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arXiv:2602.02663

Tailoring Quantum Chaos With Continuous Quantum Measurements

Preethi Gopalakrishnan, András Grabarits, Adolfo del Campo

quant-phcond-mat.stat-mechhep-thSubmitted Feb 2, 2026• Announced Feb 4, 2026• 0 citations
v1 · Feb 2, 2026v2 · Oct 8, 2026
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Abstract

We investigate the role of quantum monitoring in the dynamical manifestations of Hamiltonian quantum chaos. Specifically, we analyze the generalized spectral form factor, defined as the survival probability of a coherent Gibbs state under continuous energy measurements. We show that quantum monitoring can tailor the signatures of quantum chaos in the dynamics, such as the extension of the ramp in the spectral form factor, by varying the measurement strength and detection efficiency. In particular, a typical quantum trajectory obtained by monitoring with unit efficiency exhibits enhanced quantum chaos relative to the average dynamics and to unitary evolution without measurements.

Comments
5 pages, 2 figures + supplemental material

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We investigate the role of quantum monitoring in the dynamical manifestations of Hamiltonian quantum chaos. Specifically, we analyze the generalized spectral form factor, defined as the survival probability of a coherent Gibbs state under continuous energy measurements. We show that quantum monitoring can tailor the signatures of quantum chaos in the dynamics, such as the extension of the ramp in the spectral form factor, by varying the measurement strength and detection efficiency. In particular, a typical quantum trajectory obtained by monitoring with unit efficiency exhibits enhanced quantum chaos relative to the average dynamics and to unitary evolution without measurements.
In this work, the term “quantum chaos” refers to spectral correlations similar to those found in the random matrix theory. Quantum chaos can be diagnosed through the analysis of level statistics using, e.g., the spectral form factor, which detects both short- and long-range level correlations. The spectral form factor corresponds to the Fourier transform of the two-point spectral correlation function and exhibits a typical slope-dip-ramp-plateau structure (aka correlation hole) when the system is chaotic. We discuss how this structure could be detected through the quench dynamics of two physical quantities accessible to experimental many-body quantum systems: the survival probability and the spin autocorrelation function. The survival probability is equivalent to the spectral form factor with an additional filter. When the system is small, the dip of the correlation hole reaches sufficiently large values at times which are short enough to be detected with current experimental platforms. As the system is pushed away from chaos, the correlation hole disappears, signaling integrability or localization. We also provide a relatively shallow circuit with which the correlation hole could be detected with commercially available quantum computers.
The spectral form factor (SFF), which is the Fourier transform of the two-point spectral correlation, is a fundamental diagnostics of quantum chaos. Nevertheless, its experimental detection in quantum many-body systems was less unexplored until recently. Here, we establish a relationship between the SFF with the quantum Fisher information (QFI) by means of quantum sensing protocols. Furthermore, when the eigenvalues and eigenstates of the disordered Hamiltonian are statistically independent, we prove that QFI is typical both in short and long timescales while numerics suggest that it is also typical at intermediate timescales, which ensures self-averaging of QFI. The self-averaging property renders the experimental measurement of QFI with our protocol less noisy in contrast with the direct measurement of SFF which is known to be non-self-averaging. Our results reveal an intriguing connection between quantum noise and the signatures of quantum chaos.
The survival probability of an initial Coherent Gibbs State (CGS) is a natural extension of the Spectral Form Factor (SFF) to open quantum systems. To quantify the interplay between quantum chaos and decoherence away from the semi-classical limit, we investigate the relation of this generalized SFF with the corresponding l1 -norm of coherence. As a working example, we introduce Parametric Quantum Channels (PQC), a discrete-time model of unitary evolution mixed with the effects of measurements or transient interactions with an environment. The Energy Dephasing (ED) dynamics arises as a specific case in the Markovian limit. We demonstrate our results in a series of random matrix models.
We predict that continuously monitored quantum dynamics can be chaotic. The optimal paths between past and future boundary conditions can diverge exponentially in time when there is time-dependent evolution and continuous weak monitoring. Optimal paths are defined by extremizing the global probability density to move between two boundary conditions, and are then expressed as solutions to a Hamiltonian dynamical system. We investigate the onset of chaos in pure-state qubit systems with optimal paths generated by a periodic Hamiltonian. Specifically, chaotic quantum dynamics are demonstrated in a scheme where two noncommuting observables of a qubit are continuously monitored, and one measurement strength is periodically modulated. The optimal quantum paths in this example bear similarities to the trajectories of the kicked rotor, or standard map, which is a paradigmatic example of classical chaos. We emphasize connections with the concept of resonance between integrable optimal paths and weak periodic perturbations, as well as our previous work on ``multipaths,'' and connect the optimal path chaos to instabilities in the underlying quantum trajectories.
We establish a rigorous connection between quantum coherence and quantum chaos by employing coherence measures originating from the resource theory framework as a diagnostic tool for quantum chaos.We quantify this connection at two different levels: quantum states and quantum channels.At the level of states, we show how several well-studied quantifiers of chaos are, in fact, quantum coherence measures in disguise (or closely related to them).We further this connection for all quantum coherence measures by using tools from majorization theory.Then we numerically study the coherence of chaotic-versus-integrable eigenstates and find excellent agreement with random matrix theory in the bulk of the spectrum.At the level of channels, we show that the coherence-generating power (CGP)-a measure of how much coherence a dynamical process generates on average-emerges as a subpart of the out-of-time-ordered correlator (OTOC), a measure of information scrambling in many-body systems.Via numerical simulations of the (nonintegrable) transverse-field Ising model, we show that the OTOC and CGP capture quantum recurrences in quantitatively the same way.Moreover, using random matrix theory, we analytically characterize the OTOC-CGP connection for the Haar and Gaussian ensembles.In closing, we remark on how our coherence-based signatures of chaos relate to other diagnostics, namely, the Loschmidt echo, OTOC, and the Spectral Form Factor.
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.