We show that the recently introduced operator fidelity metric provides a natural tool to investigate the crossover to quantum chaotic behavior. This metric is an information-theoretic measure of the global stability of a unitary evolution against perturbations. We use random matrix theory arguments to conjecture that the operator fidelity metric can be used to discriminate phases with regular behavior from quantum chaotic ones. A numerical study of the onset of chaotic in the Dicke model is given in order to support the conjecture.
We show that the recently introduced operator fidelity metric provides a natural tool to investigate the crossover to quantum chaotic behavior. This metric is an information-theoretic measure of the global stability of a unitary evolution against perturbations. We use random matrix theory arguments to conjecture that the operator fidelity metric can be used to discriminate phases with regular behavior from quantum chaotic ones. A numerical study of the onset of chaotic in the Dicke model is given in order to support the conjecture.
The operator fidelity is a measure of the information-theoretic distinguishability between perturbed and unperturbed evolutions. The response of this measure to the perturbation may be formulated in terms of the operator fidelity susceptibility (OFS), a quantity which has been used to investigate the parameter spaces of quantum systems in order to discriminate their regular and chaotic regimes. In this work we numerically study the OFS for a pair of nonlinearly coupled two-dimensional harmonic oscillators, a model which is equivalent to that of a hydrogen atom in a uniform external magnetic field. We show how the two terms of the OFS, being linked to the main properties that differentiate regular from chaotic behavior, allow for the detection of this model's transition between the two regimes. In addition, we find that the parameter interval where perturbation theory applies is delimited from above by a local minimum of one of the analyzed terms.
In this dissertation, we address several problems in condensed matter physics, statistical mechanics, and quantum chaos from the perspective of the quantum information-theoretic notion of state distinguishability, as measured though evaluation of the appropriate fidelity measure. ❧ We find that the fidelity susceptibility, a measure for the response of the fidelity to infinitesimal changes of the Hamiltonian parameters, serves as a useful tool for studying the effects of disorder on quantum phase transitions. We study in detail the phase diagram of the well-known one-dimensional random quantum XY chain. Through an analysis of the fidelity susceptibility’s finite-size scaling and statistics for this model, we find that the fidelity approach reflects large variation between average and typical behavior, may detect the presence of Griffiths phases, and indicates the modifications to the phase diagram due to disorder. ❧ For this same model, though without disorder, we evaluate the dynamics following a sudden Hamiltonian quench using the Loschmidt echo, a form of the fidelity that compares the initial state with the time-evolved state. Through a central limit theorem-type argument we find that the long-time statistics of the Loschmidt echo takes one of two universal forms, even at finite temperature, depending on the system’s proximity to the quantum critical lines. ❧ Finally, for a canonical model exhibiting quantum chaos, the hydrogen atom in a uniform external magnetic field, we evaluate the operator fidelity susceptibility (OFS) between unitary operators generating the dynamics. We find that the part of the OFS which corresponds to the variation of the Hamiltonian eigenvectors serves to indicate the transition from the regular perturbative and n-mixing regimes to the quantum chaotic regime.
Using the key properties of chaos, i.e. ergodicity and exponential instability, as a resource to control classical dynamics has a long and considerable history. However, in the context of controlling "chaotic" quantum unitary dynamics, the situation is far more tenuous. The classical concepts of exponential sensitivity to trajectory initial conditions and ergodicity do not directly translate into quantum unitary evolution. Nevertheless properties inherent to quantum chaos can take on those roles: i) the dynamical sensitivity to weak perturbations, measured by the fidelity decay, serves a similar purpose as the classical sensitivity to initial conditions; and ii) paired with the fact that quantum chaotic systems are conjectured to be statistically described by random matrix theory, implies a method to translate the ergodic feature into the control of quantum dynamics. With those two properties, it can be argued that quantum chaotic dynamical systems, in principle, allow for full controllability beyond a characteristic time that scales only logarithmically with system size and ℏ−1. In the spirit of classical targeting, it implies that it is possible to fine tune the immense quantum interference with weak perturbations and steer the system from any initial state into any desired target state, subject to constraints imposed by conserved quantities. In contrast, integrable dynamics possess neither ergodicity nor exponential instability, and thus the weak perturbations apparently must break the integrability for control purposes. The main ideas are illustrated with the quantum kicked rotor. The production of revivals, cat-like entangled states, and the transition from any random state to any other random state is possible as demonstrated.
Quantum chaos entails an entropic and computational obstruction to describing a system and thus is intrinsically difficult to characterize. An understanding of quantum chaos is fundamentally related to the mechanism of thermalization in many-body systems and the quantum nature of black holes. In this thesis we adopt the view that quantum information theory provides a powerful framework in which to elucidate chaos in strongly-interacting quantum systems. We first push towards a more precise understanding of chaotic dynamics by relating different diagnostics of chaos, studying the time-evolution of random matrix Hamiltonians, and quantifying random matrix behavior in physical systems. We derive relations between out-of-time ordered correlation functions, spectral quantities, and frame potentials to relate the scrambling of quantum information, decay of correlators, and Haar-randomness. We give analytic expressions for these quantities in random matrix theory to explore universal aspects of late-time dynamics. Motivated by our random matrix results, we define k-invariance in order to capture the onset of random matrix behavior in physical systems. We then refine our diagnostics in order to study chaotic systems with symmetry by considering Haar-randomness with respect to quotients of the unitary group, and in doing so we generalize our quantum information machinery. We further consider extended random matrix ensembles in the context of strongly-interacting quantum systems dual to black holes. Lastly, we study operator growth in classes of random quantum circuits.
We demonstrate that a system's rate of fidelity decay under repeated perturbations may be measured efficiently on a quantum information processor, and analyze the conditions under which this indicator is a reliable probe of quantum chaos. The type and rate of the decay are not dependent on the eigenvalue statistics of the unperturbed system, but depend on the system's eigenvector statistics in the eigenbasis of the perturbation. For random eigenvector statistics, the decay is exponential with a rate fixed by the variance of the perturbation's energy spectrum. Hence, even classically regular models can exhibit an exponential fidelity decay under generic quantum perturbations. These results clarify which perturbations can distinguish classically regular and chaotic quantum systems.
Chaotic systems are highly sensitive to a small perturbation and are ubiquitous throughout the biological sciences, the physical sciences, and even the social sciences. Taking this as the underlying principle, we construct an operational notion for quantum chaos. Namely, we demand that the future state of a many-body isolated quantum system is sensitive to past multitime operations on a small subpart of that system. By “sensitive,” we mean that the resultant states from two different perturbations cannot easily be transformed into each other. That is, the pertinent quantity is the complexity of the effect of the perturbation within the final state. From this intuitive metric, which we call the butterfly-flutter fidelity, we use the language of multitime quantum processes to identify a series of operational conditions on chaos; in particular, the scaling of the spatiotemporal entanglement. Our criteria already contain the routine notions, as well as the well-known diagnostics for quantum chaos. This includes the Peres-Loschmidt echo, dynamical entropy, tripartite mutual information, and local-operator entanglement. We hence present a unified framework for these existing diagnostics within a single structure. We also go on to quantify how several mechanisms, such as evolution generated from random circuits, lead to quantum chaos. Our work paves the way to systematically study many-body dynamical phenomena such as many-body localization, measurement-induced phase transitions, and Floquet dynamics. Published by the American Physical Society 2024