Lorentz singular-value decomposition and its applications to pure states of three qubits

All mixed states of two qubits can be brought into normal form by the action of local operations and classical communication operations of the kind ${\rho }^{\prime }=(A\otimes B)\rho (A\otimes {B)}^{†}.$ These normal forms can be obtained by considering a Lorentz singular-value decomposition on a real parametrization of the density matrix. We show that the Lorentz singular values are variationally defined and give rise to entanglement monotones, with as a special case the concurrence. Next a necessary and sufficient criterion is conjectured for a mixed state to be convertible into another specific one with a nonzero probability. Finally the formalism of the Lorentz singular-value decomposition is applied to tripartite pure states of qubits. New proofs are given for the existence of the Greenberger-Horne-Zeilinger (GHZ) class and W class of states, and a rigorous proof for the optimal distillation of a GHZ state is derived.