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

TL;DR

This work investigates multiparameter quantum metrology using GHZ-type photon-added coherent states (PACS), deriving analytical quantum Cramér-Rao bounds for independent, linear, and nonlinear parameterization schemes. It demonstrates that simultaneous estimation generally outperforms independent strategies, particularly in nonlinear settings, and analyzes how QCRBs scale with the coherent-state amplitude $|α|^2$, the number of parameters $d$, and photon-excitation order $n$. Comparisons with ECS and NOON states show PACS-based GHZ states achieving the highest precision, improving further as $n$ and $|α|^2$ grow, while homodyne detection provides a near-quantum-limited option in linear protocols. The results offer guidance on protocol choice based on resource availability and target precision, highlighting the conditions under which linear, nonlinear, and simultaneous strategies yield the greatest gains in multiparameter quantum metrology.

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 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 $|α|^2$, the number of estimated parameters $d$, and the photon excitation order $n$ across three protocols. The results indicate that increasing $|α|^2$ 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.

Figures (5)

• Figure 1: The variation of ${|\delta\varphi|_{Ind}^{2}}$ versus the parameters ${|\alpha|^2}$ and d for various values of photon excitation number n. (a)d=5 and l=0, (b)d=5 and l=1, (c)$|\alpha|^{2}$=4 and l=0. (b)$|\alpha|^{2}$=4 and l=1.
• Figure 2: The variation of $|\delta\varphi|_{L}^{2}$ versus the parameters ${|\alpha|^2}$ and d for various values of photon excitation number n. (a)d=5 and l=0, (b)d=5 and l=1, (c)$|\alpha|^{2}$=4 and l=0. (b)$|\alpha|^{2}$=4 and n=1.
• Figure 3: The variation of $|\delta\varphi|_{NL}^{2}$ versus the parameters ${|\alpha|^2}$ and d for various values of photon excitation number n. (a)d=5 and l=0, (b)d=5 and l=1, (c)$|\alpha|^{2}$=4 and n=0. (b)$|\alpha|^{2}$=4 and l=1.
• Figure 4: The variation of $|\delta\varphi|_{Ind}^{2}$, $|\delta\varphi|_{L}^{2}$, and $|\delta\varphi|_{NL}^{2}$ as a function of the amplitude $|\alpha|^{2}$ with the total parameter number is set to d = 5 here. a.) Represent the entangled coherent state ECS. b.) Represent the case of the NOON states.
• Figure 5: The variation of $|\delta\varphi|_{Ind}^{2}$, $|\delta\varphi|_{L}^{2}$, and $|\delta\varphi|_{NL}^{2}$ as a function of the amplitude $d$ with the amplitude is set to $|\alpha|^{2}=4$ here. c.) Represent the case of an entangled coherent state ECS. d.) Represent the case of the NOON states.