Quantum metrology with an N -qubit W superposition state under noninteracting and interacting operations

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We study the performance of an $N$-qubit $W$ superposition state composed of a $W$ state and its obverse in quantum metrology. Taking advantage of the general Ising-type Hamiltonian (including noninteracting and interacting operation), we analytically present the quantum Fisher information (QFI) of an $N$-qubit $W$ superposition state under different situations and then investigate its phase sensitivity. The results show that the phase sensitivity under noninteracting operation displays a crossover from the $W$ state to Greenberger-Horne-Zeilinger (GHZ) state, where it is same as $W$ state in the few-qubit case $(N\ensuremath{\le}6)$ but asymptotically equal to the GHZ state for large-qubit cases $(N\ensuremath{\gg}1)$. Interestingly, the 4-qubit $W$ superposition state is found to have the same sensitivity as the 4-qubit GHZ state. And the optimal measurement protocols are provided for ideal metrology. Under the phase-amplitude damping channel, the phase sensitivity of the $W$ superposition state (except for $N=3$) is ultimately decreased to the standard quantum limit, while it turns worse in a depolarizing channel. Finally, the tunable phase sensitivity under interacting operation is studied, and the general Heisenberg limit is surpassed with the increasing interaction strength $\ensuremath{\gamma}$. Meanwhile, a plateau of QFI and phase sensitivity is found for all large-qubit $W$ superposition states, which is similar to the study of the GHZ state and again verifies the common feature of GHZ-type states in quantum metrology.