# Robust Phase Estimation of Gaussian States in the Presence of Outlier Quantum States

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. M-estimator

#### 2.2. Tuning Parameter

#### 2.3. Iterative Algorithm

## 3. Robustness of M-estimator

#### 3.1. Classical Contaminated Model

#### 3.2. Asymptotic Breakdown Point

#### 3.3. Finite Breakdown Point

## 4. Quantum Statistical Model with Outliers

#### 4.1. Quantum Statistical Model with Outlier Quantum States

#### 4.2. Quantum Gaussian State with Outliers

**Single outlier quantum state:**When there is one particular outlier quantum state, the model (28) is expressed as:

**Distributed outlier quantum states:**Consider the case when a center of possible outlier quantum states $\alpha $ are generated by the normal distributions on the phase space with a given dispersion ${\kappa}_{0}$. The quantum contaminated model is expressed as:

#### 4.3. Homodyne Measurement on the Noisy Quantum Gaussian States

## 5. Phase Estimation of Noisy Coherent State

#### 5.1. Numerical Simulation

#### 5.2. Single Outlier Quantum State

#### 5.3. Robustness of M-estimators

#### 5.4. Distributed Outlier Quantum States

#### 5.5. Discussion

## 6. Conclusions and Outlook

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Merkel, S.T.; Gambetta, J.M.; Smolin, J.A.; Poletto, S.; Córcoles, A.D.; Johnson, B.R.; Ryan, C.A.; Steffen, M. Self-consistent quantum process tomography. Phys. Rev. A
**2013**, 87, 062119. [Google Scholar] [CrossRef][Green Version] - Ferrie, C. Self-guided quantum tomography. Phys. Rev. Lett.
**2014**, 113, 190404. [Google Scholar] [CrossRef] [PubMed][Green Version] - Sugiyama, T.; Imori, S.; Tanaka, F. Reliable characterization of super-accurate quantum operations. arXiv
**2018**, arXiv:1806.02696. [Google Scholar] - Huber, P.J. Robust Statistics; John Wiley & Sons: Hoboken, NJ, USA, 2004; Volume 523. [Google Scholar]
- Andrews, D.; Hampel, F. Robust Estimates of Location: Survey and Advances; Princeton University Press: Princeton, NJ, USA, 2015. [Google Scholar]
- Wilcox, R.R. Introduction to Robust Estimation and Hypothesis Testing; Academic Press: Cambridge, MA, USA, 2011. [Google Scholar]
- Hampel, F.R.; Ronchetti, E.M.; Rousseeuw, P.J.; Stahel, W.A. Robust Statistics: The Approach Based on Influence Functions; John Wiley & Sons: Hoboken, NJ, USA, 2011; Volume 196. [Google Scholar]
- Maronna, R.A.; Martin, R.D.; Yohai, V.J.; Salibián-Barrera, M. Robust Statistics: Theory and Methods (with R); John Wiley & Sons: Hoboken, NJ, USA, 2019. [Google Scholar]
- Wasserstein, R.L.; Lazar, N.A. The ASA Statement on p-Values: Context, Process, and Purpose. Am. Stat.
**2016**, 70, 129–133. [Google Scholar] [CrossRef][Green Version] - Camerer, C.F.; Dreber, A.; Holzmeister, F.; Ho, T.H.; Huber, J.; Johannesson, M.; Kirchler, M.; Nave, G.; Nosek, B.A.; Pfeiffer, T.; et al. Evaluating the replicability of social science experiments in Nature and Science between 2010 and 2015. Nat. Hum. Behav.
**2018**, 2, 637–644. [Google Scholar] [CrossRef][Green Version] - Wasserstein, R.L.; Schirm, A.L.; Lazar, N.A. Moving to a World Beyond “p < 0.05”. Am. Stat.
**2019**, 73, 1–19. [Google Scholar] [CrossRef][Green Version] - Braunstein, S.L.; Van Loock, P. Quantum information with continuous variables. Rev. Mod. Phys.
**2005**, 77, 513. [Google Scholar] [CrossRef][Green Version] - Wang, X.B.; Hiroshima, T.; Tomita, A.; Hayashi, M. Quantum information with Gaussian states. Phys. Rep.
**2007**, 448, 1–111. [Google Scholar] [CrossRef][Green Version] - Serafini, A. Quantum Continuous Variables: A Primer of Theoretical Methods; CRC Press: Boca Raton, FL, USA, 2017. [Google Scholar]
- Fujisawa, H.; Eguchi, S. Robust parameter estimation with a small bias against heavy contamination. J. Multivar. Anal.
