Multiple phase estimation with photon-added multi-mode coherent states of GHZ-type

TL;DR

This paper investigates multiparameter quantum metrology using GHZ-type photon-added coherent states, deriving analytical bounds and comparing estimation strategies, highlighting the advantages of simultaneous estimation and the influence of quantum resources.

Contribution

It provides explicit analytical expressions for the quantum Cramér-Rao bound for various protocols using GHZ-type PACS, comparing strategies and quantum states for enhanced multiparameter estimation.

Findings

Simultaneous estimation outperforms independent estimation, especially in non-linear protocols.

Increasing coherent state amplitude $|eta|^2$ improves estimation precision.

Antisymmetric states show slightly better precision at higher photon excitation orders.

Abstract

This paper explores multiparameter quantum metrology using Greenberger-Horne-Zeilinger (GHZ)-type photon-added coherent states (PACS) and investigates both independent and simultaneous parameter estimation with linear and non-linear protocols, highlighting the significant potential of quantum resources to enhance precision in multiparameter scenarios. To provide a comprehensive analysis, we explicitly derive analytical expressions for the quantum Cram\'er-Rao bound (QCRB) for each protocol. Additionally, we compare the two estimation strategies, examining the behavior of their QCRBs and offering insights into the advantages and limitations of these quantum states in various contexts. Our results show that simultaneous estimation generally outperforms independent estimation, particularly in non-linear protocols. Furthermore, we analyze how the QCRB varies with the coherent state amplitude $|\alpha|^2$, the number of estimated parameters $d$, and the photon excitation order $n$ across three protocols. The results indicate that increasing $|\alpha|^2$ and decreasing $d$ improves estimation precision. For low $n$, the variation in the QCRB is similar for both symmetric and antisymmetric cases; however, at higher $n$, the antisymmetric case exhibits slightly better precision. The dependence on $d$ is comparable for both types of states. We also compare PACS-based GHZ states with NOON states and entangled coherent states, demonstrating the relative performance of each. Finally, we conclude with an analysis of homodyne detection in the context of a linear protocol, discussing its impact on estimation accuracy.