Lorentz singular-value decomposition and its applications to pure states of three qubits
Frank Verstraete, Jeroen Dehaene, Bart De Moor · Physical Review A · 2002
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 ${\ensuremath{\rho}}^{\ensuremath{'}}=(A\ensuremath{\bigotimes}B)\ensuremath{\rho}(A\ensuremath{\bigotimes}{B)}^{\ifmmode\dagger\else\textdagger\fi{}}.$ 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.