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arXiv · 1105.1556

Multipartite-entanglement monotones and polynomial invariants

Christopher Eltschka·Thierry Bastin·Andreas Osterloh·Jens Siewert
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Abstract

We show that a positive homogeneous function that is invariant under determinant-1 stochastic local operations and classical communication (SLOCC) transformations defines an N-qubit entanglement monotone if and only if the homogeneous degree is not larger than four. In particular this implies that any power larger than one of the well-known N-tangle (N > 2) is not an entanglement monotone anymore. We then describe a common basis and formalism for the N-tangle and other known invariant polynomials of degree four. This allows us to elucidate the relation of the four-qubit invariants defined by Luque and Thibon [Phys. Rev. A67, 042303 (2003)] and the reduced two-qubit density matrices of the states under consideration. Finally we prove that an analogous statement holds for any multi-partite system with even number of qudits.

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Christopher Eltschka, Thierry Bastin, Andreas Osterloh, Jens Siewert. 2012-05-22. Multipartite-entanglement monotones and polynomial invariants. https://doi.org/10.1103/physreva.85.022301

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