2026/07/07 by Katsuta Sakai, Nobuyuki Matsumoto
#quant-ph
Continuous measurements can prepare macroscopic mechanical oscillators in conditional quantum states, but their covariance is difficult to verify. The conventional retrodictive estimator assumes a forward--backward covariance symmetry and can be biased, because physical dynamics such as feedback damping reduces the observability of the state from future records. Here, we derive an exact linear-Gaussian estimator from causal, retrodictive, and smoothed trajectories. For a milligram-scale mirror, it agrees with a Riccati prediction based on parameters fixed independently, while the conventional estimate exhibits a covariance-space bias of dM≃3.5. Our method paves the way toward unbiased testing of macroscopic entanglement within a calibrated linear-Gaussian model, applicable to both tabletop mirrors and kg-scale gravitational-wave test masses.