  We consider one single copy of a mixed state of two qubits and investigate
how its entanglement changes under local quantum operations and classical
communications (LQCC) of the type $\rho'\sim (A\otimes B)\rho(A\otimes
B)^{\dagger}$. We consider a real matrix parameterization of the set of density
matrices and show that these LQCC operations correspond to left and right
multiplication by a Lorentz matrix, followed by normalization. A constructive
way of bringing this matrix into a normal form is derived. This allows us to
calculate explicitly the optimal local filterin operations for concentrating
entanglement. Furthermore we give a complete characterization of the mixed
states that can be purified arbitrary close to a Bell state. Finally we obtain
a new way of calculating the entanglement of formation.
