Characterizing complexity of many-body quantum dynamics by higher-order eigenstate thermalization

Abstract

Complexity of dynamics is at the core of quantum many-body chaos and exhibits a hierarchical feature: higher-order complexity implies more chaotic dynamics. Conventional ergodicity in thermalization processes is a manifestation of the lowest order complexity, which is represented by the eigenstate thermalization hypothesis (ETH) stating that individual energy eigenstates are thermal. Here, we propose a higher-order generalization of the ETH, named the $k$-ETH ($k=1,2,\dots$), to quantify higher-order complexity of quantum many-body dynamics at the level of individual energy eigenstates, where the lowest order ETH (1-ETH) is the conventional ETH. As a non-trivial contribution of the higher-order ETH, we show that the $k$-ETH with $k\ge 2$ implies a universal behavior of the $k$th Renyi entanglement entropy of individual energy eigenstates. In particular, the Page correction of the entanglement entropy originates from the higher-order ETH, while as is well known, the volume law can be accounted for by the 1-ETH. We numerically verify that the 2-ETH approximately holds for a nonintegrable system, but does not hold in the integrable case. To further investigate the information-theoretic feature behind the $k$-ETH, we introduce a concept named a partial unitary $k$-design (PU $k$-design), which is an approximation of the Haar random unitary up to the $k$th moment, where partial means that only a limited number of observables are accessible. The $k$-ETH is a special case of a PU $k$-design for the ensemble of Hamiltonian dynamics with random-time sampling. In addition, we discuss the relationship between the higher-order ETH and information scrambling quantified by out-of-time-ordered correlators. Our framework provides a unified view on thermalization, entanglement entropy, and unitary $k$-designs, leading to deeper characterization of higher-order quantum complexity.

References in corpus (101)

• Chaos and Quantum Thermalization
• Thermalization and its mechanism for generic isolated quantum systems
• Area laws for the entanglement entropy - a review
• Probing many-body dynamics on a 51-atom quantum simulator
• From Quantum Chaos and Eigenstate Thermalization to Statistical Mechanics and Thermodynamics
• A bound on chaos
• Average Entropy of a Subsystem
• Black holes and the butterfly effect
• Black holes as mirrors: quantum information in random subsystems
• Measuring entanglement entropy through the interference of quantum many-body twins
• Quantum thermalization through entanglement in an isolated many-body system
• Fast Scramblers
• The foundations of statistical mechanics from entanglement: Individual states vs. averages
• Probing the relaxation towards equilibrium in an isolated strongly correlated 1D Bose gas
• Quantum many-body scars
• Quantum information processing with superconducting circuits: a review
• Operator Spreading in Random Unitary Circuits
• Chaos in quantum channels
• Exact and Approximate Unitary 2-Designs: Constructions and Applications
• Localized shocks
• Integration with respect to the Haar measure on unitary, orthogonal and symplectic group
• Quantum mechanical evolution towards thermal equilibrium
• Measuring out-of-time-order correlations and multiple quantum spectra in a trapped ion quantum magnet
• Operator hydrodynamics, OTOCs, and entanglement growth in systems without conservation laws
• The approach to thermal equilibrium in quantized chaotic systems
• Foundation of Statistical Mechanics under experimentally realistic conditions
• Thermalization and prethermalization in isolated quantum systems: a theoretical overview
• Operator spreading and the emergence of dissipative hydrodynamics under unitary evolution with conservation laws
• Measuring the scrambling of quantum information
• Two-dimensional conformal field theory and the butterfly effect
• Chaos and complexity by design
• Local emergence of thermal correlations in an isolated quantum many-body system
• Testing whether all eigenstates obey the Eigenstate Thermalization Hypothesis
• Towards the fast scrambling conjecture
• Evenly distributed unitaries: on the structure of unitary designs
• Quantum Data Hiding
• Effect of Rare Fluctuations on the Thermalization of Isolated Quantum Systems
• Ergodic dynamics and thermalization in an isolated quantum system
• Local random quantum circuits are approximate polynomial-designs
• Diffusive hydrodynamics of out-of-time-ordered correlators with charge conservation
• The Second Law of Quantum Complexity
• Entanglement of Quantum Evolutions
• Finite-size scaling of eigenstate thermalization
• Chaos, Complexity, and Random Matrices
• Does a single eigenstate encode the full Hamiltonian?
• Operator space entanglement entropy in transverse Ising chain
• Unifying fast scrambling, thermalization and entanglement through the measurement of FOTOCs in the Dicke model
• Off-diagonal matrix elements of local operators in many-body quantum systems
• Eigenstate thermalization within isolated spin-chain systems
• Entanglement Entropy of Eigenstates of Quantum Chaotic Hamiltonians
• Quantum Complexity and Negative Curvature
• Canonical Typicality of Energy Eigenstates of an Isolated Quantum System
• Subsystem ETH
• The Eigenstate Thermalization Hypothesis and Out of Time Order Correlators
• Measurement of many-body chaos using a quantum clock
• Jarzynski-like equality for the out-of-time-ordered correlator
• Eigenstate thermalization in the two-dimensional transverse field Ising model
• Entanglement scaling of operators: a conformal field theory approach, with a glimpse of simulability of long-time dynamics in 1+1d
• Entanglement and matrix elements of observables in interacting integrable systems
• Bounds on chaos from the eigenstate thermalization hypothesis
• Entanglement Entropy of Eigenstates of Quadratic Fermionic Hamiltonians
• Operator entanglement entropy of the time evolution operator in chaotic systems
• Locality, Quantum Fluctuations, and Scrambling
• Global characteristics of all eigenstates of local many-body Hamiltonians: participation ratio and entanglement entropy
• Randomized benchmarking of single and multi-qubit control in liquid-state NMR quantum information processing
• Universality in volume law entanglement of pure quantum states
• Eigenstate thermalization in the Sachdev-Ye-Kitaev model
• Fluctuation Theorem for Many-Body Pure Quantum States
• Renyi Entropy of Chaotic Eigenstates
• Volume Law and Quantum Criticality in the Entanglement Entropy of Excited Eigenstates of the Quantum Ising Model
• On the Distribution of the Wave Function for Systems in Thermal Equilibrium
• Efficient unitary designs with nearly time-independent Hamiltonian dynamics
• Generalization of von Neumann's Approach to Thermalization
• Structure of chaotic eigenstates and their entanglement entropy
• Efficient decoding for the Hayden-Preskill protocol
• Eigenstate Correlations, Thermalization and the Butterfly Effect
• Finite-size scaling of out-of-time-ordered correlators at late times
• Universal eigenstate entanglement of chaotic local Hamiltonians
• Scrambling and Complexity in Phase Space
• Average eigenstate entanglement entropy of the XY chain in a transverse field and its universality for translationally invariant quadratic fermionic models
• Numerical Large Deviation Analysis of Eigenstate Thermalization Hypothesis
• Entanglement, quantum randomness, and complexity beyond scrambling
• Efficient Quantum Pseudorandomness
• Entanglement Features of Random Hamiltonian Dynamics
• Generalized Entanglement Entropies of Quantum Designs
• Tripartite information, scrambling, and the role of Hilbert space partitioning in quantum lattice models
• On the local equivalence between the canonical and the microcanonical distributions for quantum spin systems
• Generating a state $t$-design by diagonal quantum circuits
• Weingarten calculus for matrix ensembles associated with compact symmetric spaces
• Phase-random states: ensembles of states with fixed amplitudes and uniformly distributed phases in a fixed basis
• Weak eigenstate thermalization with large deviation bound
• Universal scrambling in gapless quantum spin chains
• How accurately can the microcanonical ensemble describe small isolated quantum systems?
• Chaos and random matrices in supersymmetric SYK
• Page Curves for General Interacting Systems
• General moments of matrix elements from circular orthogonal ensembles
• Random-matrix behavior of quantum nonintegrable many-body systems with Dyson's three symmetries
• Chaos and High Temperature Pure State Thermalization
• Experimental Implementation of Efficient Quantum Pseudorandomness on a 12-spin System
• Quantum Information Scrambling Through a High-Complexity Operator Mapping
• Ergodicity, eigenstate thermalization, and the foundations of statistical mechanics in quantum and classical systems

