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The fidelity susceptibility and entanglement entropy in a system of two-leg$XXZ$ spin ladder with rung coupling is investigated by using exactdiagonalization of the system. The effects of rung coupling on fidelitysusceptibility, entanglement entropy and quantum phase transition are analyzed.It is found that the quantum phase transition between two different $XY$ phasescan be well characterized by the fidelity susceptibility. Though the quantumphase transition from $XY$ phase to rung singlet phase can be hardly detectedby fidelity susceptibility, it can be predicted by the first derivative of theentanglement entropy of the system.

FIDELITY AND ENTANGLEMENT CLOSE TO QUANTUM PHASE TRANSITION IN A TWO-LEG XXZ SPIN LADDER