Irreversibility line of the MgB${}_2$ superconductor

Magneto-resistivity of a c-axis oriented MgB${}_2$ thin film was studied in perpendicular and parallel magnetic fields up to $\sim$23 T with temperatures down to 0.38 K. Resistive critical magnetic fields were determined. Large separations between irreversibility lines and upper critical magnetic fields are observed. An effective quantum parameter as a function of temperature is proposed to explain the separations under the theoretical framework of quantum fluctuations, in good agreement with our experimental and previously published experimental data.