Deterministic LOCC transformation of three-qubit pure states and entanglement transfer

Details Deterministic LOCC transformation of three-qubit pure states and entanglement transfer Deterministic LOCC transformation of three-qubit pure states and entanglement transfer

2010-12-08

arxiv.org/abs/1012.1676v2

Physics

Quantum Physics

103 pages, 18 figures

Scientific paper

A necessary and sufficient condition of the possibility of a deterministic LOCC transformation of three-qubit pure states is given. This condition is expressed as a transformation law of six entanglement parameters $j_{AB}$, $j_{AC}$, $j_{BC}$, $j_{ABC}$, $J_{5}$ and $Q_{e}$, where $j_{AB}$, $j_{AC}$ and $j_{BC}$ are bipartite entanglements, $j_{ABC}$ is a tripartite entanglement, $J_{5}$ is a tripartite parameter which means a kind of phase, and $Q_{e}$ 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 \cite{20} 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 $|\psi>$ to another arbitrary state $|\psi'>$ 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 \cite{31} to three-qubit pure states.

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