On polynomial invariants of several qubits

It is a recent observation that entanglement classification for qubits is closely related to local SL(2,C)-invariants including the invariance under qubit permutations [Dur, , Phys. Rev. A 62, 062314 (2000); Osterloh, A. and Siewert, J., Phys. Rev. A 72, 012337 (2005); O. Chterental and D. Z. okovic, Linear Algebra Research Advances (Nova Science, Hauppauge, N.Y., 2007), Chap. 4, p. 133], which has been termed SL(*) invariance. In order to single out the SL(*) invariants, we analyze the SL(2,C)-invariants of four (five) qubits and decompose them into irreducible modules for the symmetric group S(4) (S(5)) of qubit permutations. A classifying set of measures of genuine multipartite entanglement is given by the ideal of the algebra of SL(*)-invariants vanishing on arbitrary product states. We find that low degree homogeneous components of this ideal can be constructed in full by using the approach introduced by Osterloh and Siewert [Phys. Rev. A 72, 012337 (2005); Int. J. Quant. Inf. 4, 531 (2006)]. Our analysis highlights an intimate connection between this latter procedure and the standard methods to create invariants, such as the Omega-process [Luque, J.-G. and Thibon, J.-Y., J. Phys. A 39, 371 (2005)]. As the degrees of invariants increase, the alternative method proves to be particularly efficient.