  We explore the fidelity susceptibility and the quantum coherence along with
the entanglement entropy in the ground-state of one-dimensional spin-1 XXZ
chains with the rhombic single-ion anisotropy. By using the techniques of
density matrix renormalization group, effects of the rhombic single-ion
anisotropy on a few information theoretical measures are investigated, such as
the fidelity susceptibility, the quantum coherence and the entanglement
entropy. Their relations with the quantum phase transitions are also analyzed.
The phase transitions from the Y-N\'{e}el phase to the Large-$E_x$ or the
Haldane phase can be well characterized by the fidelity susceptibility. The
second-order derivative of the ground-state energy indicates all the
transitions are of second order. We also find that the quantum coherence, the
entanglement entropy, the Schmidt gap can be used to detect the critical points
of quantum phase transitions. Conclusions drawn from these quantum information
observables agree well with each other. Finally we provide a ground-state phase
diagram as functions of the exchange anisotropy $\Delta$ and the rhombic
single-ion anisotropy $E$.
