Metrological Sensitivity beyond Gaussian Limits with Cubic Phase States

TL;DR

This work shows that cubic phase states provide a robust metrological advantage in continuous-variable rotation sensing, achieving a quantum Fisher information scaling of $F_Q \sim \frac{128}{3} n^2$ which surpasses the Gaussian limit $F_Q \sim 8 n^2$. The authors establish that the enhancement is captured by nonlinear squeezing of order four, with $\xi_{(4)}^{-2}$ saturating the quantum Cramér–Rao bound, and they provide analytical expressions and asymptotics for optimal parameters $s_{opt}$ and $r_{opt}$. They further demonstrate robustness to common imperfections such as loss and detection noise, showing substantial preservation of advantage under realistic conditions. Importantly, several practical schemes for approximate cubic phase state generation—repeat-until-success with photon subtraction, Kerr-based protocols, and trisqueezing—achieve notable metrological gains, indicating the approach is viable with current or near-term CV platforms. Collectively, the results position cubic phase states as a promising non-Gaussian resource for metrology that can outperform Gaussian limits while remaining experimentally accessible.

Abstract

Cubic phase states provide the essential non-Gaussian resource for continuous-variable quantum computing. We show that they also offer significant potential for quantum metrology, surpassing the phase-sensing sensitivity of all Gaussian states at equal average photon number. Optimal sensitivity requires only moderate initial squeezing, and the non-Gaussian advantage remains robust against loss and detection noise. We identify optimal measurement strategies and show that several experimentally relevant preparation schemes surpass Gaussian limits, in some cases reaching the sensitivity of cubic phase states. Our results establish cubic phase states as a promising resource for quantum-enhanced precision measurements beyond Gaussian limits.

Figures (8)

• Figure 1: Rotation sensing with a cubic phase state. (a) Wigner function of a cubic phase state under the rotation $e^{-i \hat{n} \theta}$, corresponding to a phase-space rotation around the origin. (b) Sensitivity $F_Q(r,s)/n$ as a function of squeezing $s$ and cubicity $r$. The metrological potential increases rapidly with $r>0$, with cubic phase states significantly outperforming squeezed vacuum states ($r=0$). The white dashed line indicates the optimal squeezing $s_{\text{opt}}$.
• Figure 2: Metrological scaling laws for cubic phase states. (a) Maximum sensitivity gain for cubic phase states (red solid line) compared with squeezed vacuum states (yellow solid line). In the limit of large population $n$, cubic phase states scale as $F_Q/n \sim \frac{128}{3}n \approx 42.7n$ (red dashed line), clearly exceeding the squeezed-vacuum scaling $F_Q/n \sim 8n$ (yellow dashed line). (b) Optimal squeezing $s_{\text{opt}}$ (green solid line) and cubicity $|r_{\text{opt}}|$ (blue solid line) that achieve the maximum $F^{\max}_Q/n$. Asymptotically, they approach $s_{\text{opt}}^{n\rightarrow \infty} = \frac{1}{2} \log{\left( \sqrt{6}/2 \right)} \approx 0.1014$ (green dashed line), corresponding to only a moderate level of $\sim 0.88\,\mathrm{dB}$ of squeezing, and $|r_{\text{opt}}^{n\rightarrow \infty }| = \frac{4\sqrt{n}}{9}$ (black dashed line).
• Figure 3: Non-Gaussian sensitivity revealed by nonlinear squeezing coefficients. (a) The squeezing coefficients $\xi_{(i)}^{-2}$ obtained from nonlinear measurements of order up to $i=\{1,2,3,4\}$ for the cubic phase state with optimal parameters $(s_{\text{opt}}, |r_{\text{opt}}|)$ as a function of population $n$. In the hierarchy $\xi_{(1)}^{-2} \leq \xi_{(2)}^{-2} \leq \xi_{(3)}^{-2} \leq \xi_{(4)}^{-2}$ the fourth-order squeezing coefficient is found to exactly saturate the ultimate limit, $\xi_{(4)}^{-2} = F^{\text{max}}_Q/n$, while the third order $\xi_{(3)}^{-2}$ already follows it almost tightly. At $n=0.2$, we present the decay of the sensitivity as a function of (b) $\gamma t$, where $\gamma$ is the loss rate and the time evolution is fixed as $t=1$, and (c) the standard deviation $\sigma$ of Gaussian detection noise.
• Figure 4: Metrological potential of practical approximate cubic phase states. The maximum sensitivity gains $F_Q/n$ of practical preparation schemes--sequential photon subtractions MarshallPRA2015 (blue solid line, darker tones indicate larger iterations $N=1,\dots,5$), squeezed Kerr dynamics GuoPRA2024VenkatramanPNAS2024 (green solid line), and Kerr interaction combined with Gaussian channels YanagimotoPRL2020 (yellow solid line)--are shown as a function of population $n$. These results are compared to the benchmarks provided by ideal cubic phase states ($F_Q/n \sim 42.7 n$, red dashed line), squeezed vacuum states ($F_Q/n \sim 8 n$, black dashed line), and coherent states ($F_Q/n \sim 4$, black dotted line). The blue square indicates the sensitivity inferred from experimental tomographic data of Ref. ErikssonNC2024.
• Figure 5: The sensitivity of cubic phase states in displacement sensing.$F_Q/n$ as a function of squeezing strength $s$ for different cubicity $r$. At $r=0$, it becomes squeezed vacuum states.