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

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 6) but asymptotically equal to the GHZ state for large-qubit cases (N >> 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 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.