Acín decomposition

In a 2000 paper titled "Generalized Schmidt Decomposition and Classification of Three-Quantum-Bit States"[1] 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 ${\displaystyle |\psi \rangle =a_{000}\left|0_{A}\right\rangle \left|0_{B}\right\rangle \left|0_{C}\right\rangle +a_{001}\left|0_{A}\right\rangle \left|0_{B}\right\rangle \left|1_{C}\right\rangle +a_{010}\left|0_{A}\right\rangle \left|1_{B}\right\rangle \left|0_{C}\right\rangle +a_{011}\left|0_{A}\right\rangle \left|1_{B}\right\rangle \left|1_{C}\right\rangle +a_{100}\left|1_{A}\right\rangle \left|0_{B}\right\rangle \left|0_{C}\right\rangle +a_{101}\left|1_{A}\right\rangle \left|0_{B}\right\rangle \left|1_{C}\right\rangle +a_{110}\left|1_{A}\right\rangle \left|1_{B}\right\rangle \left|0_{C}\right\rangle +a_{111}\left|1_{A}\right\rangle \left|1_{B}\right\rangle \left|1_{C}\right\rangle }$there is no way of writing

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

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

References

1. Acín, A.; Andrianov, A.; Costa, L.; Jané, E.; Latorre, J. I.; Tarrach, R. (2000-08-14). "Generalized Schmidt Decomposition and Classification of Three-Quantum-Bit States". Physical Review Letters. 85 (7): 1560–1563. doi:10.1103/PhysRevLett.85.1560. hdl:2445/12805. ISSN 0031-9007.