Measurement-Based Estimation of Causal Conditional Variances and Its Application to Macroscopic quantum phenomenon

Abstract

We analytically investigate a quantum estimation method for a mechanical oscillator in a detuned cavity system based solely on homodyne measurement records, building on the framework developed by C.Meng et al. (Science Advances 8, 7585 (2022)). Estimation based only on measurement records is important because it enables state verification without assuming knowledge of the true system state. We construct a relative estimate operator from causal and anti-causal quantum Wiener filters and calculate its variance. The deviation from the causal conditional variance is defined as a reconstruction bias, whose magnitude is evaluated analytically. We show that, within experimentally relevant parameter regimes for typical quantum-state preparation, the reconstruction bias is sufficiently small to be neglected. As applications to state verification, we apply the method to proposals for macroscopic quantum entanglement mediated by electromagnetic interactions and for conditional momentum-squeezed states generated by homodyne detection, and clarify the conditions under which the bias remains negligible and when the reconstruction bias becomes significant.

Figures (4)

• Figure 1: These figures show the ratio between the variances obtained from estimation without access to the true system values and those from causal estimation, plotted on the plane spanned by the homodyne angle $\theta$ and the mechanical dissipation rate $\Gamma[\text{Hz}]$. The left panel corresponds to the position variance, and the right panel corresponds to the momentum variance. Values close to unity indicate that the variance obtained without access to true values closely matches the causal variance, implying that the reconstruction bias is small. Throughout the plots, we fix $\Delta/\kappa = 0.2$, $m=1[\text{mg}]$, $\Omega/2\pi=1[\text{Hz}]$, $\kappa/2\pi=10^8$[Hz] and $P_{\text{in}} = 10^{-5}\text{W}$. The regions in which the estimation becomes extremely poor correspond to the homodyne angle $\theta = \arctan{(2\Delta/\kappa)}$, at which the measurement rate vanishes. In these plots, the reconstruction bias for both the position and momentum variances remains sufficiently small.
• Figure 2: These figures show the ratio of the conditional variances reconstructed via measurement-based estimation to those obtained from causal estimation, plotted as a function of (left) the laser power $P_{\text{in}}[\mathrm{W}]$, and (right) the mechanical damping rate $\Gamma$[Hz]). The blue and red curves represent the ratios for position and momentum, respectively. Solid curves correspond to amplitude (X) measurement $(\theta = 0)$, while dashed curves correspond to phase (Y) measurement $(\theta = \pi/2)$. For the $Y$-quadrature measurement, we set $\Delta/\kappa=0$, whereas for the $X$-quadrature measurement we use $\Delta/\kappa=0.2$. For both panels, the common parameters were fixed as $m=1$[mg] and $\Omega/2\pi=1$[Hz]. In the left panel, $P_\text{in}=10^{-5}$[W] was used, whereas in the right panel, $\Gamma/2\pi=\times10^{-2}$[Hz] was fixed.
• Figure 3: This plot shows the ratio of the logarithmic negativity obtained from measurement-based estimation to that from causal estimation, plotted as a function of the effective mechanical dissipation rate $\gamma_m$. The red star marks the value of $\gamma_m$ used in the previous study Miki2023. Within the parameter range shown, the system remains entangled throughout. All other parameters are taken from Ref. Miki2023. Representative values are $\Omega/2\pi=2.2[\text{Hz}]$, $m=0.92[\text{g}]$, $\Delta/\kappa=0.2$, and $\kappa_-/2\pi=1.64\times10^6[\text{Hz}]$. The homodyne angle is set to $\theta = 0$, corresponding to an $X$-measurement.
• Figure 4: This plot shows the ratio of the momentum variance to the position variance as a function of the input laser power $P_\text{in}$ [W]. The blue curve corresponds to $X$-measurement $(\theta = 0)$ with $\Delta/\kappa = 0.02$. The green curve represents the case $\Delta/\kappa = 0.02$ with $\theta = \theta_\text{opt}$. The pink curve corresponds to $\Delta/\kappa = 0$ with $\theta = \theta_\text{opt}$. Solid curves denote the numerical results obtained from causal estimation, while dashed curves indicate those from measurement-based estimation. The parameters are fixed at $\Omega/2\pi = 1$ [Hz], $m = 100$ [mg], $\kappa/2\pi = 10^3$ [Hz], and $\Gamma/2\pi = 10^{-2}$ [Hz].