Cited by in corpus (30)

• Emergent quantum state designs from individual many-body wavefunctions
• Eigenstate Entanglement Entropy in Random Quadratic Hamiltonians
• Eigenstate thermalization hypothesis and its deviations from random-matrix theory beyond the thermalization time
• Entanglement of midspectrum eigenstates of chaotic many-body systems: Reasons for deviation from random ensembles
• Entanglement in many-body eigenstates of quantum-chaotic quadratic Hamiltonians
• Measuring Arbitrary Physical Properties in Analog Quantum Simulation
• A Maximum Entropy Principle in Deep Thermalization and in Hilbert-Space Ergodicity
• Benchmarking Quantum Simulators using Ergodic Quantum Dynamics
• Complete Hilbert-Space Ergodicity in Quantum Dynamics of Generalized Fibonacci Drives
• Chaos and Thermalization in the Spin-Boson Dicke Model
• Designs via Free Probability
• Fixing detailed balance in ancilla-based dissipative state engineering
• Free-fermion Page Curve: Canonical Typicality and Dynamical Emergence
• Hilbert-Space Ergodicity in Driven Quantum Systems: Obstructions and Designs
• Eigenstate Fluctuation Theorem in the Short and Long Time Regimes
• On the generic increase of observational entropy in isolated systems
• Typical entanglement entropy in systems with particle-number conservation
• Concentration of quantum equilibration and an estimate of the recurrence time
• Simulating Floquet scrambling circuits on trapped-ion quantum computers
• Non-Clifford Cost of Random Unitaries
• Non-Haar random circuits form unitary designs as fast as Haar random circuits
• Coherent electron splitting in interacting chiral edge channels
• Late-time ensembles of quantum states in quantum chaotic systems
• On Computational Complexity of Unitary and State Design Properties
• Toward a physically motivated notion of Gaussian complexity geometry
• Attention to Quantum Complexity
• Symmetry Properties of Quantum Dynamical Entropy
• Volume-law entanglement entropy of typical pure quantum states
• Diagnosing chaos with projected ensembles of process tensors
• Rationally independent free fermions with local hopping