Local filtering operations on two qubits
Frank Verstraete, Jeroen Dehaene, Bart DeMoor · Physical Review A · 2001
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 ${\ensuremath{\rho}}^{\ensuremath{'}}\ensuremath{\sim}(A\ensuremath{\bigotimes}B)\ensuremath{\rho}(A\ensuremath{\bigotimes}{B)}^{\ifmmode\dagger\else\textdagger\fi{}}.$ We consider a real matrix parametrization 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 filtering operations for concentrating entanglement. Furthermore, we give a complete characterization of the mixed states that can be purified arbitrarily close to a Bell state. Finally, we obtain a new way of calculating the entanglement of formation.