# Quantum Probes for the Characterization of Nonlinear Media

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## Abstract

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## 1. Introduction

## 2. Local Multiparameter Quantum Estimation Theory

## 3. QFI Matrix for Optical Non-Linearities

## 4. Optimal Probes for Individual Estimation

## 5. Optimal Probes for Joint Estimation

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**First line: The QFI ${\mathcal{F}}_{\lambda \lambda}^{\left(q\right)}$ of Equation (19) as a function of the squeezing phase $\theta $ and coherent amplitude phase $\varphi $ for $N=3$ and for different values of the order of nonlinearity $\zeta $: from bottom to top $\zeta =2,3$ and 4. Second line: The QFI ${\mathcal{F}}_{\zeta \zeta}^{\left(q\right)}$ of Equation (21) rescaled by ${\lambda}^{2}$ as a function of the squeezing parameter phase $\theta $ and coherent amplitude phase $\varphi $ for $N=3$ and for different values of the order of nonlinearity $\zeta $: from bottom to top $\zeta =2,3$ and 4. On both lines, the plots refer to different values of the squeezing ratio: (left panels) $\gamma =0.01$, (middle panels) $\gamma =0.5$ and (right panels) panel: $\gamma =0.99$. Notice that the quantity ${\mathcal{F}}_{\zeta \zeta}^{\left(q\right)}/{\lambda}^{2}$ is independent of $\lambda $.

**Figure 2.**Upper plots: ${\mathcal{F}}_{\lambda \lambda}^{\left(q\right)}$ and ${\mathcal{F}}_{\zeta \zeta}^{\left(q\right)}/{\lambda}^{2}$ for a coherent probe, i.e., $\gamma =0$, as a function of the coherent state phase $\varphi $ for $N={\left|\alpha \right|}^{2}=2$ (dashed lines) and $N={\left|\alpha \right|}^{2}=3$ (solid lines) and different values of the order of nonlinearity: form bottom to top $\zeta =2,3$ and 4. Note that for $\zeta =2$ we have ${\mathcal{F}}_{\zeta \zeta}^{\left(q\right)}/{\lambda}^{2}=16$ (lower line the right panel). Lower plots: ${\mathcal{F}}_{\lambda \lambda}^{\left(q\right)}$ and ${\mathcal{F}}_{\zeta \zeta}^{\left(q\right)}/{\lambda}^{2}$ for a squeezed vacuum probe, i.e., $\gamma =1$, as functions of the squeezing phase $\theta $ for $N={sinh}^{2}r=2$ (dashed lines) and $N={sinh}^{2}r=3$ (solid lines) and different values of the order of nonlinearity: form bottom to top $\zeta =2,3$ and 4.

**Figure 3.**The optimal squeezing fraction ${\gamma}_{\mathrm{opt}}$ maximizing ${\mathcal{F}}_{\lambda \lambda}^{\left(q\right)}$ and ${\mathcal{F}}_{\zeta \zeta}^{\left(q\right)}$ for different values of the nonlinearity order $\zeta $. The horizontal lines corresponds to the asymptotic value given in Equation (49). See the text for details.

**Figure 4.**Plot of ${\mathcal{F}}_{\lambda \lambda}^{\left(q\right)}$ as a function of $\gamma $ and N for $\zeta =3$. The right panel is a magnification of the left one to highlight the behaviour of the QFI in the regime $N\ll 1$. The blue line refers to the maximum of the QFI (see also the right panel of Figure 3). Analogous results can be obtained for ${\mathcal{F}}_{\lambda \lambda}^{\left(q\right)}$ and other values of $\zeta $. See the text for details.

**Figure 5.**Left panel: optimal value of the fraction of squeezing $\gamma $ for the scalar bound ${C}_{S}^{-1}(\mathit{I},\{\lambda ,\zeta \})$ as a function of N and for $\lambda =0.01$ (solid lines), $\lambda =1$ (dashed lines) and $\lambda =100$ (dotted lines). Right panel: threshold value ${N}_{\mathrm{th}}$ we observe in the left panel. If $N<{N}_{\mathrm{th}}$ the squeezed vacuum is optimal, otherwise the optimal probe has ${\gamma}_{\mathrm{opt}}<1$.

**Figure 6.**Plot of ${C}_{S}^{-1}(\mathit{I},\{\lambda ,\zeta \})$ as a function of $\gamma $ and N for $\zeta =3$. The right panel is a magnification of the left one to highlight the behaviour of the QFI in the regime $N\ll 1$. The blue line refers to the maximum of the QFI at fixed N.

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**MDPI and ACS Style**

Candeloro, A.; Razavian, S.; Piccolini, M.; Teklu, B.; Olivares, S.; Paris, M.G.A.
Quantum Probes for the Characterization of Nonlinear Media. *Entropy* **2021**, *23*, 1353.
https://doi.org/10.3390/e23101353

**AMA Style**

Candeloro A, Razavian S, Piccolini M, Teklu B, Olivares S, Paris MGA.
Quantum Probes for the Characterization of Nonlinear Media. *Entropy*. 2021; 23(10):1353.
https://doi.org/10.3390/e23101353

**Chicago/Turabian Style**

Candeloro, Alessandro, Sholeh Razavian, Matteo Piccolini, Berihu Teklu, Stefano Olivares, and Matteo G. A. Paris.
2021. "Quantum Probes for the Characterization of Nonlinear Media" *Entropy* 23, no. 10: 1353.
https://doi.org/10.3390/e23101353