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

TL;DR

This paper develops an analytical quantum estimation method using homodyne measurements for mechanical oscillators, enabling state verification without prior knowledge, and evaluates its accuracy and applicability to macroscopic quantum states.

Contribution

It introduces a measurement-based estimation framework using quantum Wiener filters and analytically assesses the estimation bias in relevant quantum systems.

Findings

Reconstruction bias is negligible in typical experimental regimes.

The method effectively verifies macroscopic quantum entanglement.

Bias becomes significant under certain parameter conditions.

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.