Quantum metrology with superposition of GHZ state and Twin-Fock state

We study the metrological performance of the superposition of Greenberger–Horne–Zeilinger state and Twin Fock state (SGT) in quantum phase estimation. The analytical quantum Fisher information (QFI) of an N-qubit SGT state in noninteracting environment has been derived and with the increasing ratio β in SGT state it emerges a turning point at critical βt=1−1N⋅23, where the smallest QFI FQ=(N2+2N)/3 is achieved. Interestingly, we find the 4-qubit SGT state performs better than TF state in all ratio region and further achieves the Heisenberg limit at β=3/2. Working in the Mach–Zehnder interferometry, we have provided the optimal metrology schemes saturated by quantum CRLB for all kinds of SGT states and found the same βt also exists in the selection of phase generator and measured parity operator. Moreover, we have considered the decoherence effects and with the increasing decay the QFI in amplitude damping and phase damping channel decrease to the number of qubits involved, while it decreases to zero in depolarizing channel. Specially, the 4-qubit SGT state of β=3/2 is found to perform better than GHZ state in phase damping channel. Our work enriches the metrological useful quantum resources.