Entanglement and quantum phase transition in matrix product spin-one chains

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 has scaling behavior near the critical point.