# Estimation with Heisenberg-Scaling Sensitivity of a Single Parameter Distributed in an Arbitrary Linear Optical Network

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## Abstract

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## 1. Introduction

## 2. Quantum Estimation Based on Single-Homodyne Measurements

#### 2.1. Setup

#### 2.2. Heisenberg Scaling

- It is possible to loosen the optimal conditions found in literature, which still allow us to reach the Heisenberg scaling, at the price of a multiplying factor $0<\varrho (k,\ell )\le 1$ which does not depend on N and hence does not ruin the scaling of the precision;
- These conditions are explicitly expressed in terms of the average number N of photons in the probe and, therefore, in terms of the precision we want to achieve. In Section 2.3, we will discuss how this allows us to assess the precision needed to engineer suitable auxiliary stages ${V}_{\mathrm{in}}$ and ${V}_{\mathrm{out}}$ to reach the Heisenberg scaling, showing that it is possible to avoid an iterative adaptation of the optical network.

#### 2.3. Conditions for the Heisenberg Scaling

#### 2.4. A Two-Channel Network

## 3. Quantum Estimation Based on Multi-Homodyne Measurements

#### 3.1. Setup

#### 3.2. Heisenberg Scaling

#### 3.3. Conditions for the Heisenberg Scaling

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Probability Distributions from Homodyne Measurements

## Appendix B. Fisher Information for Gaussian Probabilities

## Appendix C. Asymptotic Analyses of Gaussian Metrology

#### Appendix C.1. Single Homodyne

#### Appendix C.2. Multiple Homodyne

## Appendix D. Maximum-Likelihood Estimators for Gaussian Distributions

## Appendix E. Formulas for the Determinant of a Sum of Two Matrices

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**Figure 1.**Example of a passive and linear network ${\widehat{U}}_{\phi}$ which depends on a single global parameter $\phi $. The parameter can be thought of as a physical property of an external agent (e.g., temperature, electromagnetic field) which affects multiple components, possibly of different natures, of the network [42,43]. Reprinted with permission from ref. [42], © 2021 The Author(s).

**Figure 2.**Schematic diagram of the setup described in Section 2.1. The squeezed vacuum state in Equation (2) is injected in the first channel of a network composed of a first auxiliary stage ${\widehat{V}}_{\mathrm{in}}$, a network ${\widehat{U}}_{\phi}$ which depends on a generally distributed parameter $\phi $ we want to estimate, and a second auxiliary stage ${\widehat{V}}_{\mathrm{out}}$, before being detected through homodyne measurements in the first output port. The role of the two auxiliary stages ${\widehat{V}}_{\mathrm{in}}$ and ${\widehat{V}}_{\mathrm{out}}$ is to respectively distribute the photons of the probe through multiple channels, and then to refocus them into the only observed channel. We will show that only one auxiliary network needs to be optimized to reach the Heisenberg scaling, while, for networks with a large number of channels, the effect of the non-optimized network is typically irrelevant on the overall precision of the estimation [42]. Reprinted with permission from ref. [42], © 2021 The Author(s).

**Figure 3.**Polar plot of the standard deviation ${\sigma}_{\phi}$ (see Equation (6)) in blue, and of the Fisher information $\mathcal{F}\left(\phi \right)$ in Equation (16) in orange as functions of the phase $\theta $ of the quadrature ${\widehat{x}}_{\theta}$ measured, for ${P}_{\phi}=1$. The large values of $\mathcal{F}\left(\phi \right)$ are reached for $\theta $, satisfying condition (14). Interestingly, for $\theta ={\gamma}_{\phi}\pm \pi /2$, namely when measuring the quadrature with minimum variance, ${\sigma}_{\phi}$ reaches its minimum, but the Fisher information drops to zero: as a squeezing-encoding estimation scheme, this model relies on the information about $\phi $ inscribed in the variance of the quadrature measured. On the other hand, the minimum variance is a stationary point as a function of $\phi $, and thus is locally insensitive to the variations of the parameter. Reprinted with permission from ref. [34], © 2021 The Author(s).

**Figure 4.**Schematic diagram of the two-channel network described in Section 2.4. The linear network ${\widehat{U}}_{\phi}$ is composed of a beam splitter with coefficient ${\eta}_{\phi}$ and two phase-shifts of magnitudes ${\lambda}_{\phi}$ and ${\lambda}_{\phi}^{\prime}$. The auxiliary stage ${\widehat{V}}_{\mathrm{in}}$ at the input is $\phi $-independent, while the output stage ${\widehat{V}}_{\mathrm{out}}$ is optimized after a classical prior estimation ${\phi}_{\mathrm{cl}}$ of the parameter. In particular, the quantity ${\alpha}_{{\phi}_{\mathrm{cl}}}=({\lambda}_{{\phi}_{\mathrm{cl}}}-{\lambda}_{{\phi}_{\mathrm{cl}}}^{\prime})/2-\pi /4$ depends on ${\phi}_{\mathrm{cl}}$ only through the phase-shifts ${\lambda}_{{\phi}_{\mathrm{cl}}}$ and ${\lambda}_{{\phi}_{\mathrm{cl}}}^{\prime}$. Reprinted with permission from ref. [42], © 2021 The Author(s).

**Figure 5.**Scheme of the setup described in Section 3. A squeezed coherent state is injected in the first input port of a network ${\widehat{U}}_{\phi}$ which depends on a parameter $\phi $ that is generally distributed among multiple components of the network. Homodyne detection is performed at each of the output ports. Differently from the setup in Figure 2, no auxiliary stage is required to reach the Heisenberg scaling. Reprinted with permission from ref. [43], © 2022 The Author(s).

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**MDPI and ACS Style**

Triggiani, D.; Tamma, V.
Estimation with Heisenberg-Scaling Sensitivity of a Single Parameter Distributed in an Arbitrary Linear Optical Network. *Sensors* **2022**, *22*, 2657.
https://doi.org/10.3390/s22072657

**AMA Style**

Triggiani D, Tamma V.
Estimation with Heisenberg-Scaling Sensitivity of a Single Parameter Distributed in an Arbitrary Linear Optical Network. *Sensors*. 2022; 22(7):2657.
https://doi.org/10.3390/s22072657

**Chicago/Turabian Style**

Triggiani, Danilo, and Vincenzo Tamma.
2022. "Estimation with Heisenberg-Scaling Sensitivity of a Single Parameter Distributed in an Arbitrary Linear Optical Network" *Sensors* 22, no. 7: 2657.
https://doi.org/10.3390/s22072657