible regions, we show that the Unruh effect and the Page curve can be understood as manifestations of a universal thermal filtering mechanism governed by the modular Hamiltonian. This structure leads to a Gibbs-weighted hierarchy of information transmission modes, with entanglement entropy corresponding to spectral activation and thermal capacity. We reinterpret the first law of entanglement as a Clausius relation for modular flow and derive Einstein's equations by requiring its local validity across Rindler horizons. In the case of black hole evaporation, we find that the modular channel undergoes an informational phase transition at the Page time, marked by a shift from entanglement preservation to recoverability, quantified through fidelity and channel capacity. Based on these results, we propose the Modular Channels Flow Correspondence (MCFC): a minimal holographic principle whereby the area of a causal screen measures the storage of filtered quantum information. Our framework offers a unified operational account of holography, entropy, and curvature, grounded in the spectral dynamics of modular channels.
We develop an operator-algebraic framework for modular self-duality, symmetrized relative entropy, and Bogoliubov--Kubo--Mori (BKM) susceptibility of local states in quantum field theory. In finite dimensions, modular self-duality singles out fixed points at which a state coincides with its modularly reflected partner. At such points, the natural comparison functional is the symmetrized Umegaki relative entropy. It vanishes at coincidence, and its Hessian is governed by the BKM quantum Fisher information along the reflected tangent direction. We then extend this fixed-point construction to the local type~III von Neumann algebras that arise in quantum field theory. Here, a local state is compared with the modular pullback of its commutant restriction, and the intrinsic comparison functional is the symmetrized Araki relative entropy. For sufficiently regular state deformations, the fixed-localization Hessian at the self-dual point defines a type~III BKM susceptibility. This coefficient is obtained by evaluating the BKM bilinear form on the tangent selected by the modular pairing. Exact coherent-state realizations are obtained for the free scalar field on wedge algebras and for the chiral U(1) current on half-line algebras. In both examples, the comparison functional is exactly quadratic in the deformation parameter, and the susceptibility coefficients admit explicit boost-energy, stress-tensor, or half-line integral representations. We also apply the construction to a ball-shaped boundary region in the vacuum state of a conformal field theory with a semiclassical anti-de Sitter (AdS) gravitational dual. For linear bulk coherent excitations in an exact complementary-recovery model of this sector, bulk--boundary modular intertwining identifies the modular partner with the AdS--Rindler-reflected coherent state, and the symmetrized relative entropy is exactly quadratic.