The Lorentz singular value decomposition and its applications to pure states of 3 qubits

Abstract

All mixed states of two qubits can be brought into normal form by the action of SLOCC 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 parameterization 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 non-zero 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 GHZ- and W-class of states, and a rigorous proof for the optimal distillation of a GHZ-state is derived.

References in corpus (2)

• Classification of mixed three-qubit states
• Local filtering operations on two qubits

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