  Motivated by recent development in quantum entanglement, we study relations
among concurrence $C$, SU$_q$(2) algebra, quantum phase transition and
correlation length at the zero temperature for the XXZ chain. We find that at
the SU(2) point, the ground state possess the maximum concurrence. When the
anisotropic parameter $\Delta$ is deformed, however, its value decreases. Its
dependence on $\Delta$ scales as $C=C_0-C_1(\Delta-1)^2$ in the XY metallic
phase and near the critical point (i.e. $1<\Delta<1.3$) of the Ising-like
insulating phase. We also study the dependence of $C$ on the correlation length
$\xi$, and show that it satisfies $C=C_0-1/2\xi$ near the critical point. For
different size of the system, we show that there exists a universal scaling
function of $C$ with respect to the correlation length $\xi$.
