Partial K -way negativities and three-tangle for three-qubit states
S. Shelly Sharma, Naresh Kumar Sharma · Physical Review A · 2008
We obtain, analytically, the global negativity, partial $K$-way negativities $(K=2,3)$, Wootter's tangle, and three-tangle for the generic three-qubit canonical state. It is found that the product of global negativity and partial three-way negativity is equal to the three-tangle, while the partial two-way negativity is related to the tangle of qubit pairs. We also calculate similar quantities for the state canonical to a single-parameter $(0<q<1)$ pure state which is a linear combination of a Greenberger-Horne-Zeilinger state and a $W$ state. In this case, for $q=0.626\phantom{\rule{0.2em}{0ex}}85$, the state has zero three-tangle and zero three-way negativity, having only $W$-like entanglement. The difference between the product of global and partial three-way negativities and the three-tangle for a given state is a quantitative measure of two-qubit coherences transformed by unitary transformations on the canonical state into three-qubit coherences. The global negativity and partial $K$-way negativities, obtained by selective partial transpositions on multiqubit state operators, satisfy inequalities which for three qubits are equivalent to the Coffman-Kundu-Wootter inequality.