In this work we revisit the problem of equilibration in isolated many-body interacting quantum systems. We pay particular attention to quantum chaotic Hamiltonians, and rather than focusing on the properties of the asymptotic states and how they adhere to the predictions of the Eigenstate Thermalization Hypothesis, we focus on the equilibration process itself, i.e., the road to equilibrium. Along the road to equilibrium the diagonal ensembles obey an emergent form of the second law of thermodynamics and we provide an information theoretic proof of this fact. With this proof at hand we show that the road to equilibrium is nothing but a hierarchy in time of diagonal ensembles. Furthermore, introducing the notions of statistical complexity and the entropy-complexity plane, we investigate the uniqueness of the road to equilibrium in a generic many-body system by comparing its trajectories in the entropy-complexity plane to those generated by a random Hamiltonian. Finally, by treating the random Hamiltonian as a perturbation we analyzed the stability of entropy-complexity trajectories associated with the road to equilibrium for a chaotic Hamiltonian and different types of initial states.
In this work we revisit the problem of equilibration in isolated many-body interacting quantum systems. We pay particular attention to quantum chaotic Hamiltonians, and rather than focusing on the properties of the asymptotic states and how they adhere to the predictions of the Eigenstate Thermalization Hypothesis, we focus on the equilibration process itself, i.e., the road to equilibrium. Along the road to equilibrium the diagonal ensembles obey an emergent form of the second law of thermodynamics and we provide an information theoretic proof of this fact. With this proof at hand we show that the road to equilibrium is nothing but a hierarchy in time of diagonal ensembles. Furthermore, introducing the notions of statistical complexity and the entropy-complexity plane, we investigate the uniqueness of the road to equilibrium in a generic many-body system by comparing its trajectories in the entropy-complexity plane to those generated by a random Hamiltonian. Finally, by treating the random Hamiltonian as a perturbation we analyzed the stability of entropy-complexity trajectories associated with the road to equilibrium for a chaotic Hamiltonian and different types of initial states.
As recently manifested [ 1 ], the quench dynamics of isolated quantum systems consisting of a finite number of particles, is characterized by an exponential spreading of wave packets in the many-body Hilbert space. This happens when the inter-particle interaction is strong enough, thus resulting in a chaotic structure of the many-body eigenstates considered in the non-interacting basis. The semi-analytical approach used here, allows one to estimate the rate of the exponential growth as well as the relaxation time, after which the equilibration (thermalization) emerges. The key ingredient parameter in the description of this process is the width Γ of the local density of states (LDoS) defined by the initially excited state, the number of particles and the interaction strength. In this paper we show that apart from the meaning of Γ as the decay rate of survival probability, the width of the LDoS is directly related to the diagonal entropy and the latter can be linked to the thermodynamic entropy of a system equilibrium state emerging after the complete relaxation. The analytical expression relating the two entropies is derived phenomenologically and numerically confirmed in a model of bosons with random two-body interaction, as well as in a deterministic model which becomes completely integrable in the continuous limit.
The question of how statistical physics can be seen to emerge from unitary quantum dynamics goes back to Schrödinger and von Neumann, however has been reaffirmed as a topic of importance due to recent advances in experimental capabilities, allowing the observation of the unitary evolution of many-particle systems over long periods of time. Understanding why such a system evolves to the specific thermal equilibrium state, effectively described by a relevant thermodynamic ensemble, has thus seen significantly more interest in recent years. Furthermore, observations of the dynamics of such systems open questions on the route to equilibrium, and subsequent fluctuations. In this thesis I will develop an approach to this problem, taking as a starting point the non-integrability of the system under study, which leads to a generic model in terms of random matrix theory (RMT), and more generally, chaotic wavefunctions. In the formulation of these methods, special attention is paid to the form of local observables of quantum systems, and an approach to their description is developed in terms of correlation functions of chaotic wavefunctions. From this approach a key conjecture in the understanding of thermalization, the eigenstate thermalization hypothesis (ETH), is derived in full, and the dynamical behaviour of such observables is obtained analytically. Further, I will show that emergent classical behaviour can be observed in the form of a fluctuation-dissipation theorem (FDT) of Brownian processes. This is exploited to develop an experimental proposal to measure the ‘complexity’ of a quantum device, by experimentally obtaining its effective Hilbert space dimension in terms of a fully connected system. Finally, quantum jump trajectories, describing stroboscopic projective measurements of local observables are studied, and shown to display classical Brownian dynamics in the form of a Markov process. Throughout this thesis, exact diagonalization calculations of realistic quantum spin systems are used to confirm analytical results.
