Holevo Cram\'{e}r-Rao bound for multi-parameter estimation in nonlinear interferometers

TL;DR

This paper derives the Holevo Cramér-Rao Bound for multi-parameter estimation in nonlinear interferometers, demonstrating its tightness and comparing it with other bounds and measurement results to advance quantum precision measurement understanding.

Contribution

It provides the first derivation of the HCRB for multi-parameter estimation in nonlinear interferometers and compares it with existing bounds and measurement techniques.

Findings

HCRB equals dual homodyne measurement result

SLD-CRB is not saturable at small squeezing

HCRB is shown to be tight and accurate

Abstract

Due to the potential of quantum advantage to surpass the standard quantum limit (SQL), the nonlinear interferometers have garnered significant attention from researchers in the field of precision measurement. However, many practical applications require multi-parameter estimation. In this work, we discuss the precision limit of multi-parameter estimation of pure Gaussian states based on nonlinear interferometers, and derive the Holevo Cram\'{e}r-Rao Bound (HCRB) for the case where both modes undergo displacement estimation. Furthermore, we compare our analytical results with the quantum Cram\'er-Rao Bound based on the symmetric logarithmic derivative (SLD-CRB), and with the result of the dual homodyne measurement. Through numerical analysis, we find that the HCRB equals the result of the dual homodyne measurement, whereas SLD-CRB is not saturable at small squeezed parameters. Therefore, this indicates that the HCRB is tight. Additionally, we provide intuitive analysis and visual representation of our numerical results in phase space.