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

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

2001-08-09

arxiv.org/abs/quant-ph/0108043v1

Phys. Rev. A (\bf 65), 032308 (2002).

Physics

Quantum Physics

Scientific paper

10.1103/PhysRevA.65.032308

All mixed states of two qubits can be brought into normal form by the action of SLOCC operations of the kind $\rho'=(A\otimes B)\rho(A\otimes B)^\dagger$. 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.

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