# Parameter Estimation of Wormholes beyond the Heisenberg Limit

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## Abstract

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## 1. Introduction

## 2. Quantum Metrology

## 3. Traversable Wormholes

## 4. Parameter Estimation of Wormholes through Nonlinear Effects

#### 4.1. Quantum Fisher Information

#### 4.2. Nonlinear Interferometer

#### 4.3. Parameter Estimation

#### 4.4. Rotation and Translation of the Interferometer

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Homodyne detection: The unknown signal, b, is introduced into a beam splitter with a local oscillator, a, whose phase $\theta $ is controlled. The difference of photons is proportional to the quadrature ${X}_{b}$.

**Figure 3.**Sensitivity of the Quadrature Measurement with a nonlinear medium (Equation (34)) together with theoretical SQL vs. number of photons N for $L=1\mathrm{km}$, $\omega ={10}^{15}\mathrm{Hz}$, $\frac{r}{{b}_{0}}={10}^{8}$, ${n}^{\prime}=1.5$ and $\chi ={10}^{-8}\mathrm{Hz}$. We see that, in the high-number regime shown in the plot, the sensitivity goes in principle significantly beyond the standard quantum limit. This range for the number of photons would correspond to a laser with a wavelength around a thousand of nm and a few W of power, as in LIGO.

**Figure 4.**Theoretical sensitivity for the Quantum Fisher Information and Quadrature Measurement in the nonlinear cases, with the same parameters as in Figure 3. We see that, in the high-number regime discussed in Figure 3, our realistic measurement protocol achieves a precision extremely close to the fundamental limit, exhibiting super-Heisenberg scaling.

**Figure 5.**Improvement of sensitivity with respect to the SQL vs. number of photons N for several values of the nonlinear parameter, $L=1\mathrm{m}$, $M=1\mathrm{GHz}={10}^{9}\mathrm{Hz}$ and $\frac{r}{{b}_{0}}={10}^{9}$, with the rest of the parameters as in Figure 3. The nonlinear case becomes relevant for $\chi <{10}^{-4}\mathrm{Hz}$ in the high-number regime.

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**MDPI and ACS Style**

Sanchidrián-Vaca, C.; Sabín, C.
Parameter Estimation of Wormholes beyond the Heisenberg Limit. *Universe* **2018**, *4*, 115.
https://doi.org/10.3390/universe4110115

**AMA Style**

Sanchidrián-Vaca C, Sabín C.
Parameter Estimation of Wormholes beyond the Heisenberg Limit. *Universe*. 2018; 4(11):115.
https://doi.org/10.3390/universe4110115

**Chicago/Turabian Style**

Sanchidrián-Vaca, Carlos, and Carlos Sabín.
2018. "Parameter Estimation of Wormholes beyond the Heisenberg Limit" *Universe* 4, no. 11: 115.
https://doi.org/10.3390/universe4110115