We investigate the generic emergence of entropy in isolated quantum systems from the spectral structure of their long-time dephased states. Using a resolvent-based hierarchy, we separate the smooth energy-shell envelope of eigenstate intensities from their intra-shell fluctuations. The single-resolvent sector determines the coarse-grained intensity envelope and yields an entropy contribution that is quantitatively captured by a Lorentzian-Gaussian form. We then show that the difference between this envelope entropy and the von Neumann entropy is governed exactly by the conditional statistics of the residual intensity fluctuations. The two-resolvent covariance sector identifies their leading variance, while higher correlation sectors control the full non-Gaussian correction. A Gaussian-amplitude closure provides a benchmark entropy deficit, whereas exact diagonalization of an interacting Ising chain exhibits substantial finite-size deviations from this benchmark. These results establish a hierarchical spectral framework in which entropy beyond the smooth envelope is directly tied to multi-resolvent correlations, providing a systematic route toward its microscopic characterization.
We investigate the generic emergence of entropy in isolated quantum systems from the spectral structure of their long-time dephased states. Using a resolvent-based hierarchy, we separate the smooth energy-shell envelope of eigenstate intensities from their intra-shell fluctuations. The single-resolvent sector determines the coarse-grained intensity envelope and yields an entropy contribution that is quantitatively captured by a Lorentzian-Gaussian form. We then show that the difference between this envelope entropy and the von Neumann entropy is governed exactly by the conditional statistics of the residual intensity fluctuations. The two-resolvent covariance sector identifies their leading variance, while higher correlation sectors control the full non-Gaussian correction. A Gaussian-amplitude closure provides a benchmark entropy deficit, whereas exact diagonalization of an interacting Ising chain exhibits substantial finite-size deviations from this benchmark. These results establish a hierarchical spectral framework in which entropy beyond the smooth envelope is directly tied to multi-resolvent correlations, providing a systematic route toward its microscopic characterization.
The eigenstate thermalization hypothesis (ETH) parametrizes off-diagonal matrix elements by a smooth function whose microscopic origin remains largely phenomenological. We develop a multi-resolvent hierarchy that derives this smooth structure from the microscopic Hamiltonian. Starting from exact projection identities, we express the ETH variance as fji2=Dji+gji, where Dji is the diagonal-overlap baseline and gji=∑r≥2gji(r) is a systematically improvable hierarchy of multi-channel interference processes. The leading r=3 sector generates an odd-parity component in the energy difference, inaccessible to parity-preserving single-resolvent closures. The same resolvent construction yields an exact covariance representation of eigenstate fluctuations. Under amplitude isotropy, decorrelation, and regularity, it reduces to the Gaussian limit, with q−3=κ4/⟨c2⟩2; normalization further fixes the cross-channel covariance Cˉi, including its energy-resolved form. Exact diagonalization verifies these relations in random-matrix and structured systems, while the latter exhibit controlled breakdown of the isotropic Gaussian closure. The framework thus provides a microscopic hierarchy for both the smooth and fluctuation sectors of subsystem ETH.
We develop a nonperturbative framework for generic nonintegrable many-body systems that reorganizes the expansion of diagonal Green's functions. Starting from exact projection identities and the spectral representation of the resolvent, we derive a recursive hierarchy for the self-energy in which cross-correlated propagation processes are systematically rewritten in terms of diagonal resolvents. Under a diagonal closure approximation, the hierarchy becomes formally closed yet remains systematically improvable.
The framework combines two nonperturbative mechanisms. First, a Lanczos continued-fraction representation provides a recursive single-resolvent structure that naturally produces non-Lorentzian spectral features beyond self-consistent Born approximations. Second, an exact projected multi-resolvent hierarchy introduces nonlocal frequency couplings through products of resolvents and their Hilbert transforms. These contributions mix parity sectors under energy reflection and generate spectral skewness, which is absent in single-resolvent closures.
To solve the resulting equations, we employ a hierarchy of Lorentzian, Gaussian, and Voigt-type ansätze together with an effective Faddeeva self-energy representation ensuring analyticity and causality. Spectral broadening, distribution tails, and higher-order fluctuations emerge from the interplay between continued-fraction recursion and multi-resolvent correlations.
The framework requires no small expansion parameters or diagrammatic truncations, relying instead on ETH-type statistical assumptions appropriate for dense chaotic spectra. It provides a unified route from microscopic interactions to emergent spectral structure, revealing a progression from single-pole self-consistent dynamics to continued-fraction renormalization and finally to multi-resolvent interference effects.
