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Privacy in Quantum Estimation^{ †}

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## Abstract

**:**

## 1. Introduction

## 2. Private Quantum Estimation

## 3. Application to Two-Qubit Unitaries

- (i)
**Estimation of ${\alpha}_{z}$**. We took 325 points in the region $0\le {\alpha}_{y}\le {\alpha}_{x}\le \frac{\pi}{2}$ and for each point we estimated ${\alpha}_{z}$ through ${\rho}_{B}$ and independently through ${\rho}_{F}$. This is done by also optimizing the privacy (5) over the probe’s state, i.e., by considering$${\mathcal{P}}_{e}({\alpha}_{x},{\alpha}_{y})=max\left(\right)open="\{"\; close="\}">\underset{x}{max}\left(\right)open="["\; close="]">{\overline{C}}_{min}^{F}({\alpha}_{x},{\alpha}_{y},x)-{\overline{C}}_{min}^{B}({\alpha}_{x},{\alpha}_{y},x),$$On the line ${\alpha}_{x}+{\alpha}_{y}=\frac{\pi}{2}$ we have ${\overline{C}}_{min}^{B}={\overline{C}}_{min}^{F}$ and this divides the region $0\le {\alpha}_{y}\le {\alpha}_{x}\le \frac{\pi}{2}$ into two triangles. Only in the lower one the estimation is private (in the upper one ${\overline{C}}_{min}^{F}$ results smaller than ${\overline{C}}_{min}^{B}$). Furthermore there is a specific and small region where the privacy increases with respect to the background.- (ii)
**Estimation of ${\alpha}_{y}$**. In this case we took 325 points in the region $0\le {\alpha}_{z}\le {\alpha}_{x}\le \frac{\pi}{2}$, and for each point we estimated ${\alpha}_{y}$ through ${\rho}_{B}$ and independently through ${\rho}_{F}$ likewise the previous case. Then we evaluated the privacy$${\mathcal{P}}_{e}({\alpha}_{x},{\alpha}_{z})=max\left(\right)open="\{"\; close="\}">\underset{x,\phi}{max}\left(\right)open="["\; close="]">{\overline{C}}_{min}^{F}({\alpha}_{x},{\alpha}_{z},x,\phi )-{\overline{C}}_{min}^{B}({\alpha}_{x},{\alpha}_{z},x,\phi ),$$- (iii)
**Estimation of ${\alpha}_{x}$**. In this last case we took 325 points in the region $0\le {\alpha}_{z}\le {\alpha}_{y}\le \frac{\pi}{2}$ and for each point we estimated ${\alpha}_{x}$ through ${\rho}_{B}$ and independently through ${\rho}_{F}$. We actually computed$${\mathcal{P}}_{e}({\alpha}_{y},{\alpha}_{z})=max\left(\right)open="\{"\; close="\}">\underset{x,\phi}{max}\left(\right)open="["\; close="]">{\overline{C}}_{min}^{F}({\alpha}_{y},{\alpha}_{z},x,\phi )-{\overline{C}}_{min}^{B}({\alpha}_{y},{\alpha}_{z},x,\phi ),$$

## 4. Conclusion

## Author Contributions

## Funding

## Conflicts of Interest

## References

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1 | This choice by virtue of (7) forces ${\alpha}_{y}$ to be exactly determined. |

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**MDPI and ACS Style**

Rexiti, M.; Mancini, S.
Privacy in Quantum Estimation. *Proceedings* **2019**, *12*, 13.
https://doi.org/10.3390/proceedings2019012013

**AMA Style**

Rexiti M, Mancini S.
Privacy in Quantum Estimation. *Proceedings*. 2019; 12(1):13.
https://doi.org/10.3390/proceedings2019012013

**Chicago/Turabian Style**

Rexiti, Milajiguli, and Stefano Mancini.
2019. "Privacy in Quantum Estimation" *Proceedings* 12, no. 1: 13.
https://doi.org/10.3390/proceedings2019012013