# Optimal estimation of parameters encoded in coherent states quadratures

## TL;DR

This paper develops and analyzes optimal linear schemes for estimating multiple classical parameters encoded in the quadratures of quantum coherent states, achieving the quantum Cramér-Rao bound under energy constraints.

## Contribution

It introduces a method for constructing optimal multiparameter estimation schemes using coherent states, including phase-conjugate states, and demonstrates their optimality.

## Key findings

- Optimal scheme with phase-conjugate states saturates quantum Cramér-Rao bound.
- General n-mode schemes can encode and estimate n parameters simultaneously.
- Analysis under global energy constraints confirms scheme optimality.

## Abstract

In the context of multiparameter quantum estimation theory, we investigate the construction of linear schemes in order to infer two classical parameters that are encoded in the quadratures of two quantum coherent states. The optimality of the scheme built on two phase-conjugate coherent states is proven with the saturation of the quantum Cram\'er--Rao bound under some global energy constraint. In a more general setting, we consider and analyze a variety of $n$-mode schemes that can be used to encode $n$ classical parameters into $n$ quantum coherent states and then estimate all parameters optimally and simultaneously.