Quantum Fisher information of N-qubit W and GHZ superposition state under nonlocal operation

From the perspective of nonlocal operation, we investigate the quantum Fisher information of N-qubit W and GHZ superposition state. Taking advantage of the general Ising model and Lipkin–Meshkov–Glick model, we analytically present the QFI of N-qubit superposition state. Evidently the traditional fixed QFI is changed and largely increased. As an paradigm, we numerically study the QFI of 3-qubit superposition state with respect to ratio α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} and interaction strength ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document}, and find the trends of QFI in both models are similar. There always exists a turning point for QFI, and the α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} is around 0.7. Interestingly, for the interaction strength ε=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon =1$$\end{document} in the Ising model, the well-known Heisenberg limit is almost achieved for all α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}. Moreover, under the same interaction strength, we compare the QFI of N-qubit superposition state in both models. With the increasing qubits involved, the α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} for maximal QFI in the Ising model is changed from close to 1 to the middle region, while it is different in Lipkin–Meshkov–Glick model and the α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} is close to 1.

The Greenberger-Horne-Zeilinger (GHZ) state and W state denote two important but different class of multipartite entanglement. Generally, the both cannot be converted under local operations and classical communications (LOCC), while in some situations the conversion can be realized [36]. Therefore, the engineering of the W and GHZ superposition state in experiment is possible and may exhibit some superiority in quantum information processing. The superposition state has been studied by many authors [37][38][39][40][41], but it is researched under local operation and few to be addressed by nonlocal operation [42]. Traditionally, the nonlocal operation (or nonlinear Hamiltonian) is used to create entanglement for quantum metrology [43], accompanying the study of decoherence effects [44,45], and rarely be applied to quantum entangled state. However, with the large-scale multipartite entanglement state created in various experiments, the inevitable interaction or redundant nonlocal operation (e.g., cross talk between qubits [46]) induced by local operation has emerged and affected the high-precision metrology [47].
In this paper, we hence study the performance of QFI from the N -qubit W and GHZ superposition state under nonlocal operation. Among all types of nonlocal operation [48,49], the Ising-type nonlocal operation is well known and well studied, not only representing the simplest two-body interaction but also more relevant to experimental realization. Therefore, we investigate the QFI of N -qubit superposition state in the Ising model [50] and Lipkin-Meshkov-Glick (LMG) model [51]. We analytically present the QFI and analyze it with respect to ratio α and interaction strength ε. The results show that the QFI of superposition state is increased in both models and is beneficial to quantum metrology. In the case of 3-qubit superposition state, an interesting phenomenon is discovered that the conventional Heisenberg limit (the inverse of the number of qubits employed) is almost achieved for all α as ε = 1 in the Ising model. We also find that there always exists a turning point for QFI in both models and the α is around 0.7. Besides, we compare the QFI of N -qubit superposition state under the same ε, and the maximal QFI with respect to ratio α is discussed. a e-mail: li8989971@163.com (corresponding author) The paper is structured as follows: In Sect. 2, we briefly review the N -qubit W and GHZ superposition state and the formula to calculate QFI of a general quantum state under two-body nonlocal operation. Then, in Sect. 3 we, respectively, study the QFI of N -qubit superposition state in the Ising model and LMG model. Specially, we in detail investigate the QFI of 3-qubit superposition state. Finally, we summarize our results in Sect. 4.

Model and method
Nonlocal operation means a many-body interaction process, and usually it is difficult to be tackled. For simplicity, we take two-body interaction as our subject of study, where n represent the local operation and nonlocal operation, respectively. Here μ i denotes inhomogeneous linear couplings and V i j = V ji , σ (i) n = σ (i) · n is the Pauli matrix for the ith particle and n is a vector. The N -qubit W and GHZ superposition state are expressed as: where the coefficients satisfy |α| 2 + |β| 2 = 1. Assuming that the qubit system is denoted by |0 and |1 , the N -qubit W state and N -qubit GHZ state are given by: and According to the formula of QFI for pure state under a unitary transformation e −iĤ θ [4], where and ν 2 2 = 4Δ 2 H 1 separately denote the local QFI, covariant QFI and nonlocal QFI. With respect to a general quantum state and the nonlocal operation Eq. (1), they are specifically given by [52] and The QFI can therefore be obtained through the calculation of first-order moment of the interactive qubits, i.e., σ n , etc. In the following, we will focus on the QFI of N -qubit W and GHZ superposition state in the Ising model and LMG model and further analyze its performance in quantum metrology.

