  Spin-orbital entanglement in quantum spin-orbital systems is quantified by a
reduced von Neumann entropy, and is calculated for the ground state of a
coupled spin-orbital chain with $SU(2)\times SU(2)$ symmetry. By analyzing the
discontinuity and local extreme of the reduced entropy as functions of the
model parameters, we deduce a rich phase diagram to describe the quantum phase
transitions in the model. Our approach provides an efficient and powerful
method to identify phase boundaries in a system with complex correlation
between multiply degrees of freedom.
