# Optimal Estimation of Parameters Encoded in Quantum Coherent State Quadratures

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## Abstract

**:**

## 1. Introduction

## 2. Two-Mode Coherent-State Parameter Communication Scheme with Linear Encoding

## 3. Parameter Estimation

#### 3.1. Quantum Cramér–Rao Bound

#### 3.2. Optimal Lower Bound

#### 3.3. Attainability of the Quantum Cramér–Rao Bound

## 4. Two-Mode Encoding and Estimation Schemes

#### 4.1. Identical Encoding

#### 4.2. Optimal Scheme

## 5. Arbitrary Number of Modes

#### 5.1. Three-Mode Encoding and Estimation Scheme

#### 5.2. n-Mode Extension

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

- Alice chooses real a, b, and $0\le T\le 1$ by imposing that energies of the “optimal input states” given by Equation (29) are equal to the energies of the given coherent states in corresponding modes. This leads to the following equations:$$\left(\right)$$$$\left(\right)$$If the energies of the given coherent states are not equal, we can replace the second equation by$$\frac{{x}_{1}^{2}+{p}_{1}^{2}-{x}_{2}^{2}-{p}_{2}^{2}}{{a}^{2}-{b}^{2}}=2(2T-1),$$$$2|{a}^{2}-{b}^{2}|\ge |{x}_{1}^{2}+{p}_{1}^{2}-{x}_{2}^{2}-{p}_{2}^{2}|.$$$$T=\frac{1}{2}\left(\right)open="("\; close=")">\frac{{x}_{1}^{2}+{p}_{1}^{2}-{x}_{2}^{2}-{p}_{2}^{2}}{2({a}^{2}-{b}^{2})}+1$$
- In both cases considered above, Equation (29) determines the parameters of two input states, which would provide the desired optimal measurement,$$\begin{array}{cc}{x}_{1}^{\left(\mathrm{opt}\right)}=\hfill & {\displaystyle x\sqrt{\frac{{x}_{1}^{2}+{p}_{1}^{2}-{x}_{2}^{2}-{p}_{2}^{2}}{2({a}^{2}-{b}^{2})}+1},}\hfill \\ {p}_{1}^{\left(\mathrm{opt}\right)}=\hfill & {\displaystyle p\sqrt{\frac{{x}_{1}^{2}+{p}_{1}^{2}-{x}_{2}^{2}-{p}_{2}^{2}}{2({a}^{2}-{b}^{2})}-1},}\hfill \\ {x}_{2}^{\left(\mathrm{opt}\right)}=\hfill & {\displaystyle x\sqrt{\frac{{x}_{1}^{2}+{p}_{1}^{2}-{x}_{2}^{2}-{p}_{2}^{2}}{2({a}^{2}-{b}^{2})}-1},}\hfill \\ {p}_{2}^{\left(\mathrm{opt}\right)}=\hfill & {\displaystyle -p\sqrt{\frac{{x}_{1}^{2}+{p}_{1}^{2}-{x}_{2}^{2}-{p}_{2}^{2}}{2({a}^{2}-{b}^{2})}+1}.}\hfill \end{array}$$$$cos{\theta}_{i}=\frac{{x}_{i}^{\left(\mathrm{opt}\right)}{x}_{i}+{p}_{i}^{\left(\mathrm{opt}\right)}{p}_{i}}{{x}_{i}^{2}+{p}_{i}^{2}}.$$
- After performing the calculations described above, Alice provides to Bob with the measurement settings $\{T,\theta ,\varphi \}$ and the given coherent states.
- Upon receiving the measurement settings, Bob applies local rotations to the input modes followed by the beam splitter transformation and homodyne measurements in the output modes, thus realizing an optimal extraction of encoded x and p variable.

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**Figure 1.**Optimal joint estimation scheme for two modes using a beam splitter of transmittance T and two homodyne detectors. The phases are all set to zero in Equation (28).

**Figure 2.**Enhancement of the measurement precision of the joint (compared to individual) measurement as a function of the beam splitter transmissivity T. This is calculated as the reduction of the error variance for the joint measurement attaining the QCRB with respect to the error variance of the optimal individual measurements.

**Figure 3.**Family of optimal encoding schemes. For a given value of the parameters a and b (here, $a=2$ and $b=1$), each optimal encoding consists of preparing two coherent states (see two black points, two yellow points, two red points, etc.) according to Equation (29).

**Figure 4.**Optimal joint estimation scheme for three modes using two beam splitters of transmittance ${T}_{1}$ and ${T}_{2}$ and a phase shifter of angle $\varphi $.

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**MDPI and ACS Style**

Arnhem, M.; Karpov, E.; Cerf, N.J.
Optimal Estimation of Parameters Encoded in Quantum Coherent State Quadratures. *Appl. Sci.* **2019**, *9*, 4264.
https://doi.org/10.3390/app9204264

**AMA Style**

Arnhem M, Karpov E, Cerf NJ.
Optimal Estimation of Parameters Encoded in Quantum Coherent State Quadratures. *Applied Sciences*. 2019; 9(20):4264.
https://doi.org/10.3390/app9204264

**Chicago/Turabian Style**

Arnhem, Matthieu, Evgueni Karpov, and Nicolas J. Cerf.
2019. "Optimal Estimation of Parameters Encoded in Quantum Coherent State Quadratures" *Applied Sciences* 9, no. 20: 4264.
https://doi.org/10.3390/app9204264