**2008**, 99, 2053–2081. [Google Scholar] [CrossRef][Green Version] - Helstrom, C.W. The minimum variance of estimates in quantum signal detection. IEEE Trans. Inf. Theory
**1968**, 14, 234–242. [Google Scholar] [CrossRef] - Helstrom, C.W. Quantum detection and estimation theory. J. Stat. Phys.
**1969**, 1, 231–252. [Google Scholar] [CrossRef][Green Version] - Helstrom, C.W.; Kennedy, R. Noncommuting observables in quantum detection and estimation theory. IEEE Trans. Inf. Theory
**1974**, 20, 16–24. [Google Scholar] [CrossRef][Green Version] - Yuen, H.; Lax, M. Multiple-parameter quantum estimation and measurement of nonselfadjoint observables. IEEE Trans. Inf. Theory
**1973**, 19, 740–750. [Google Scholar] [CrossRef] - Helstrom, C.W. Quantum Detection and Estimation Theory; Academic Press: Cambridge, MA, USA, 1976. [Google Scholar]
- Holevo, A.S. Probabilistic and Statistical Aspects of Quantum Theory; Edizioni della Normale: Pisa, Italy, 2011. [Google Scholar]
- Fujiwara, A.; Nagaoka, H. An estimation theoretical characterization of coherent states. J. Math. Phys.
**1999**, 40, 4227–4239. [Google Scholar] [CrossRef] - D’ariano, G.M.; Paris, M.G.; Sacchi, M.F. Parameter estimation in quantum optics. Phys. Rev. A
**2000**, 62, 023815. [Google Scholar] [CrossRef][Green Version] - Paris, M.G.A.; Řeháček, J.E. Quantum State Estimation; Springer: Berlin, Germany, 2004. [Google Scholar]
- Hayashi, M. Quantum Information Theory. In Graduate Texts in Physics; Springer: Berlin, Germany, 2017. [Google Scholar]
- Suzuki, J. Nuisance parameter problem in quantum estimation theory: Tradeoff relation and qubit examples. J. Phys. A: Math. Theor.
**2020**, 53, 264001. [Google Scholar] [CrossRef] - Suzuki, J.; Yang, Y.; Hayashi, M. Quantum state estimation with nuisance parameters. J. Phys. A Math. Theor.
**2020**. [Google Scholar] [CrossRef] - Aspachs, M.; Calsamiglia, J.; Muñoz-Tapia, R.; Bagan, E. Phase estimation for thermal Gaussian states. Phys. Rev. A
**2009**, 79, 033834. [Google Scholar] [CrossRef][Green Version] - Pinel, O.; Jian, P.; Treps, N.; Fabre, C.; Braun, D. Quantum parameter estimation using general single-mode Gaussian states. Phys. Rev. A
**2013**, 88, 040102. [Google Scholar] [CrossRef][Green Version] - Bradshaw, M.; Lam, P.K.; Assad, S.M. Ultimate precision of joint quadrature parameter estimation with a Gaussian probe. Phys. Rev. A
**2018**, 97, 012106. [Google Scholar] [CrossRef][Green Version] - Oh, C.; Lee, C.; Rockstuhl, C.; Jeong, H.; Kim, J.; Nha, H.; Lee, S.Y. Optimal Gaussian measurements for phase estimation in single-mode Gaussian metrology. Npj Quantum Inf.
**2019**, 5, 1–9. [Google Scholar] [CrossRef] - Lee, C.; Oh, C.; Jeong, H.; Rockstuhl, C.; Lee, S.Y. Using states with a large photon number variance to increase quantum Fisher information in single-mode phase estimation. J. Phys. Commun.
**2019**, 3, 115008. [Google Scholar] [CrossRef] - Arnhem, M.; Karpov, E.; Cerf, N.J. Optimal Estimation of Parameters Encoded in Quantum Coherent State Quadratures. Appl. Sci.
**2019**, 9, 4264. [Google Scholar] [CrossRef][Green Version] - Oh, C.; Lee, C.; Lie, S.H.; Jeong, H. Optimal distributed quantum sensing using Gaussian states. Phys. Rev. Res.
**2020**, 2, 023030. [Google Scholar] [CrossRef][Green Version] - Assad, S.M.; Li, J.; Liu, Y.; Zhao, N.; Zhao, W.; Lam, P.K.; Ou, Z.; Li, X. Accessible precisions for estimating two conjugate parameters using Gaussian probes. Phys. Rev. Res.
**2020**, 2, 023182. [Google Scholar] [CrossRef] - Donoho, D.L.; Huber, P.J. The notion of breakdown point. In A Festschrift for Erich L. Lehmann; CRC Press: Boca Raton, FL, USA, 1983; pp. 157–184. [Google Scholar]
- Huber, P.J. Finite sample breakdown of M-and P-estimators. Ann. Stat.
**1984**, 12, 119–126. [Google Scholar] [CrossRef]

