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Abstract: The invariant-comb approach is a method to construct entanglement measuresfor multipartite systems of qubits. The essential step is the construction ofan antilinear operator that we call {\em comb} in reference to the {\emhairy-ball theorem}. An appealing feature of this approach is that for qubitsor spins 1-2 the combs are automatically invariant under $SL2,\CC$, whichimplies that the obtained invariants are entanglement monotones byconstruction. By asking which property of a state determines whether or not itis detected by a polynomial $SL2,\CC$ invariant we find that it is thepresence of a {\em balanced part} that persists under local unitarytransformations. We present a detailed analysis for the maximally entangledstates detected by such polynomial invariants, which leads to the concept of{\em irreducibly balanced} states. The latter indicates a tight connection withSLOCC classifications of qubit entanglement. \\ Combs may also help to definemeasures for multipartite entanglement of higher-dimensional subsystems.However, for higher spins there are many independent combs such that it isnon-trivial to find an invariant one. By restricting the allowed localoperations to rotations of the coordinate system i.e. again to the$SL2,\CC$ we manage to define a unique extension of the concurrence togeneral half-integer spin with an analytic convex-roof expression for mixedstates.



Author: A. Osterloh, J. Siewert

Source: https://arxiv.org/







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