Quantum Information Processing, Vol. 7, No. 1, February 2008 (© 2008)
Geometrical Structure of Entangled States
and the Secant Variety
Received August 29, 2007; accepted November 15, 2007; Published online: January 13, 2008
We investigate the geometrical structure of entangled and separable bipartite and
multipartite states based on the secant variety of the Segre variety. We show that
the Segre variety coincides with the space of separable multipartite state and the
higher secant variety of the Segre variety coincides with the space of entangled
KEY WORDS: multipartite quantum system; quantum entanglement; complex
projective variety; secant variety of the serge variety.
PACS: 03.65.Ta; 03.65.Ud; 03.67.Mn; 02.40.Dr.
Recently, the geometry and topology of entanglement has got more atten-
tion and we know more about the geometrical structure of pure multipar-
tite entangled quantum states. We have also managed to construct some
useful measures of entanglement based on these underlying geometrical
structures. However, we know less about the geometrical structure of an
arbitrary multipartite quantum state and there is a need for further investi-
gation on these states. Concurrence is a measure of entanglement which is
directly related to the entanglement of formation.
Its geometrical struc-
ture is hidden in a map called Segre embedding.
The Segre variety is
generated by the quadratic polynomials that correspond to the separable
set of pure multipartite states. We can construct a measure of entangle-
ment for bipartite and three-partite states based on the Segre variety.
can also construct a measure of entanglement for general pure multipar-
tite states based on a modiﬁcation of the Segre variety by adding similar
Institute of Quantum Science, Nihon University, 1-8 Kanda-Surugadai, Chiyoda-ku,
Tokyo 101-8308, Japan. E-mail: email@example.com
1570-0755/08/0200-0043/0 © 2008 Springer Science+Business Media, LLC