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

TL;DR

This paper derives an analytical expression for the quantum Fisher information in chaotic quantum systems, revealing how entanglement scales with system size and the structure of Krylov spaces, with implications for quantum metrology.

Contribution

It introduces a universal formula for the time-averaged quantum Fisher information in chaotic systems, linking entanglement generation to Krylov space structure and quantum chaos.

Findings

QFI ranges from N^2/3 to N depending on Krylov space

Chaotic systems can generate scalable multipartite entanglement

QFI distinguishes chaotic from integrable dynamics

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.