A bipartite system whose subsystems are fully quantum chaotic and coupled by a perturbative interaction with a tunable strength is a paradigmatic model for investigating how isolated quantum systems relax toward an equilibrium. It is found that quantum coherence of the initial product states in the energy eigenbasis of the subsystems-quantified by the off-diagonal elements of the subsystem density matrices-can be viewed as a resource for equilibration and thermalization as manifested by the entanglement generated. Results are given for four distinct perturbation strength regimes, the ultraweak, weak, intermediate, and strong regimes. For each, three types of tensor product states are considered for the initial state: uniform superpositions, random superpositions, and individual subsystem eigenstates. A universal timescale is identified involving the interaction strength parameter. In particular, maximally coherent initial product states (a form of uniform superpositions) thermalize under time evolution for any perturbation strength in spite of the fact that in the ultraweak perturbative regime the underlying eigenstates of the system have a tensor product structure and are not at all thermal-like; though the time taken to thermalize tends to infinity as the interaction vanishes. Moreover, it is shown that in the ultraweak regime the initial entanglement growth of the system whose initial states are maximally coherent is quadratic-in-time, in contrast to the widely observed linear behavior.
Thermalization conventionally describes local properties of isolated many-body systems at equilibrium. Magic, or nonstabilizerness $\unicode{x2013}$ the resource enabling universal quantum computation $\unicode{x2013}$ is by contrast encoded in the global structure of the many-body wavefunction. We show that, despite its global nature, the magic of equilibrium pure states of chaotic many-body systems, including late-time evolved states and energy eigenstates, is universal: it is captured by the thermal Scrooge ensemble, the minimally informative ensemble of pure states consistent with the Gibbs state at the same effective temperature. Therefore, for systems with no conserved quantities other than the total energy, equilibrium magic is a function of temperature alone, independent of the initial state and other microscopic features of the equilibrium state. This yields concrete universal predictions for the stabilizer Rényi entropies (SREs). At infinite temperature, the SRE is set by Haar-like fluctuations of the Pauli spectrum, while at finite temperature energy conservation induces a volume-law thermodynamic correction controlled by the thermal Pauli spectrum. We support these predictions with analytical arguments and extensive numerical simulations. We further show that chaotic many-body systems at high temperatures possess long-range magic and entanglement that cannot be removed by finite-depth local quantum circuits. Our results establish magic as a thermodynamic property of chaotic many-body systems and suggest that Scrooge ensembles may provide a unified framework for quantum many-body resources.
We investigate the role of a statistical complexity measure to assign equilibration in isolated quantum systems. While unitary dynamics preserve global purity, expectation values of observables often exhibit equilibration-like behavior, raising the question of whether a measure of complexity can track this process. In addition to examining observable equilibration, we extend our analysis to study how the complexity of the quantum states evolves, providing insight into the transition from initial coherence to equilibrium. We define a classical statistical complexity measure based on observable entropy and deviation from equilibrium, which captures the dynamical progression towards equilibration and effectively distinguishes between complex and non-complex trajectories. In particular, our measure is sensitive to non-complex dynamics. Such dynamics include the quasi-periodic behavior exhibited by low-dimensional initial states, where the system explores a limited region of Hilbert space while preserving coherence. Numerical simulations of an Ising-like non-integrable Hamiltonian spin-chain model support these findings. Our work provides new insight into the emergence of equilibrium behavior from unitary dynamics and advances complexity as a meaningful tool in the study of the emergence of classicality in microscopic systems.
Understanding equilibration times in closed quantum systems is essential for characterising their approach to equilibrium. Chaotic many-body systems are paradigmatic in this context: they are expected to thermalise according to the eigenstate thermalisation hypothesis and exhibit spectral properties well described by random matrix theory (RMT). While RMT successfully captures spectral correlations, its ability to provide quantitative predictions for equilibration timescales has remained largely unexplored. Here, we study equilibration within RMT using the framework of equilibration as dephasing, focusing on closed systems whose Hamiltonians are drawn from the Gaussian unitary ensemble (GUE). We derive an analytical expression that approximates the average equilibration time of the GUE and show that it is independent of both the initial state and the choice of observable, a consequence of the rotational invariance of the GUE. Numerical simulations confirm our analytical expression and demonstrate that our approximation is in close agreement with the true average equilibration time of the GUE. We find that the equilibration time decreases with system size and vanishes in the thermodynamic limit. This unphysical result indicates that the true equilibration timescale of realistic chaotic many-body systems must be dominated by physical features not captured by random matrix ensembles -- the GUE in particular.