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

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 Cramér-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 [Formula: see text], the number of estimated parameters [Formula: see text], and the photon excitation order [Formula: see text] across three protocols. The results indicate that increasing [Formula: see text] and decreasing [Formula: see text] improves estimation precision. For low [Formula: see text], the variation in the QCRB is similar for both symmetric and antisymmetric cases; however, at higher [Formula: see text], the antisymmetric case exhibits slightly better precision. The dependence on [Formula: see text] 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.