QFI of N -qubit superposition state in Ising model
We first consider the N -qubit superposition state that is operated under the homogeneous nearest-neighbor nonlocal operation, i.e., m · n = 1, It is usually difficult to be managed because there are many a permutation needed to be calculated in a large-scale quantum state. However, due to symmetry of the superposition state Eq. (2), all the terms required for QFI are same under the exchange of qubits. As a result, we just need to calculate any one of these permutations and then assemble them for final QFI. After calculations, we find the explicit form of the first-order moment of interactive qubits is different when the number of qubits is N < 6. Next we take 3-qubit W and GHZ as a paradigm to elaborate it.
At the beginning, we present the QFI of N -qubit superposition state in the Ising model, Given the 3-qubit W and GHZ superposition state, the averaged terms are calculated as: Replacing the above equations into Eq. (9), the QFI of 3-qubit superposition state is obtained. Similarly, we have access to the QFI of 4, 5-qubit superposition state. As the number of qubits increases larger than 5, a general formula is found to represent the QFI of superposition state |ψ N . The averaged terms are calculated as: and the QFI of N -qubit superposition state is achieved by substituting above equations into Eq. (9). Due to the lengthy of the formula, it is not shown here. In Fig. 1, we numerically show the QFI of 3-qubit superposition state |ψ 3 with respect to α. It is interesting that the QFI is almost unchanged as ε = 1 in Fig. 1a, which implies that the superposition state can be used to achieve the conventional Heisenberg limit in all α region. This may be helpful to the experimental research. Meanwhile, we find that the QFI under local operation (ε = 0) is in line with the result shown in Ref. [38], where the RMQFI is plotted with respect to α. In Fig. 1b, with the increase of ε, the trends of QFI are similar and there always exists a turning point for QFI, where the α is around 0.7. For the case of N -qubit superposition state, we present the QFI with respect to α in Fig. 2, in which the interaction strength is same and ε = 2. It is easy to found that with the increasing qubits involved, the ratio α for the maximal QFI is changed from the left side (close to 1) to the middle region. Meanwhile, the position of α for the turning point appeared in the 3-qubit superposition state is changed, as it is shown by the blue line and other colorful lines. Additionally, there also exists a sudden change of QFI for all N -qubit superposition state and the α is close to 1.

QFI of N -qubit superposition state in LMG model
In this section, we consider the fully-connected nonlocal operation, Lipkin-Meshkov-Glick model, originated from the nuclear physics, and used to describe many-body problem approximation [51]. Likewise, m · n = 1 and V i j = ε, the QFI of N -qubit superposition state is expressed as: where all the interactive terms required are same under the exchange of qubits. Replacing Eqs. (11,12,13) into Eq. (14), we have access to the QFI of 3-qubit superposition state in the LMG model. In Fig. 3, we numerically plot the QFI with respect to ratio α. Obviously it looks like an enlarged version of Fig. 1b, except for the case ε = 1. Because all the interactive terms are contributed to final QFI, it gives a larger QFI in the LMG model than the Ising model. Similarly, there still exists a turning point for all N -qubit superposition states.   To describe it clearly, we in detail record the value of turning points (tps) from both models in Table 1, where the results are numerically obtained under the different ε. It shows that the tps for 3-qubit superposition state is relatively stable and the ratio α is around 0.7, except for the case of ε = 1 in the Ising model.
We then investigate the QFI of N -qubit superposition state in LMG model. Under the same interaction strength ε = 2, the QFI of N = 3, . . . , 8 qubits superposition state with respect to α is shown in Fig. 4. Apparently, with the increase in the number of qubits, the QFI is largely enhanced and the turn points are almost disappeared. However, the sudden change of QFI is still displayed when the ratio α approaches 1, and this indicates that if one chooses the large-scale superposition state for quantum metrology, the optimal α for the maximal QFI is close to 1 but not 1.

Conclusions
In summary, we have studied the QFI of N -qubit W and GHZ superposition state under nonlocal operation. By utilizing the formula to calculate QFI under two-body nonlocal operation, we explicitly present the QFI of superposition state in the Ising model and LMG model. The results show that the conventional fixed QFI is evidently changed and becomes a parameter-controlled QFI. This greatly facilitates the quantum precision measurement and other quantum information processing. To learn more about the QFI in both models, we in detail investigate the QFI of 3-qubit superposition state with respect to ratio α and interaction strength ε. There always exists a turning point for QFI in both models, and it is around α = 0.7. Meanwhile, for ε = 1 in the Ising model, the well-known Heisenberg limit is almost attained for all α. Then, we compare the QFI of N -qubit superposition state under the same interaction strength and find the position of α for the maximal QFI is different in both models. With the increase in the number of qubits, the optimal α in the Ising model is changed from close to 1 to the middle region, whereas it is unchanged in the LMG model. We hope that our work promotes the development of nonlinear quantum metrology and further provides help for quantum information processing.