Quantification of multipartite entanglement with the use of LOCC transformation
Hiroyasu Tajima
Entanglement is known to be a promising resource which enables us to execute various quantum tasks. Its quantification has been successful for bipartite pure states. However, the quantification of entanglement for multipartite states has been in a difficult situation. The difficulty is due to the difference of structures between bipartite and multipartite states. In the present thesis, we introduce a new approach for this problem. We give a necessary and sufficient condition of the possibility of a deterministic LOCC transformation of three-qubit pure states. This condition is expressed as a transformation law of six entanglement parameters jAB, jAC , jBC , jABC , J5 and Qe, where jAB, jAC and jBC are bipartite entanglements, jABC is a tripartite entanglement, J5 is a tripartite parameter which means a kind of phase, and Qe is a new tripartite parameter which means a kind of charge. This fact shows that three-qubit pure states are a partially ordered set parametrized by the six entanglement parameters. The order of the partially ordered set is defined by the possibility of a deterministic LOCC transformation from a state to another state. In this sense, the present condition is an extension of Nielsen’s work [14] to three-qubit pure states. We also clarify the rules of transfer and dissipation of entanglement. These rules guarantee that the tripartite entanglement can be transformed into bipartite entanglements, but that the bipartite entanglements cannot be transformed into the tripartite entanglement. This implies that the tripartite entanglement is a higher entity than the bipartite entanglements. With a new combination of the six entanglement parameters, the present condition can be simplified enough to determine easily whether a deterministic LOCC transformation from an arbitrary state |ψ〉 to another arbitrary state |ψ′〉 is possible or not. Moreover, the minimum number of times of measurements to reproduce an arbitrary deterministic LOCC transformation is given. This is an extension of Horodecki et al.’s work [24] to three-qubit pure states.
•Jan 1, 2012•0 citations