We theoretically investigate parameter quantum estimation in quantum chaotic systems. Our analysis is based on an effective description of non-integrable quantum systems in terms of a random matrix Hamiltonian. Based on this approach we derive an analytical expression for the time evolution of the quantum Fisher information. We test our random matrix theory prediction with the exact diagonalization of a non-integrable spin system, focusing on the estimation of a local magnetic field by measurements of the many-body state. Our numerical calculations agree with the effective random matrix theory approach and show that the information on the local Hamiltonian parameter is distributed throughout the quantum system during the quantum thermalization process. Our analysis shows a first stage in which the initial information spread is quadratic in time which quickly passes into linear increase with slope determine by the decay rate of the measured spin observable. When the information is fully spread among all degrees of freedom a second quadratic time scale determines the long time behaviour of the quantum Fisher information.
We theoretically investigate parameter quantum estimation in quantum chaotic systems. Our analysis is based on an effective description of non-integrable quantum systems in terms of a random matrix Hamiltonian. Based on this approach we derive an analytical expression for the time evolution of the quantum Fisher information. We test our random matrix theory prediction with the exact diagonalization of a non-integrable spin system, focusing on the estimation of a local magnetic field by measurements of the many-body state. Our numerical calculations agree with the effective random matrix theory approach and show that the information on the local Hamiltonian parameter is distributed throughout the quantum system during the quantum thermalization process. Our analysis shows a first stage in which the initial information spread is quadratic in time which quickly passes into linear increase with slope determine by the decay rate of the measured spin observable. When the information is fully spread among all degrees of freedom a second quadratic time scale determines the long time behaviour of the quantum Fisher information.
We theoretically investigate quantum parameter estimation in quantum chaotic systems. Our analysis is based on an effective description of quantum ergodic systems in terms of a random matrix Hamiltonian. Based on this approach, we derive an analytical expression for the time evolution of the quantum Fisher information (QFI), which we find to have three distinct timescales. Initially, the QFI increase is quadratic in time, characterizing the timescale over which initial information is extractable from local measurements only. This quickly passes into linear increase with slope determined by the decay rate of the measured spin observable. When the information is fully spread among all degrees of freedom, a second quadratic timescale determines the long-time behavior of the QFI. We test our random matrix theory prediction with the exact diagonalization of a nonintegrable spin system, focusing on the estimation of a local magnetic field by measurements of the many-body state. Our numerical calculations agree with the effective random matrix theory approach and show that the information on the local Hamiltonian parameter is distributed throughout the quantum system during the quantum thermalization process.
Phenomena in quantum chaotic systems such as spectral fluctuations are known to be described remarkably well by the random matrix theory (RMT) in small energy intervals and over large time scales. However, in exchange for analytical tractability, RMT has been constructed to leave out an important element of realistic physical systems { the notion of locality. Since quantum information propagates at a finite speed in a spatially-extended quantum chaotic system, there exists time scales before which RMT is effective. How do we describe the non-equilibrium quantum dynamics at intermediate timescales which are much longer than the local relaxation time but too short for RMT to apply? What are the signatures of many-body quantum chaotic systems beyond the description of RMT? In this thesis, we address these questions by analysing minimal models that incorporate a notion of locality and at the same time retain some analytical tractability of RMT. More specifically, we study Floquet quantum circuits that are random in space but periodic in time. These circuits consist of local and random quantum gates that act on a q-state `spin' at each site. After a brief review of classical chaos, quantum chaos, and quantum circuits in Chapter 1, we present three main results. In Chapter 2, we formulate a diagrammatical approach for computing ensemble averages of observables over random unitary circuits, and derive exact results on spectral statistics and dynamical observables for a Floquet circuit in the large-q limit. In Chapter 3, we further investigate the spectral statistics in spatiallyextended many-body quantum chaotic systems by computing the spectral form factor K(t) analytically and numerically. We show that K(t) follows the RMT description at times longer than the Thouless time tTh and identify the scaling of tTh at the large-q limit of a Floquet circuit. In Chapter 4, we discuss eigenstate correlations for spatially-extended many-body quantum chaotic systems, in terms of the statistical properties of matrix elements of local observables. While the eigenstate thermalization hypothesis (ETH) is known to give an excellent description of these quantities, the phenomenon of scrambling and the butterfly effect imply structure beyond ETH. We determine analytically and verify numerically the universal form of this structure at long distances and small eigenvalue separations for Floquet systems.
Quantum many-body systems can exhibit distinct regimes where dynamics is either ergodic, dynamically exploring an extensive region of available state-space, or non-ergodic, where the dynamics may be restricted. An example is the many-body localization (MBL) transition, where disorder induces non-ergodic behaviour. Most measures of ergodicity notably rely on global quantities, such as level spacing statistics. We explore the ability for a subsystem to probe the ergodicity of dynamics via measurement of local observables, and use expected results from random matrix theory (RMT) as a benchmark for the ergodic regime. We exploit two predictions from RMT as ergodicity is broken: the time evolution of the quantum Fisher information, and a fluctuation-dissipation relation. These are investigated in three different ergodicity breaking mechanisms, namely, as a consequence of transition to integrability, MBL, and Quantum Many-Body Scars (QMBS). We show that the predicted behaviour from RMT can be used as a potential witness for transition to non-ergodic behaviour from the measurement of local observables alone.
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.
Many-body quantum chaos at ℏ∼1: Analytic connection to random matrix theory
A key goal of quantum chaos is to establish a relationship between widely observed universal spectral fluctuations of clean quantum systems and random matrix theory (RMT). For single particle systems with fully chaotic classical counterparts, the problem has been partly solved by Berry (1985) within the so-called diagonal approximation of semiclassical periodic-orbit sums. Derivation of the full RMT spectral form factor K(t) from semiclassics has been completed only much later in a tour de force by Mueller et al (2004). In recent years, the questions of long-time dynamics at high energies, for which the full many-body energy spectrum becomes relevant, are coming at the forefront even for simple many-body quantum systems, such as locally interacting spin chains. Such systems display two universal types of behaviour which are termed as `many-body localized phase' and `ergodic phase'. In the ergodic phase, the spectral fluctuations are excellently described by RMT, even for very simple interactions and in the absence of any external source of disorder. Here we provide the first theoretical explanation for these observations. We compute K(t) explicitly in the leading two orders in t and show its agreement with RMT for non-integrable, time-reversal invariant many-body systems without classical counterparts, a generic example of which are Ising spin 1/2 models in a periodically kicking transverse field.
Quantum metrology enhances the sensitivity of parameter estimation using the distinctive resources of quantum mechanics such as entanglement. It has been shown that the precision of estimating an overall multiplicative factor of a Hamiltonian can be increased to exceed the classical limit, yet little is known about estimating a general Hamiltonian parameter. In this paper, we study this problem in detail. We find that the scaling of the estimation precision with the number of systems can always be optimized to the Heisenberg limit, while the time scaling can be quite different from that of estimating an overall multiplicative factor. We derive the generator of local parameter translation on the unitary evolution operator of the Hamiltonian, and use it to evaluate the estimation precision of the parameter and establish a general upper bound on the quantum Fisher information. The results indicate that the quantum Fisher information generally can be divided into two parts: one is quadratic in time, while the other oscillates with time. When the eigenvalues of the Hamiltonian do not depend on the parameter, the quadratic term vanishes, and the quantum Fisher information will be bounded in this case. To illustrate the results, we give an example of estimating a parameter of a magnetic field by measuring a spin-21 particle and compare the results for estimating the amplitude and the direction of the magnetic field.