# Scattering as a Quantum Metrology Problem: A Quantum Walk Approach

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## Abstract

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## 1. Introduction

## 2. Quantum Walks with Inhomogeneous On-Site Energies

## 3. Tools of Local Quantum Estimation Theory

## 4. Scattering in the Presence of an Obstacle

#### 4.1. Scattering States

#### 4.2. Gaussian Wave Packets

#### 4.3. Scattering with Gaussian Wave Packets

## 5. Quantum Estimation of a Scattering Potential

#### Dichotomic Position Measurement

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CTQW | Continuous-time quantum walk |

CR | Cramér–Rao |

FI | Fisher information |

QFI | Quantum Fisher information |

QSNR | Quantum signal-to-noise ratio |

## Appendix A. Gaussian Wave Packet in K-Space

## Appendix B. The Explicit Expression of the Functions g_{H} (Δ, k_{0}) and g_{F} (Δ, k_{0})

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**Figure 2.**Transmission probability $\tau \left(t\right)$. The left plot is for a fixed value of $\Delta =1$ and for decreasing values of ${k}_{0}={k}_{1},\phantom{\rule{0.166667em}{0ex}}{k}_{2},\phantom{\rule{0.166667em}{0ex}}{k}_{3}$, with ${k}_{1}=\pi /2$ (black), ${k}_{2}=\pi /4$ (red), ${k}_{3}=\pi /7$ (blue). In the right plot, ${k}_{0}=\pi /2$ is kept fixed while varying the disorder $\Delta =1$ (black), $\Delta =2$ (red), $\Delta =3$ (blue). The dashed lines correspond to the value of the transmission coefficient ${\tau}_{\mathcal{G}}({k}_{0},\Delta )$ in Equation (25). In both plots, we considered $\sigma =15$.

**Figure 3.**Left: QFI ${H}_{\mathcal{G}}({k}_{0},\Delta )$ for an initial Gaussian wave packet with $\sigma =5$. Right: QSNR ${R}_{\mathcal{G}}({k}_{0},\Delta )$ for the same initial Gaussian wave packet.

**Figure 4.**Comparison between the QFI (upper panel) and the QSNR (lower panel) with a large and a narrow initial wave packet in k-space, as a function of $\Delta $ and for three different values of ${k}_{0}$. The black solid lines are for $\sigma =20$, while the dashed red lines are for $\sigma =5$. The considered values of initial momentum are ${k}_{0}=\frac{\pi}{4},\frac{\pi}{3},\frac{\pi}{2}$ for the left, center, and right column, respectively.

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**MDPI and ACS Style**

Zatelli, F.; Benedetti, C.; Paris, M.G.A.
Scattering as a Quantum Metrology Problem: A Quantum Walk Approach. *Entropy* **2020**, *22*, 1321.
https://doi.org/10.3390/e22111321

**AMA Style**

Zatelli F, Benedetti C, Paris MGA.
Scattering as a Quantum Metrology Problem: A Quantum Walk Approach. *Entropy*. 2020; 22(11):1321.
https://doi.org/10.3390/e22111321

**Chicago/Turabian Style**

Zatelli, Francesco, Claudia Benedetti, and Matteo G. A. Paris.
2020. "Scattering as a Quantum Metrology Problem: A Quantum Walk Approach" *Entropy* 22, no. 11: 1321.
https://doi.org/10.3390/e22111321