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Variational Characterisations of Separability and Entanglement of Formation
In this paper we develop a mathematical framework for the characterisation of
separability and entanglement of formation (EoF) of general bipartite states.
These characterisations are of the variational kind, meaning that separability
and EoF are given in terms of a function which is to be minimized over the
manifold of unitary matrices. A major benefit of such a characterisation is
that it directly leads to a numerical procedure for calculating EoF. We present
an efficient minimisation algorithm and an apply it to the bound entangled 3X3
Horodecki states; we show that their EoF is very low and that their distance to
the set of separable states is also very low. Within the same variational
framework we rephrase the results by Wootters (W. Wootters, Phys. Rev. Lett.
80, 2245 (1998)) on EoF for 2X2 states and present progress in generalising
these results to higher dimensional systems.Comment: 11 pages RevTeX, 4 figure
Four qubits can be entangled in nine different ways
We consider a single copy of a pure four-partite state of qubits and
investigate its behaviour under the action of stochastic local quantum
operations assisted by classical communication (SLOCC). This leads to a
complete classification of all different classes of pure states of four-qubits.
It is shown that there exist nine families of states corresponding to nine
different ways of entangling four qubits. The states in the generic family give
rise to GHZ-like entanglement. The other ones contain essentially 2- or 3-qubit
entanglement distributed among the four parties. The concept of concurrence and
3-tangle is generalized to the case of mixed states of 4 qubits, giving rise to
a seven parameter family of entanglement monotones. Finally, the SLOCC
operations maximizing all these entanglement monotones are derived, yielding
the optimal single copy distillation protocol
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