In terms of quantum Fisher information, a quantity $\chi^{2}$ was introduced by Pezz\'{e} and Smerzi, which is a multiparticle entanglement measure, and provides a necessary and sufficient condition for sub-shot-noise phase estimation sensitivity. We derive a general expression of $\chi ^{2}$ for arbitrary symmetric multiqubit states with nonzero mean spins. It is shown that the entangled symmetric states are useful for phase sensitivity beyond the shot-noise limit. Using the expression, we explicitly examine a series of superpositions of spin states. We find that the superpositions of Dicke states perform better than Dicke states themselves in phase esitmation. Although the spin coherent states themselves only have a shot-noise limit phase sensitivity, their superpositions may reach the Heisenberg limit.
QUANTUM INTERFEROMETRY AT ZERO AND FINITE TEMPERATURE WITH TWO-MODE BOSONIC JOSEPHSON JUNCTIONS
