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 $E$.
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