# Optomechanics-Based Quantum Estimation Theory for Collapse Models

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## Abstract

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## 1. Introduction

## 2. Parameter Estimation Theory

## 3. System and Collapse Mechanism

- The extra input light fields, given by the operators $\{{\widehat{X}}_{i{n}_{i}},{\widehat{Y}}_{i{n}_{i}}\}$ for each cavity $i\in \{1,2\}$.
- The Brownian noise, described by the noise operator $\widehat{\xi}$, characterized by the Markovian correlation functions $\langle \widehat{\xi}\left(t\right)\widehat{\xi}\left({t}^{\prime}\right)\rangle =2{\displaystyle \frac{{\gamma}_{m}{k}_{B}T}{\hslash {\omega}_{m}}}\delta (t-{t}^{\prime})$. Here, ${k}_{B}$ is the Boltzmann constant, while T is the temperature of the surrounding thermal environment.
- The CSL collapse model described by ${\widehat{f}}_{\mathsf{\Lambda}}$, acting as an extra source of decoherence.

## 4. Dynamical Analysis

## 5. Steady-State Analysis

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Bell, J.S.; Aspect, A. Speakable and Unspeakable in Quantum Mechanics: Collected Papers on Quantum Philosophy, 2nd ed.; Cambridge University Press: Cambridge, UK, 2004. [Google Scholar] [CrossRef]
- Zurek, W.H. Decoherence and the transition from quantum to classical. Phys. Today
**1991**, 44, 36–44. [Google Scholar] [CrossRef] - Schlosshauer, M. Decoherence, the measurement problem, and interpretations of quantum mechanics. Rev. Mod. Phys.
**2005**, 76, 1267. [Google Scholar] [CrossRef] [Green Version] - Ghirardi, G.C.; Rimini, A.; Weber, T. Unified Dynamics for Microscopic and Macroscopic Systems. Phys. Rev. D
**1986**, 34, 470. [Google Scholar] [CrossRef] - Bassi, A.; Lochan, K.; Satin, S.; Singh, T.P.; Ulbricht, H. Models of wave-function collapse, underlying theories, and experimental tests. Rev. Mod. Phys.
**2013**, 85, 471. [Google Scholar] [CrossRef] - Ghirardi, G.C.; Pearle, P.; Rimini, A. Markov processes in Hilbert space and continuous spontaneous localization of systems of identical particles. Phys. Rev. A
**1990**, 42, 78. [Google Scholar] [CrossRef] - Bassi, A.; Ghirardi, G. Dynamical reduction models. Phys. Rep.
**2003**, 379, 257–426. [Google Scholar] [CrossRef] [Green Version] - Carlesso, M.; Donadi, S.; Ferialdi, L.; Paternostro, M.; Ulbricht, H.; Bassi, A. Present status and future challenges of non-interferometric tests of collapse models. Nat. Phys.
**2022**, 18, 243. [Google Scholar] [CrossRef] - McMillen, S.; Brunelli, M.; Carlesso, M.; Bassi, A.; Ulbricht, H.; Paris, M.G.; Paternostro, M. Quantum-limited estimation of continuous spontaneous localization. Phys. Rev. A
**2017**, 95, 012132. [Google Scholar] [CrossRef] [Green Version] - Marchese, M.M.; Belenchia, A.; Pirandola, S.; Paternostro, M. An optomechanical platform for quantum hypothesis testing for collapse models. New J. Phys.
**2021**, 23, 043022. [Google Scholar] [CrossRef] - Branford, D.; Gagatsos, C.N.; Grover, J.; Hickey, A.J.; Datta, A. Quantum enhanced estimation of diffusion. Phys. Rev. A
**2019**, 100, 022129. [Google Scholar] [CrossRef] [Green Version] - Genoni, M.G.; Duarte, O.S.; Serafini, A. Unravelling the noise: The discrimination of wave function collapse models under time-continuous measurements. New J. Phys.
**2016**, 18, 103040. [Google Scholar] [CrossRef] - Schrinski, B.; Nimmrichter, S.; Hornberger, K. Quantum-classical hypothesis tests in macroscopic matter-wave interferometry. Phys. Rev. Res.
**2020**, 2, 033034. [Google Scholar] [CrossRef] - Giovannetti, V.; Lloyd, S.; Maccone, L. Quantum metrology. Phys. Rev. Lett.
**2006**, 96, 010401. [Google Scholar] [CrossRef] [Green Version] - Giovannetti, V.; Lloyd, S.; Maccone, L. Advances in quantum metrology. Nat. Photonics
**2011**, 5, 222–229. [Google Scholar] [CrossRef] [Green Version] - Helstrom, C.W. Quantum detection and estimation theory. J. Stat. Phys.
**1969**, 1, 231–252. [Google Scholar] [CrossRef] [Green Version] - Paris, M.G. Quantum estimation for quantum technology. Int. J. Quantum Inf.
**2009**, 7, 125–137. [Google Scholar] [CrossRef] - Braunstein, S.L.; Caves, C.M. Statistical distance and the geometry of quantum states. Phys. Rev. Lett.
**1994**, 72, 3439. [Google Scholar] [CrossRef] [PubMed] - Liu, J.; Yuan, H.; Lu, X.M.; Wang, X. Quantum Fisher information matrix and multiparameter estimation. J. Phys. Math. Theor.
**2020**, 53, 023001. [Google Scholar] [CrossRef] - Weedbrook, C.; Pirandola, S.; García-Patrón, R.; Cerf, N.J.; Ralph, T.C.; Shapiro, J.H.; Lloyd, S. Gaussian quantum information. Rev. Mod. Phys.
**2012**, 84, 621. [Google Scholar] [CrossRef] - Adesso, G.; Illuminati, F. Gaussian measures of entanglement versus negativities: Ordering of two-mode Gaussian states. Phys. Rev. A
**2005**, 72, 032334. [Google Scholar] [CrossRef] [Green Version] - Pirandola, S. Quantum reading of a classical digital memory. Phys. Rev. Lett.
**2011**, 106, 090504. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Genoni, M.G.; Lami, L.; Serafini, A. Conditional and unconditional Gaussian quantum dynamics. Contemp. Phys.
**2016**, 57, 331–349. [Google Scholar] [CrossRef] [Green Version] - Mari, A.; Eisert, J. Gently Modulating Optomechanical Systems. Phys. Rev. Lett.
**2009**, 103, 213603. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Nimmrichter, S.; Hornberger, K.; Hammerer, K. Optomechanical sensing of spontaneous wave-function collapse. Phys. Rev. Lett.
**2014**, 113, 020405. [Google Scholar] - Bassi, A.; Ulbricht, H. Collapse models: From theoretical foundations to experimental verifications. J. Phys. Conf. Ser.
**2014**, 504, 012023. [Google Scholar] [CrossRef] - Gasbarri, G.; Belenchia, A.; Carlesso, M.; Donadi, S.; Bassi, A.; Kaltenbaek, R.; Paternostro, M.; Ulbricht, H. Testing the foundations of quantum physics in space Interferometric and non-interferometric tests with Large Particles. arXiv
**2021**, arXiv:2106.05349. [Google Scholar] - Adler, S.L. Lower and upper bounds on CSL parameters from latent image formation and IGM heating. J. Phys. A Math. Theor.
**2007**, 40, 2935. [Google Scholar] [CrossRef] - Paternostro, M.; Vitali, D.; Gigan, S.; Kim, M.S.; Brukner, C.; Eisert, J.; Aspelmeyer, M. Creating and probing macroscopic entanglement with light. Phys. Rev. Lett.
**2007**, 99, 250401. [Google Scholar] [CrossRef] [Green Version] - Ferraro, A.; Olivares, S.; Paris, M.G. Gaussian states in continuous variable quantum information. arXiv
**2005**, arXiv:quant-ph/0503237. [Google Scholar] - Mirkhalaf, S.S.; Mehboudi, M.; Rahimi-Keshari, S. Operational significance of nonclassicality in nonequilibrium Gaussian quantum thermometry. arXiv
**2022**, arXiv:2207.10742. [Google Scholar]

