Quantum Fisher information of multi-qubit WV entangled state under Lipkin-Meshkov-Glick model
Authors: Li Yan , Ren Zhi-Hong ,   Department of Physics, Taiyuan Normal University, Jinzhong 030619, China,   Institute of Computational and Applied Physics, Taiyuan Normal University, Jinzhong 030619, China,   School of Physics and Information Engineering, Shanxi Normal University, Taiyuan 030031, China
Published: September 2023

As an important quantity in the field of parameter estimation theory and quantum precision measurement, quantum Fisher information (QFI) can not only be used to set the theoretical limit of measurement precision in quantum system, but also be exploited to witness metrological useful quantum entanglement. Recently, it has also been broadly used in many aspects of quantum information science, including quantum metrology, multipartite entanglement structure detection, quantum phase transition, quantum chaos, quantum computation and etc. In this work, from the perspective of quantum measurement, we study the quantum Fisher information of an <i>N</i>-qubit WV state (<inline-formula><tex-math id="M4">\begin{document}$\alpha \left\vert W_N \right\rangle +\sqrt{1-\alpha^2}\left\vert 00\cdots0\right\rangle$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="22-20231179_M4.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="22-20231179_M4.png"/></alternatives></inline-formula>) under local operation and Lipkin-Meshkov-Glick (LMG) model. Furthermore, with the general Cramér-Rao lower bound (CRLB) we analyze its performance in high-precision phase measurement. The results show that, under the local operation, the QFI of an <i>N</i>-qubit WV state becomes larger with the increase of parameter <i>α</i>. This not only means the enhanced quantum entanglement, but also implies the powerful ability in high-precision quantum measurement. In the LMG model, as the increase of interactional strength <i>γ</i> the QFI of <inline-formula><tex-math id="M5">\begin{document}$N=3$\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="22-20231179_M5.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="22-20231179_M5.png"/></alternatives></inline-formula> qubits WV state gradually tends to be stable and almost not be affected by parameter <i>α</i>, which relaxes the requirement in the preparation of target state and indicates a great potential in achieving the relatively stable measurement precision. When the number of qubits from WV state is larger than 3, the QFI of WV state increases with the increase of parameter <i>α</i>. In the case of fixed parameter <i>α</i>, we investigate the QFI of an <i>N</i>-qubit WV state with respect to interaction strength <i>γ</i>. It is found that the QFI of WV state will increase with the increasing interaction strength, which implies that the greater the interaction strength, the stronger the quantum measurement ability of the WV state. Our work will promote the development of high-precision quantum metrology and especially the interaction-enhanced quantum measurement, and further provide new insights in quantum information processing.