Phase damping destroys quantum Fisher information of W states

Abstract

We study the quantum Fisher information (QFI) of W states analytically with respect to SU(2) rotations in the basic decoherence channels i.e. depolarizing (DPC), amplitude damping (ADC) and phase damping (PDC), and present the interesting behavior of QFI of W states, especially when compared to that of GHZ states [Ma et al., Phys. Rev. A, 84, 022302 (2011)]. We find that when initially pure W states are under decoherence, i) DPC: as decoherence starts and increases, QFI smoothly decays; ii) ADC: just as decoherence starts, QFI exhibits a sudden drop to the shot noise level and as decoherence increases, QFI continues to decrease to zero and then increases back to the shot noise level; iii) PDC: just as decoherence starts, a sudden death of QFI occurs and QFI remains zero for any rate of decoherence, therefore W states in phase damping channel do not provide phase sensitivity. We also find that, on the contrary to GHZ states, pure or decohered W states are not sensitive with respect to rotations in z direction and the sensitivities with respect to rotations in x and y directions are equal to each other, implying no sudden change points of QFI due to competition between directions.