  We study the fidelity and the entanglement entropy for the ground states of
quantum systems that have infinite-order quantum phase transitions. In
particular, we consider the quantum O(2) model with a spin-$S$ truncation,
where there is an infinite-order Gaussian (IOG) transition for $S = 1$ and
there are Berezinskii-Kosterlitz-Thouless (BKT) transitions for $S \ge 2$. We
show that the height of the peak in the fidelity susceptibility ($\chi_F$)
converges to a finite thermodynamic value as a power law of $1/L$ for the IOG
transition and as $1/\ln(L)$ for BKT transitions. The peak position of $\chi_F$
resides inside the gapped phase for both the IOG transition and BKT
transitions. On the other hand, the derivative of the block entanglement
entropy with respect to the coupling constant ($S^{\prime}_{vN}$) has a peak
height that diverges as $\ln^{2}(L)$ [$\ln^{3}(L)$] for $S = 1$ ($S \ge 2$) and
can be used to locate both kinds of transitions accurately. We include
higher-order corrections for finite-size scalings and crosscheck the results
with the value of the central charge $c = 1$. The crossing point of $\chi_F$
between different system sizes is at the IOG point for $S = 1$ but is inside
the gapped phase for $S \ge 2$, while those of $S^{\prime}_{vN}$ are at the
phase-transition points for all $S$ truncations. Our work elaborates how to use
the finite-size scaling of $\chi_F$ or $S^{\prime}_{vN}$ to detect
infinite-order quantum phase transitions and discusses the efficiency and
accuracy of the two methods.
