On polynomial invariants of several qubits

TL;DR

This paper studies polynomial invariants related to entanglement classification in multi-qubit systems, focusing on $SL^*$ invariants and their algebraic structure, providing new methods for constructing these invariants.

Contribution

It analyzes $SL(2, ext{CC})$-invariants for four and five qubits, decomposes them into symmetric group modules, and introduces efficient methods for constructing invariants of higher degree.

Findings

Decomposition of invariants into irreducible modules for symmetric groups.

Identification of the ideal of invariants vanishing on product states.

Efficient construction methods for high-degree invariants.

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 $\Omega$-process. As the degrees of invariants increase, the alternative method proves to be particularly efficient.