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arXiv · 0804.1661

On polynomial invariants of several qubits

Andreas Osterloh·Dragomir Z. Djokovic
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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 $SL^*$ invariance. In order to single out the $SL^*$ 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 $SL^*$-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 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 $Ω$-process. As the degrees of invariants increase, the alternative method proves to be particularly efficient.

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Andreas Osterloh, Dragomir Z. Djokovic. 2009-02-17. On polynomial invariants of several qubits. https://doi.org/10.1063/1.3075830

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