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

Frank Verstraete·Jeroen Dehaene·Bart De Moor
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Abstract

All mixed states of two qubits can be brought into normal form by the action of SLOCC operations of the kind $ρ'=(A\otimes B)ρ(A\otimes B)^\dagger$. 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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Frank Verstraete, Jeroen Dehaene, Bart De Moor. 2001-08-09. The Lorentz singular value decomposition and its applications to pure states of 3 qubits. https://doi.org/10.1103/physreva.65.032308

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