On polynomial invariants of several qubits

Abstract

It is a recent observation that entanglement classification for qubits is closely related to local $SL(2,\CC)$-invariants including the invariance under qubit permutations, which has been termed $S{L}^{\ast }$ invariance. In order to single out the $S{L}^{\ast }$ invariants, we analyze the $SL(2,\CC)$-invariants of four resp. five qubits and decompose them into irreducible modules for the symmetric group $S_4$ resp. $S_5$ of qubit permutations. A classifying set of measures of genuine multipartite entanglement is given by the ideal of the algebra of $S{L}^{\ast }$-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 in [Phys. Rev. A 72, 012337 (2005)]. Our analysis highlights an intimate connection between this latter procedure and the standard methods to create invariants, such as the $\Omega$-process. As the degrees of invariants increase, the alternative method proves to be particularly efficient.

29 pages, 3 eps figures, aipproc. Minor modifications and corrections. Length change only due to style change

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