We develop a systematic theory of spectral decimation for quantum many-body Hamiltonians and show that it provides a quantitative probe of emergent symmetries in statistically mixed spectra. Building on an analytical description of statistical mixtures, we derive an explicit expression for the size of a characteristic symmetry sector (CSS), defined as the largest subsequence of levels exhibiting non-Poissonian correlations. The CSS dimension is shown to be the size-biased average of the underlying symmetry sectors, establishing a direct link between spectral statistics and Hilbert-space structure. We apply this framework to two paradigmatic settings: Hilbert-space fragmentation and disorder-induced many-body localization (MBL). In fragmented systems, the CSS reproduces the mixture prediction and isolates correlated subsectors even when the full spectrum appears nearly Poissonian. In the disordered Heisenberg chain, spectral decimation reveals the gradual emergence of integrability through a shrinking CSS, whose statistics exhibit signatures consistent with local integrals of motion. We introduce a characteristic symmetry entropy (CSE) as a finite-size scaling observable and extract, within accessible system sizes, the crossover exponents. Our results establish spectral decimation as a controlled, unbiased and computationally inexpensive diagnostic of hidden structure in many-body spectra, capable of distinguishing between chaotic dynamics, statistical mixtures, and emergent integrability.
We build a framework for the thermodynamics of macroscopic quantum systems. In contrast with approaches requiring access to the full density matrix, our framework relies on a coarse-grained description, based on measurement statistics of a few observables. When these observables commute, the outcomes define classical macrostates whose entropy is quantified by observational entropy, accounting for uncertainty about both the macrostate and the microstate within it. We extend this notion to non-commuting observables forming a subalgebra of the operator space, and use Jaynes' principle to define an algebra-dependent entropy interpolating between von Neumann and observational entropies. Given initial and final measurement sets, connected by internal and/or environment-induced dynamics, we derive a second law for the coarse-grained dynamics. Unlike formulations based on von Neumann entropy, our inequality captures irreversibility from both non-unitary environment-induced dynamics and internal equilibration. It takes the usual form of a positive entropy production when the system is initially at internal equilibrium, while correction terms capture nonequilibrium resources ignored by the coarse-graining. We also derive fluctuation theorems for coarse-grained thermodynamic quantities. Along a quasi-static path of measurement schemes, we identify quantum macroscopic notions of work and heat fulfilling the first and second laws, including an additional work contribution from manipulating the algebra to which the system is confined, through external constraints or quantum measurement backaction. Finally, we apply our framework to examples illustrating the impact of varying the coarse-graining scheme. Our approach unifies macroscopic and stochastic thermodynamics in a genuinely quantum framework, laying the basis for a versatile, experimentally friendly toolbox to analyze complex quantum dynamics.
Despite two decades of work on decoherence and entanglement robustness, no quantitative principle explains why equally entangled quantum states decohere at radically different rates under identical noise. Here we identify the missing variable: a single integer D∗that measures the minimum geometric distance of quantum coherence in Hilbert space. We show that this spectral bottleneck controls irreversible entropy production, while entanglement depth, purity, and coherence norms do not. The principle emerges from an exact spectral decomposition of coherence under local noise, combined with a controlled asymptotic expansion of von Neumann entropy. States with identical entanglement but different D∗exhibit paramet-rically different collapse times—explaining the stark contrast between fragile GHZ states (D∗= n, fast collapse) and robust cluster states (D∗= 1, slow collapse). This upgrades previous empirical observations to a derived theorem, with a prefactor Acomputed exactly from the Kubo–Mori kernel rather than fitted. The spectral bottleneck is not a diagnostic but a control variable: it determines when classical statistics emerge from quantum superposition. This principle resolves a foundational gap in decoherence theory and identifies the correct macroscopic variable governing the quantum-to-classical transition under local noise.
Abstract Entanglement entropy has long served as a central diagnostic of complexity and information scrambling in quantum many-body systems, and its behavior in random quantum circuits is by now well characterized. However, as a single scalar, entanglement entropy captures only the total uncertainty of the reduced density matrix spectrum while discarding the detailed structure of that spectrum — including the shape of the eigenvalue distribution, level repulsion signatures, and the transition from integrable to chaotic behavior. In this work, we propose spectral entropy — the Shannon entropy of the eigenvalue distribution of the reduced density matrix — as a structured diagnostic that goes beyond standard entanglement entropy. Using numerical simulations, we compute the spectral entropy of subsystem reduced states in Haar-random and Clifford circuits as a function of circuit depth, and compare its behavior against von Neumann entanglement entropy, Rényi-2 entropy, and entanglement spectrum level-spacing statistics. We expect to find three key results: (i) spectral entropy exhibits a growth timescale distinct from that of entanglement entropy, potentially reflecting spectral restructuring more sensitively; (ii) in the depth regime where entanglement entropy has already saturated, spectral entropy may continue to evolve, revealing late-time complexification that standard entanglement measures cannot resolve; (iii) spectral entropy may display a distinctive response to measurement-induced or noise-induced disentanglement that differs qualitatively from entanglement entropy. This work positions spectral entropy as a complementary — not replacement — tool to entanglement entropy, offering a finer spectral lens on information scrambling, thermalization, and complexity growth in random quantum circuits.