  We consider a one-parameter family of matrix product states of spin one
particles on a periodic chain and study in detail the entanglement properties
of such a state. In particular we calculate exactly the entanglement of one
site with the rest of the chain, and the entanglement of two distant sites with
each other and show that the derivative of both these properties diverge when
the parameter $g$ of the states passes through a critical point. Such a point
can be called a point of quantum phase transition, since at this point, the
character of the matrix product state which is the ground state of a
Hamiltonian, changes discontinuously. We also study the finite size effects and
show how the entanglement depends on the size of the chain. This later part is
relevant to the field of quantum computation where the problem of initial state
preparation in finite arrays of qubits or qutrits is important. It is also
shown that entanglement of two sites have scaling behavior near the critical
point.
