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 $|\psi \rangle =a_{000}\left|0_{A}\right\rangle \left|0_{B}\right\rangle \left|0_{C}\right\rangle +a_{001}\left|0_{A}\right\rangle \left|0_{B}\right\rangle \left|1_{C}\right\rangle +a_{010}\left|0_{A}\right\rangle \left|1_{B}\right\rangle \left|0_{C}\right\rangle +a_{011}\left|0_{A}\right\rangle \left|1_{B}\right\rangle \left|1_{C}\right\rangle +a_{100}\left|1_{A}\right\rangle \left|0_{B}\right\rangle \left|0_{C}\right\rangle +a_{101}\left|1_{A}\right\rangle \left|0_{B}\right\rangle \left|1_{C}\right\rangle +a_{110}\left|1_{A}\right\rangle \left|1_{B}\right\rangle \left|0_{C}\right\rangle +a_{111}\left|1_{A}\right\rangle \left|1_{B}\right\rangle \left|1_{C}\right\rangle$there is no way of writing

$\left|\psi _{A,B,C}\right\rangle \neq {\sqrt {\lambda _{0}}}\left|0_{A}^{\prime }\right\rangle \left|0_{B}^{\prime }\right\rangle \left|0_{C}^{\prime }\right\rangle +{\sqrt {\lambda _{1}}}\left|1_{A}^{\prime }\right\rangle \left|1_{B}^{\prime }\right\rangle \left|1_{C}^{\prime }\right\rangle$

but there is a general transformation to $|\psi \rangle =\lambda _{1}|0_{A}^{}\rangle |0_{B}^{}\rangle |0_{C}^{}\rangle +|1_{A}^{}\rangle (\lambda _{2}e^{i\phi }|0_{B}^{}\rangle |0_{C}^{}\rangle +\lambda _{3}|0_{B}^{}\rangle |1_{C}^{}\rangle +\lambda _{4}|1_{B}^{}\rangle |0_{C}^{}\rangle +\lambda _{5}|1_{B}^{}\rangle |1_{C}^{}\rangle )$where $\lambda _{i}\geq 0,\sum _{i=1}^{5}\lambda _{i}^{2}=1$.