  The Holevo quantity and the $SU(2)$-invariant states have particular
importance in quantum information processing. We calculate analytically the
Holevo quantity for bipartite systems composed of spin-$j$ and
spin-$\frac{1}{2}$ subsystems with $SU(2)$ symmetry, when the projective
measurements are performed on the spin-$\frac{1}{2}$ subsystem. The relations
among the Holevo quantity, the maximal values of the Holevo quantity and the
states are analyzed in detail. In particular, we show that the Holevo quantity
increases in the parameter region $F<F_d$ and decreases in region $F>F_d$ when
$j$ increases, where $F$ is function of temperature in thermal equilibrium and
$F_d=j/(2j+1)$, and the maximum value of the Holevo quantity is attained at
$F=1$ for all $j$. Moreover, when the dimension of system increases, the
maximal value of the Holevo quantity decreases.
