Acín decomposition

In a 2000 paper titled "Generalized Schmidt Decomposition and Classification of Three-Quantum-Bit States" Acín et al. described a way of separating out one of the terms of a general tripartite quantum state. This can be useful in considering measures of entanglement of quantum states.

General decomposition

For a general three-qubit state $|\psi ⟩=a_{000}|0_A⟩|0_B⟩|0_C⟩+a_{001}|0_A⟩|0_B⟩|1_C⟩+a_{010}|0_A⟩|1_B⟩|0_C⟩+a_{011}|0_A⟩|1_B⟩|1_C⟩+a_{100}|1_A⟩|0_B⟩|0_C⟩+a_{101}|1_A⟩|0_B⟩|1_C⟩+a_{110}|1_A⟩|1_B⟩|0_C⟩+a_{111}|1_A⟩|1_B⟩|1_C⟩$there is no way of writing

$|{\psi }_{A,B,C}⟩\ne \sqrt{{\lambda }_0}|0_A^{\prime }⟩|0_B^{\prime }⟩|0_C^{\prime }⟩+\sqrt{{\lambda }_1}|1_A^{\prime }⟩|1_B^{\prime }⟩|1_C^{\prime }⟩$

but there is a general transformation to $|\psi ⟩={\lambda }_1|0_A^{}⟩|0_B^{}⟩|0_C^{}⟩+|1_A^{}⟩({\lambda }_2{e}^{i\varphi }|0_B^{}⟩|0_C^{}⟩+{\lambda }_3|0_B^{}⟩|1_C^{}⟩+{\lambda }_4|1_B^{}⟩|0_C^{}⟩+{\lambda }_5|1_B^{}⟩|1_C^{}⟩)$where ${\lambda }_i\ge 0,\sum _{i=1}^5{\lambda }_i^2=1$.