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

Reply to the Report of Referee A:
Report
I thank the authors for the corrections made. I may again suggest to apply various different chaos measures and compare them directly. This would make the message of this paper much clearer. Possible measures could be entropy growth, spectral quantum chaos measures (distribution of ratios of gaps, higher-order spectral correlation functions), correlations functions, ..., please see any book on quantum chaos.

Reply: We thank the Referee for the interesting suggestion. However, our manuscript is focused on the analysis of the quantum Fisher information. We believe that extending the analysis to entropy growth, spectral quantum chaos measures, and correlation functions-although highly valuable-would blur the focus of our work. We believe that the message of our work is quite clear: unlike regular, stable, as well as unstable dynamics, the chaotic dynamics provide a quantum Fisher information (a quantity related to entanglement and optimal sensitivity in quantum sensing) that is constant in time. The constant value reached after a transient Ehrenfest time is equal to N^2/3, which characterizes Heisenberg scaling, and is recovered by an analytical method. We believe, supported also by the other positive reviewer, that these results are of high interest to a broad community to guarantee publication in SciPost Physics.

Reply to the Report of Referee B:
Reply: We thank the Referee for patiently explaining his/her point of view. However, we respectfully disagree with this interpretation. The misunderstanding arises from the notion of averaging. In our setting, we consider the state $|\Psi_{\rm chaos}(t)\rangle$ generated by chaotic dynamics and then calculate its quantum Fisher information (QFI), $F_Q[|\Psi_{\rm chaos}(t)\rangle;J_\alpha]$, with respect to different measurement directions $\alpha = x,y,z$. In Fig.6 (a-d), we plot the evolution of $F_Q(t)$ under chaotic dynamics [panels (a-c)] and integrable dynamics [panel (d)], considering different measurement directions $\alpha$. Our first main result shows that, in all cases of collective chaotic spin dynamics, the long-time averaged QFI approaches N2/3, i.e.,
\begin{equation}
\lim_{T \to \infty} \frac{1}{T - t^*} \int_{t^*}^T dt \,
F_Q[|\psi_{\mathrm{chaos}}(t)\rangle, \hat{J}_\alpha]
= \frac{N^2}{3},
\end{equation}

where $t^*$ denotes the Ehrenfest time. Equation (1) implies that for $t > t^*$

\begin{equation}
F_Q[|\psi_{\mathrm{chaos}}(t)\rangle, \hat{J}_\alpha]
= \frac{N^2}{3} + \mathcal{O}(N),
\end{equation}

where $\mathcal{O}(N)$ represents the deviation from the dominant term,
i.e., the averaged QFI $N^2/3$. For example, it is obviously shown in Fig.~1(a) when $t > 10$

\begin{equation}
\frac{3 F_Q[|\psi_{\mathrm{chaos}}(t)\rangle, \hat{J}_\alpha]}{N^2} \simeq 1.
\end{equation}

Equation~(2) thus clarifies the meaning of the averaging in Eq.~(1).
In contrast, the Referee’s example involves averaging over a mixture of different pure states.
Indeed, it is generally true that

\begin{equation}
\sum_{k} p_k F_Q[|\psi_k\rangle, \hat{O}]
\;\neq\;
F_Q\!\left[\sum_{k} p_k |\psi_k\rangle \langle \psi_k|, \hat{O}\right].
\end{equation}

We emphasize, however, that our averaging is performed
\textbf{over time, not over ensembles of different quantum states}.

Secondly, since Eq.~(2) is independent of the measurement direction
$\alpha = x, y, z$, we conclude that, compared with the GHZ state, chaotic states
offer an advantage in that no specific measurement direction is required.
This conclusion is supported not only by our analytic derivations but also
by numerical simulations: as shown in Fig.~6(a), the three curves corresponding
to different directions converge to the same value for $t > t^*$.
By contrast, in the integrable model, this behavior is absent, as illustrated in Fig.~6(d).

Changed parts are marked in red.