On the geometry of a class of N-qubit entanglement monotones

Details On the geometry of a class of N-qubit entanglement monotones On the geometry of a class of N-qubit entanglement monotones

2005-07-07

arxiv.org/abs/quant-ph/0507070v1

Journal of Physics A38 (2005) 9075-9085

Physics

Quantum Physics

7 pages RevTex, Submitted to Physical Review A

Scientific paper

10.1088/0305-4470/38/41/016

A family of N-qubit entanglement monotones invariant under stochastic local operations and classical communication (SLOCC) is defined. This class of entanglement monotones includes the well-known examples of the concurrence, the three-tangle, and some of the four, five and N-qubit SLOCC invariants introduced recently. The construction of these invariants is based on bipartite partitions of the Hilbert space in the form ${\bf C}^{2^N}\simeq{\bf C}^L\otimes{\bf C}^l$ with $L=2^{N-n}\geq l=2^n$. Such partitions can be given a nice geometrical interpretation in terms of Grassmannians Gr(L,l) of l-planes in ${\bf C}^L$ that can be realized as the zero locus of quadratic polinomials in the complex projective space of suitable dimension via the Plucker embedding. The invariants are neatly expressed in terms of the Plucker coordinates of the Grassmannian.

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