In this work, we develop a quantum metrological framework for quantum chaos by showing that local subsystems of information scrambling systems naturally function as quantum stopwatches. The reduced quantum state of a subsystem encodes the passage of time through its growing distinguishability from the initial preparation. Treating time as the estimation parameter, we then derive a generalized quantum Cramer-Rao bound that directly relates the precision of time estimation to the decay of out-of-time ordered correlators (OTOCs) and subsystem quantum Fisher information (QFI). As a main result, we obtain continuity bounds for quantum Lyapunov exponent in terms of the subsystem QFI in quantumly chaotic dynamics. Furthermore, using a scaling analysis based on imaginary-time correlators, we show that the subsystem QFI exhibits universal critical amplification near quantum phase transitions. Our results are demonstrated and verified by a numerical analysis of the dynamics of a chaotic Ising chain.
In this work, we develop a quantum metrological framework for quantum chaos by showing that local subsystems of information scrambling systems naturally function as quantum stopwatches. The reduced quantum state of a subsystem encodes the passage of time through its growing distinguishability from the initial preparation. Treating time as the estimation parameter, we then derive a generalized quantum Cramer-Rao bound that directly relates the precision of time estimation to the decay of out-of-time ordered correlators (OTOCs) and subsystem quantum Fisher information (QFI). As a main result, we obtain continuity bounds for quantum Lyapunov exponent in terms of the subsystem QFI in quantumly chaotic dynamics. Furthermore, using a scaling analysis based on imaginary-time correlators, we show that the subsystem QFI exhibits universal critical amplification near quantum phase transitions. Our results are demonstrated and verified by a numerical analysis of the dynamics of a chaotic Ising chain.
Fast scrambling, quantified by the exponential initial growth of out-of-time-ordered correlators (OTOCs), is the ability to efficiently spread quantum correlations among the degrees of freedom of interacting systems and constitutes a characteristic signature of local unstable dynamics. As such, it may equally manifest both in systems displaying chaos or in integrable systems around criticality. Here we go beyond these extreme regimes with an exhaustive study of the interplay between local criticality and chaos right at the intricate phase-space region where the integrability-chaos transition first appears. We address systems with a well-defined classical (mean-field) limit, as coupled large spins and Bose-Hubbard chains, thus allowing for semiclassical analysis. Our aim is to investigate the dependence of the exponential growth of the OTOCs, defining the quantum Lyapunov exponent λ_{q} on quantities derived from the classical system with mixed phase space, specifically the local stability exponent of a fixed point λ_{loc} as well as the maximal Lyapunov exponent λ_{L} of the chaotic region around it. By extensive numerical simulations covering a wide range of parameters we give support to a conjectured linear dependence 2λ_{q}=aλ_{L}+bλ_{loc}, providing a simple route to characterize scrambling at the border between chaos and integrability.
We investigate the integrability-to-chaos transition and information scrambling in Ising spin networks via a graph-theoretic formulation. Modeling spins as vertices and interactions as edges encoded by adjacency matrices across path, Erdos-Renyi, and Watts-Strogatz topologies, we demonstrate that long-range couplings and heterogeneous degree distributions markedly accelerate quantum information propagation. The Hamiltonian comprises local and normalized non-local interactions; tuning the non-local coupling and field heterogeneity drives integrability breaking. To quantify scrambling, we employ bipartite mutual and tripartite information. Increasing non-local interactions drives tripartite information to large negative values, signaling deep information scrambling. The squared commutators constructed from out-of-time-order correlators (OTOCs) exhibit early-time exponential growth; equivalently, the OTOC four-point functions themselves decay exponentially, yielding quantum Lyapunov exponents that scale systematically with parameters governing the chaotic regime. Complementing this, Krylov complexity reveals rapid operator growth in the chaotic phase, synchronizing with OTOC and mutual information dynamics. Spectrally, the transition manifests as a shift from Poissonian to Wigner-Dyson level spacing statistics. The spectral form factor (SFF) exhibits the characteristic slope-dip-ramp-plateau structure, enabling the extraction of Thouless and Heisenberg times. Crucially, a reduced Thouless time strongly correlates with accelerated information and operator scrambling. Ultimately, this work establishes a unified framework bridging network topology with information-theoretic, operator, and spectral diagnostics, offering insights into thermalization and non-equilibrium dynamics in quantum many-body systems.
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.
The out-of-time-ordered correlator (OTOC) is central to the understanding of information scrambling in quantum many-body systems. In this work, we show that the OTOC in a quantum many-body system close to its critical point obeys dynamical scaling laws which are specified by a few universal critical exponents of the quantum critical point. Such scaling laws of the OTOC imply a universal form for the butterfly velocity of a chaotic system in the quantum critical region and allow one to locate the quantum critical point and extract all universal critical exponents of the quantum phase transitions. We numerically confirm the universality of the butterfly velocity in a chaotic model, namely, the transverse axial next-nearest-neighbor Ising model, and show the feasibility of extracting the critical properties of quantum phase transitions from OTOC using the Lipkin-Meshkov-Glick model.
The rate at which information scrambles in a quantum system can be quantified using out-of-time-ordered correlators. A remarkable prediction is that the associated Lyapunov exponent λ that quantifies the scrambling rate in chaotic systems obeys a universal bound λ < 2πkBT/h. Previous numerical and analytical studies have indicated that this bound has a quantum-statistical origin. Here, we use path-integral techniques to show that a minimal theory to reproduce this bound involves adding contributions from quantum thermal fluctuations (describing quantum tunneling and zero-point energy) to classical dynamics. By propagating a model quantum-Boltzmannconserving classical dynamics for a system with a barrier, we show that the bound is controlled by the stability of thermal fluctuations around the barrier instanton (a delocalized structure which dominates the tunneling statistics). This stability requirement appears to be general, implying that there is a close relation between the formation of instantons, or related delocalized structures, and the imposition of the quantum-chaos bound.