Chaos in classical systems can be characterized by Lyapunov exponents that measure the exponential divergence of nearby trajectories, but directly extending this framework to quantum mechanics has been a persistent challenge. The wavelike nature of quantum states and the non-commutative geometry of quantum phase space obstruct a straightforward generalization of classical chaos theory. Here we develop a rigorous approach to quantum chaos by leveraging quantum optimal transport theory, which provides the missing geometric foundation for measuring distances between extended quantum distributions. We define quantum Lyapunov exponents that naturally avoid divergences encountered when naïvely generalizing classical exponents, and show that in the semiclassical limit they recover the classical global expansion rate, and hence the usual maximal Lyapunov exponent when they coincide. Our framework provides a tight connection between the divergence of classical trajectories and semiclassical phase space evolution, and additionally clarifies the role of out-of-time-order correlators as diagnostics of quantum chaos. These results establish quantum optimal transport as a unifying mathematical foundation for quantum chaos theory, providing new tools to characterize dynamical behavior across the full range of quantum dynamics from simple few-body models to complex many-body systems.
Out-of-time-order correlators being explored as a measure of quantum chaos, are studied here in a coupled bipartite system.Each of the subsystems can be chaotic or regular and lead to very different out-of-time-order correlators growths both before and after the scrambling or the Ehrenfest time.We present preliminary results then on weakly coupled subsystems which have very different Lyapunov exponents.We also review the case when both the subsystems are strongly chaotic when a random matrix model can be pressed into service to derive an exponential relaxation to saturation.
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.
Out-of-Time-Ordered Commutators (OTOCs), representing a key diagnostic for scrambling as a facet of short-time quantum chaos, have attracted wide-ranging interest, from many-body physics to quantum gravity. By means of a suitable form of the Wigner-Moyal expansion, and invoking ensemble equivalence in statistical physics, we provide a consistent approach to the growth rate of the OTOC for many-body systems with chaotic classical limit where both the classical Lyapunov exponent and the quantum nature of the density of states enter. Applying this construction to quantized high-dimensional hyperbolic motion, i.e., a quantum chaotic system that exhibits gravity-like correlation functions in the late-time regime, we compute the OTOC growth rate
Λ as a function of the number of degrees of freedom,
f, and inverse temperature,
β.
We show that the scaled growth rate,
Λ/f, can be described by a universal function of
fβ and displays a cross-over from classical to quantum behavior as we increase
f and/or lower the temperature. In the deep quantum regime of infinite
f, we find maximally fast scrambling in the sense of the Maldacena-Shenker-Stanford bound on chaos. This elucidates the non-perturbative mechanism underlying the saturation of the bound via quantum contributions to the mean density of states, and it provides further support for this dynamical system as a dual to two-dimensional quantum gravity. In this way, we present first evidence of maximally fast scrambling in a quantum chaotic system with a well-defined classical Hamiltonian limit, without invoking any external mechanism such as (disorder) averaging.
In this work, we show, that relative oscillations in asymptotic times of microcanonical Out-of-Time-Order correlators (OTOCs) are suppressed by chaos with the size of the sys- tem. Therefore they can serve as an indicator of quantum chaos in quantum systems. We demonstrate this phenomenon on numerical results of the Out-of-Time-Order correlators from a model with hamiltonian based on the u(3) algebra. Firstly we prove, that this model can be used not only for the description of molecular vibrations, but also Bose- Einstein condensate. Then we employ several methods for the analysis of both quantum and classical chaos, namely spectral statistics such as Nearest Neighbour Spacing Distri- bution and Brody distribution, Inverse Participation Ratio, Poincaré sections, Lyapunov exponent, and the chaotic fraction of classical phase space. We compare these results with relative oscillations of microcanonical Out-of-Time-Order correlators and show, that their suppression corresponds with the chaoticity of the quantum system and the fraction of chaotic regions of the classical phase space. 1
Krylov complexity has recently emerged as a new paradigm to characterize quantum chaos in many-body systems. However, which features of Krylov complexity are a prerogative of quantum chaotic systems and how they relate to more standard probes, such as spectral statistics or out-of-time-order correlators (OTOCs), remain open questions. Recent insights have revealed that in quantum chaotic systems Krylov state complexity exhibits a distinct peak during time evolution before settling into a well-understood late-time plateau. In this work we propose that this Krylov complexity peak (KCP) is a hallmark of quantum chaotic systems and suggest that its height could serve as an order parameter for quantum chaos. We demonstrate that the KCP effectively identifies chaotic-integrable transitions in two representative quantum-mechanical models at both infinite and finite temperature: the mass-deformed Sachdev-Ye-Kitaev model and the sparse Sachdev-Ye-Kitaev model. Our findings align with established results from spectral statistics and OTOCs while introducing an operator-independent diagnostic for quantum chaos, offering more universal insights and a deeper understanding of the general properties of quantum chaotic systems.