Probing parameters estimation with Gaussian non-commutative measurements

TL;DR

This work addresses parameter estimation for continuous-variable Gaussian channels by preparing the probe with two noncommuting Gaussian measurements on the canonical observables $q$ and $p$. The authors derive the probe covariance after preparation, formulate the Gaussian-channel QFI in terms of the output covariance, and show that tuning the measurement uncertainties can boost the QFI beyond the thermal-probe limit; asymmetry between the measurements also generates coherence in the energy basis, whose parameter-sensitivity correlates with QFI enhancement. Focusing on the quantum attenuator and amplifier channels, they provide explicit QFI expressions and demonstrate a power-law improvement with an asymmetry parameter $\epsilon$ (best-fit exponent $n\approx3$). The results highlight coherence as a resource in metrology, with practical relevance for quantum-optical implementations and potential extensions to gravimetry and magnometry.

Abstract

Gaussian quantum states and channels are pivotal across many branches of quantum science and their applications, including the processing and storage of quantum information, the investigation of thermodynamics in the quantum regime, and quantum computation. The great advantage is that Gaussian states are experimentally accessible via their first and second statistical moments. In this work, we investigate parameter estimation for Gaussian states, in which the probe-state preparation stage involves two noncommutative Gaussian measurements on the position and momentum observables, introducing tunable parameters. The influence of these noncommutative Gaussian measurements is investigated through the quantum Fisher information (QFI). We showed that the QFI for characterizing Gaussian channels can be increased by adjusting the uncertainty parameters in the preparation of the probe state. Furthermore, if the probe is initially in a thermal state, probe-state preparation may generate quantum coherence in its energy basis. We showed that not only does the amount of coherence affect the improvement of the QFI, but also the rate of change of the coherence with respect to the parameter to be estimated. The proposed probe-state protocol is applied to two paradigmatic single-mode Gaussian channels, the attenuator and amplification channels, which are building blocks of Gaussian quantum information. Our results contribute to the use of coherence in quantum metrology and are experimentally feasible in quantum-optical devices.

Figures (6)

• Figure 1: Illustration of the proposed probe state protocol. The single-mode Gaussian state is initially in a thermal state with inverse temperature $\beta$. The first Gaussian measurement is performed in the position observable with an uncertainty $\sigma_q$, followed by a second Gaussian measurement in the momentum observable, with an uncertainty $\sigma_p$. Then, the probe interacts with a single-mode Gaussian channel with a parameter vector $\vec{\theta}$, and we use the final probe state to interrogate a specific channel parameter.
• Figure 2: Behavior of the symplectic eigenvalues and quantum Fisher information for the QAttC. (a) and (b) illustrate the behavior of the product between the symplectic eigenvalues, $\nu_1\nu_2$, as a function of $\varphi$ and $\bar{m}$, respectively. (c) shows the QFI for the estimation of $\varphi$, with a fixed value of $\bar{m} = 0.5$, while (d) shows the QFI for the estimation of $\bar{m}$, with a fixed value of $\varphi = \pi/4$. We set $\beta = \omega = 1$, and the values of $\sigma_q$ and $\sigma_p$ such that the product $\sigma_q\sigma_p$ satisfies the uncertainty principle for the probe state.
• Figure 3: Behavior of the symplectic eigenvalues and quantum Fisher information for the QAmpC. (a) and (b) illustrate the behavior of the product between the symplectic eigenvalues, $\mu_1\mu_2$, as a function of $r_g$ and $\bar{m}$, respectively. (c) shows the QFI for the estimation of $r_g$, with a fixed value of $\bar{m} = 0.5$, while (d) shows the QFI for the estimation of $\bar{m}$, with a fixed value of $r_g = 1.0$. We have considered $\beta = \omega = 1$, and the values of $\sigma_q$ and $\sigma_p$ such that the product $\sigma_q\sigma_p$ satisfies the uncertainty principle for the probe state.
• Figure 4: Quantum coherence of the probe state as a function of the uncertainties $\sigma_q$ and $\sigma_p$ after the two Gaussian measurements in the probe state preparation and before any quantum channel. The dashed black straight line represents the symmetric scenario in which $\sigma_q = \sigma_p$, with no coherence being produced. The probe initial parameters are the same as in Fig. \ref{['QFI_att01']} and Fig. \ref{['QFI_amp01']}.
• Figure 5: Derivative of the quantum coherence with respect to different channel parameters. (a) Derivative of the coherence as a function of the parameter $\varphi$ for the attenuator channel, with the inset the corresponding derivative for the channel average thermal number $\bar{m}$. (b) Derivative of the coherence as a function of the parameter $r_g$ for the amplifier channel, with the inset the corresponding derivative channel average thermal number $\bar{m}$. The probe initial parameters are the same as in Fig. \ref{['QFI_att01']} Fig. \ref{['QFI_amp01']}.