2011/05/31 by Christopher Eltschka, Thierry Bastin, Andreas Osterloh +1
Physics and Astronomy · #quant-ph
paper · pdf · doi:10.1103/physreva.85.022301
published as Phys. Rev. A 85, 022301 (2012) and Phys. Rev. A 85, 059903 (2012) · this version contains a complete proof of Theorem 1
arxiv created 2012/05/22 · arxiv updated 2013/05/29
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.