## Bounding the three-tangle    [PDF]

Samuel Rodriques, Nilanjana Datta, Peter J. Love
We prove that any three-qubit mixed state can be written as a convex combination of pure states, at most one of which has non-zero three-tangle. We construct an upper bound on the three tangle using these ensembles and a non-uniform continuity bound for any non-negative convex function of density matrices. The existence of such ensembles also motivates the definition of the best W-class approximation (BWA) of a density matrix, a natural generalization of the best separable approximation (BSA) introduced by Lewenstein and Sanpera. By analogy with the BSA, we prove that any three-qubit density matrix has a unique decomposition in terms of its BWA and a pure state with possibly non-zero three-tangle. The upper bound may be computed with a cost linear in the rank d of the density matrix and has accuracy comparable to a steepest descent algorithm for the convex roof whose cost scales as \$d^8 \log d\$. We compare the upper bound to a lower bound algorithm given by Eltschka and Siewert for the three-tangle, and find that on random rank-two three-qubit density matrices, the difference between the upper and lower bounds is 0.14 on average. We also find that the three-tangle of random three qubit density matrices is less than 0.023 on average.
View original: http://arxiv.org/abs/1307.2323