Narrow Results

Using the Hilbert–Schmidt (HS) decomposition we suggest new possible choices of Bell operators and entanglement witnesses (EWs) for $n$ ($>2$) qubits systems for (full/bi) separability. The latter give upper bounds for (full/bi) separability. Also using the HS decomposition, we find explicitly (full/bi) separable forms for some qubits states which give lower bounds for (full/bi) separability. When the lower bounds and upper bounds coincide it means that the EW is optimal. In the case of full separability, the positive transpose method can sometimes give optimal results. As concrete examples, we give results for the GHZ(3), $W(3)$ and cluster $Cl(4)$ states.