Quantum Chaos, Randomness and Universal Scaling of Entanglement in Various Krylov Spaces

TL;DR

The paper addresses which chaotic quantum systems can generate scalable multipartite entanglement and how to detect it using the quantum Fisher information (QFI). It develops a universal, Krylov-space–dependent expression for the time-averaged QFI, derived via random-matrix theory and ergodicity for Dyson ensembles, showing the QFI scales as $\bar{F}_{\rm chaos} \sim N^2/3$ in the permutation-symmetric subspace and as $\bar{F}_{\rm chaos} \sim N$ for dynamics spanning the full Hilbert space or symmetry-restricted variants. These results are validated in chaotic kicked-top models (COE, CUE, CSE) and chaotic Ising-like systems, and they reveal how symmetry and Krylov-space structure enable Heisenberg-scaling entanglement in certain regimes while linear scaling persists in others. The work also contrasts the QFI-based chaos criterion with OTOCs and semiclassical indicators, showing that QFI provides a robust long-time signature of chaos and offers a practical metrological perspective on entanglement generation in many-body quantum systems. Overall, it strengthens the connection between randomness, chaos, and scalable entanglement, with potential implications for quantum sensing and information processing in complex quantum systems.

Abstract

Multipartite entanglement is a crucial resource for advancing quantum technologies, with considerable research efforts directed toward achieving its rapid and scalable generation. In this work, we derive an analytical expression for the time-averaged quantum Fisher information (QFI), enabling the detection of scalable multipartite entanglement dynamically generated by all quantum chaotic systems governed by Dyson's ensembles. Our approach integrates concepts of randomness and quantum chaos, demonstrating that the QFI is universally determined by the structure and dimension of the Krylov space that confines the chaotic dynamics. In particular, the QFI ranges from $N^2/3$ for $N$ qubits in the permutation-symmetric subspace (e.g. for chaotic kicked top models with long-range interactions), to $N$ when the dynamics extend over the full Hilbert space with or without bit reversal symmetry or parity symmetry (e.g. in chaotic models with short-range Ising-like interactions). In the former case, the QFI reveals multipartite entanglement among $N/3$ qubits and highlights the power of chaotic collective spin systems in generating scalable multipartite entanglement. Interestingly this result can be related to isotropic substructures in the Wigner distribution of chaotic states and demonstrates the efficacy of quantum chaos for Heisenberg-scaling quantum metrology. Finally, our general expression for the QFI agrees with that obtained for random states and, differently from out-of-time-order-correlators, it can also distinguish chaotic from integrable unstable spin dynamics.

Figures (6)

• Figure 1: Panels (a)-(c) show the Wigner distributions (colormap) of the evolved state $|\psi_{\rm chaos}(t)\rangle$ for the COE chaotic model Eq. (\ref{['Floquet-COE']}): (a) $t\!=\!0$, corresponding to the initial coherent spin state aligned along the $y$ axis; (b) $t\!<\!t^*$; and (c) $t\!\gg\!t^*$. Panel (d) shows the Wigner distribution of a typical example of random state with QFI close to $\bar{F}_{\rm rand}$, Eq. (\ref{['Main2']}).
• Figure 2: Level spacing distributions $P(S)$ [panels (a), (e), and (g)], LE $\lambda_{\rm LE}$ [panel (c)], and long-time averaged QFI $\bar{F}_{\rm chaos}$ [panels (b), (d), (f) and (h)] for COE [panels (a)-(d)], CUE [panels (e)-(f)], and CSE [panels (g)-(h)] chaotic kicked top models (\ref{['Floquet-COE']})-(\ref{['Floquet-CSE']}). Panels (a), (e), and (g) compare numerically obtained level spacing distributions (histograms) for models Eqs. (\ref{['Floquet-COE']})-(\ref{['Floquet-CSE']}) with analytic expressions Eq. (\ref{['LS']}) [line, dashed line, and dots, respectively], and verify their classification under RMT. Panels (b), (f), and (h) confirm the consistency between numerically obtained $\bar{F}_{\rm chaos}$ (blue crosses) and the universal scaling $N^2/3$ of QFI from Eq. (\ref{['Main']}) (red line). Their insets depict the dynamic evolution of QFI (blue circles). The short-time behavior is accurately described by exponential growth (red line) $\propto \exp(\lambda_{\rm LE}t)$ with $\lambda_{\rm LE}=1.874$ for $U_{\rm COE}$ and $\lambda_{\rm LE}=1.937$ for $U_{\rm CUE}$. After the transient Ehrenfest time, the QFI exhibits barely noticeable fluctuations around the constant value given by Eq. (\ref{['Main']}). For COE model, the red region in panel (d), given by Eq. (\ref{['Main']}) corresponds to the region in panel (c) with a positive LE. In all panels, we start with a coherent spin state along the $-y$ direction and evolve it under the following chaotic conditions: $A\!=\!1.7, C\!=\!10$ for model Eq. (\ref{['Floquet-COE']}), $p\!=\! 1.7,\lambda\!=\!10,\lambda'\!=\!0.5$ for model Eq. (\ref{['Floquet-CUE']}), and $\lambda_0\!=\! \lambda_1\!=\!2.5,\lambda_2\!=\!5, \lambda_3\!=\! 7.5$ for model Eq. (\ref{['Floquet-CSE']}).
• Figure 3: Level spacing distributions [panel (a)] and long-time averaged QFI [panels (b)] for the chaotic Ising model, Eq. (\ref{['cIsing']}) with parameters $J\!=\!h\!=\!\lambda=1$.
• Figure 4: Time-evolution of QFI (a) and long-time averaged QFI (b) in the integrable LMG model for the initial unstable coherent spin state along the $-z$ direction. The red straight line in (a) (semi-logarithmic coordinate) corresponds to the exponential curve with the rate given by four times LE, $\exp(4\lambda_{\rm LE}t)$. Blue dots in (b) are numerical results, which are fitted by a black solid line with scaling of $F_Q\simeq0.112N^{1.950}$. This behavior deviates from the universal scaling for chaotic collective spin models $N^2/3$, as marked by the solid red line.
• Figure 5: Evolution of QFI with respect to $J_z^2$ in the chaotic COE model with parameters $A=1.7$ and $C=10$.