Semiclassical Approach to Quantum Fisher Information

TL;DR

The paper introduces a semiclassical method to compute the quantum Fisher information (QFI) by linking it to the Loschmidt echo and expressing the echo via the Van Vleck-Gutzwiller propagation. The central result, $I_{ ext{sc}}(oldsymbol{z}_0,t) = \frac{4}{ħ^2} \text{var}igl(\partial S/\partial β\bigr)$, expresses QFI as a phase-space variance of the classical action derivative, enabling efficient, accurate QFI access for chaotic and mixed systems. The authors demonstrate high accuracy and computational efficiency in the quantum kicked top, aligning with exact results up to the Heisenberg time and extending smoothly to larger Hilbert spaces; they also validate the approach on the quantum kicked rotor and Henon-Heiles system. This work provides a practical bridge between classical dynamics and quantum metrological performance, offering a tool for designing quantum sensors in regimes where full quantum calculations are prohibitive.

Abstract

Quantum sensors driven into the quantum chaotic regime can have dramatically enhanced sensitivity, which, however, depends intricately on the details of the underlying classical phase space. Here, we develop an accurate semiclassical approach that provides direct and efficient access to the phase-space-resolved quantum Fisher information (QFI), the central quantity that quantifies the ultimate achievable sensitivity. This approximation reveals, in very concrete terms, that the QFI is large whenever a specific dynamical quantity tied to the sensing parameter displays a large variance over the course of the corresponding classical time evolution. Applied to a paradigmatic system of quantum chaos, the kicked top, we show that the semiclassical description is accurate already for modest quantum numbers, i.e., deep in the quantum regime, and it extends seamlessly to very high quantum numbers that are beyond the reach of other methods.

Figures (9)

• Figure 1: Comparison of the semiclassical QFI, Eq. \ref{['eq:mainVVG1']}, with the exact QFI for the quantum kicked top as a function of the position of the initial $\mathrm{SU}(2)$ coherent state. The initial coherent state is parametrized in terms of the phase space variables $\phi\in(0,2\pi)$ and $z\in(-1,1)$, which are discretized using 220 and 150 equidistant points, respectively, in the $\phi$ and $z$ directions. The calculations are performed for $J=4096$, $t=8$, $\beta=1.5$, and $k=3$. The portraits of the exact QFI in (a) and the semiclassical result in (b) are in excellent agreement. Both quantities are plotted on a logarithmic scale to cover their wide range of values and verify agreement into regions where they are both small. (c) Quantifies the difference between the two results in terms of the relative deviation $\Delta_I=\left|{\bar{I}}-{\bar{I}}_\mathrm{sc}\right|/({\bar{I}} + {\bar{I}}_\mathrm{sc})$.
• Figure 2: Comparison of the exact and semiclassical results for the QFI of the QKT in terms of the phase space average $\bar{I}$ and the phase space variance $\mathrm{var}\, I$, as a function of $J$ and $t$ for fixed $k=4$. All quantities are plotted on a logarithmic scale to cover a wide range of values and parameters. (a),(c) Heat maps of $\bar{I}$ and $\mathrm{var}\,I$ from the exact QFI. (b),(d) Deviation of the corresponding results of the semiclassical approach from the exact values. According to panels (a) and (c), the average QFI and its fluctuations in phase space both increase with $t$ and $J$. The white areas in panels (b) and (d) demonstrate excellent agreement of the semiclassical method in the semiclassical regime of large $J$. The red dashed lines indicate scaling with the Heisenberg time $t_{H}\propto J$.
• Figure 3: Convergence of the averaged semiclassical QFI and its variance as a function of the phase space resolution parameter, $r$, for two different conditions in Fig. \ref{['fig:ave_var_comp_fixed_kz']} (where $k=4$). In (a), $J=1024$ and $t=4$, as indicated by the blue dot in Figs. \ref{['fig:ave_var_comp_fixed_kz']}(b) and \ref{['fig:ave_var_comp_fixed_kz']}(d). This corresponds to the regime where the semiclassical description is highly accurate. Panel (b) contrasts this with the case $J=64$ and a larger time $t=22$, indicated by the red dot in Figs. \ref{['fig:ave_var_comp_fixed_kz']}(b) and \ref{['fig:ave_var_comp_fixed_kz']}(d), where the semiclassical method provides a less accurate, but still reasonable, estimate.
• Figure 4: Illustration of the construction of grid points utilized to evaluate the semiclassial QFI $I_{sc}$ around a central point (white dot). Denoting the arc distance between neighboring points as $d_0$, and assuming that the radius of the shown cap--- measured in arc length--- equals the width $\sigma$ of the evaluated Gaussian distribution, the figure corresponds to a phase space resolution parameter $r=\sigma/d_0=6$. In the numerical implementation the cap extends to an arc length $R_\mathrm{eff}=5\sigma$, while $r=50$, unless stated otherwise, resulting in approximately $\approx 1.96\times 10^5$ grid points.
• Figure 5: Comparison of the exact and semiclassical results for the QFI in terms of the phase space average $\bar{I}$ and the phase space variance $\mathrm{var}\, I$, in analogy to Fig. 2 of the main text, but as a function of $J$ and $k$ for fixed time $t=12$. Excellent agreement is obtained in the semiclassical regime of large $J$, and this agreement is more readily attained for small and moderate values of $k$.

Theorems & Definitions (1)

• proof