**Figure 2.**Performances of M-estimators (bisquare and gamma) and the standard estimators (mean and median) as functions of the sample size.

**Figure 3.**Performances of M-estimators for phase estimation of a coherent state in the presence of a single outlier quantum state.

$\mathit{\epsilon}$ | 0 | 0.025 | 0.05 | 0.075 | 0.1 | 0.125 | 0.15 | 0.175 | 0.2 | 0.225 | 0.25 | 0.275 | 0.3 | 0.325 | 0.35 |

bisquare | 3.21 | 5.00 | 5.01 | 5.00 | 5.00 | 5.11 | 5.99 | 6.00 | 6.08 | 7.11 | 9.07 | 20.02 | 11.00 | 6.67 | 5.29 |

gamma | 7.97 | 7.99 | 7.99 | 8.00 | 8.02 | 8.04 | 8.12 | 8.28 | 8.57 | 8.85 | 9.00 | 9.20 | 9.89 | 11.60 | 17.77 |

$\mathit{\epsilon}$ | 0 | 0.025 | 0.05 | 0.075 | 0.1 | 0.125 | 0.15 | 0.175 | 0.2 | 0.225 | 0.25 | 0.275 | 0.3 | 0.325 | 0.35 |

bisquare | 3.19 | 5.00 | 5.00 | 4.96 | 4.05 | 4.67 | 5.00 | 5.00 | 5.41 | 6.06 | 7.07 | 9.64 | 20.06 | 9.99 | 6.50 |

gamma | 7.97 | 7.99 | 8.01 | 8.01 | 8.02 | 8.02 | 8.05 | 8.09 | 8.09 | 8.12 | 8.18 | 8.20 | 8.21 | 8.23 | 8.38 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Mototake, Y.; Suzuki, J.
Robust Phase Estimation of Gaussian States in the Presence of Outlier Quantum States. *Appl. Sci.* **2020**, *10*, 5475.
https://doi.org/10.3390/app10165475

**AMA Style**

Mototake Y, Suzuki J.
Robust Phase Estimation of Gaussian States in the Presence of Outlier Quantum States. *Applied Sciences*. 2020; 10(16):5475.
https://doi.org/10.3390/app10165475

**Chicago/Turabian Style**

Mototake, Yukito, and Jun Suzuki.
2020. "Robust Phase Estimation of Gaussian States in the Presence of Outlier Quantum States" *Applied Sciences* 10, no. 16: 5475.
https://doi.org/10.3390/app10165475