Quantum retrodiction in Gaussian systems and applications in optomechanics

TL;DR

This paper explores quantum retrodiction in Gaussian systems, providing a practical framework for optomechanical applications, enabling near-ideal measurements of mechanical quadratures for quantum state tomography.

Contribution

It introduces a practical retrodictive POVM framework for Gaussian states in optomechanics, enabling direct access to mechanical quadratures and advancing quantum state tomography methods.

Findings

Retrodictive POVMs can be practically implemented in optomechanical systems.

Near-ideal quadrature measurements are achievable, facilitating quantum state reconstruction.

The framework applies to both resonant and off-resonant driving conditions.

Abstract

What knowledge can be obtained from the record of a continuous measurement about the quantum state the measured system was in at the beginning of the measurement? The task of quantum state retrodiction, the inverse of the more common state prediction, is rigorously and elegantly addressed in quantum measurement theory through retrodictive Positive Operator Valued Measures. This article provides an introduction to this general framework, presents its practical formulation for retrodicting Gaussian quantum states using continuous-time homodyne measurements, and applies it to optomechanical systems. We identify and characterise achievable retrodictive POVMs in common optomechanical operating modes with resonant or off-resonant driving fields and specific choices of local oscillator frequencies in homodyne detection. In particular, we demonstrate the possibility of a near-ideal measurement of the quadrature of the mechanical oscillator, giving direct access to the position or momentum distribution of the oscillator at a given time. This forms the basis for complete quantum state tomography, albeit in a destructive manner.