**Figure 1.**Schematic set-up considered in the main text. Two cavities, one of which has a movable end mirror, are injected with noise described by two Gaussian modes with quadratures $\{{\widehat{X}}_{i{n}_{i}},{\widehat{Y}}_{i{n}_{i}}\}$ with $i=\{1,2\}$. The movable end-mirror in cavity 1 represents a Gaussian mechanical mode with quadratures $\{\widehat{Q},\widehat{P}\}$.

**Figure 2.**Classical and quantum Fisher information for the quantum and the classical schemes. The top panel shows the comparison of the classical Fisher information for the quantum and classical schemes. The bottom panel shows the ratio between the quantum Fisher information and the classical one. This second panel shows that, consistently, the quantum Fisher information upper bounds the classical one in both instances. For the quantum scheme, we set the squeezing angle ${\psi}_{S}=\pi $ for the input TMS light, and ${\phi}_{BS}=\pi /4$ for the beam-splitter angle employed in the EPR measurement. We used $\mathsf{\Lambda}={10}^{6}$, which results from assuming ${r}_{CSL}=100\phantom{\rule{0.166667em}{0ex}}\mathrm{nm}$ and Adler’s collapse rate [28] ${\lambda}_{CSL}\equiv {\lambda}_{A}={10}^{-9}\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}$ for a spherical micrometer mechanical oscillator with mass $m\sim 150$ ng and frequency ${\omega}_{m}\sim {10}^{5}$ Hz. The parameters for the measurement covariant matrix ${\sigma}_{\mathrm{m}}$ are set to $l=1$ and $\theta =0$. Only at small times, up to $t\sim 0.25$ $\mathsf{\mu}\mathrm{s}$, the quantum scheme brings an advantage over the classical one.

**Figure 3.**Classical Fisher information at the steady state against the squeezing parameter r of the TMS input light. We compare two measurement schemes: local measurements (solid curve) and EPR measurements (dashed curve). The squeezing angle of the input TMS state is set to be ${\psi}_{S}=\pi $. However, for the local measurements, this does not change the Fisher information. The EPR measurement scheme uses a beam-splitter angle ${\phi}_{BS}=\pi /4$ to combine the two optical cavity modes. In both cases, the classical Fisher information vanishes with increasing the squeezing parameter r. The inset shows the same plots for the quantum Fisher information, which qualitatively gives the same results.

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**MDPI and ACS Style**

Marchese, M.M.; Belenchia, A.; Paternostro, M.
Optomechanics-Based Quantum Estimation Theory for Collapse Models. *Entropy* **2023**, *25*, 500.
https://doi.org/10.3390/e25030500

**AMA Style**

Marchese MM, Belenchia A, Paternostro M.
Optomechanics-Based Quantum Estimation Theory for Collapse Models. *Entropy*. 2023; 25(3):500.
https://doi.org/10.3390/e25030500

**Chicago/Turabian Style**

Marchese, Marta Maria, Alessio Belenchia, and Mauro Paternostro.
2023. "Optomechanics-Based Quantum Estimation Theory for Collapse Models" *Entropy* 25, no. 3: 500.
https://doi.org/10.3390/e25030500