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