Abstract
It is well known that the support of an optimal decomposable entanglement witness is completely entangled. We add two more necessary conditions for the optimality: The orthogonal complement of the support must have a nonzero product vector; another one will be given in terms of related faces of a convex cone. With these necessary conditions, we show that there exist examples of non-optimal decomposable entanglement witnesses which are the partial transposes of positive semi-definite matrices supported on completely entangled spaces, whenever both of the local dimensions are greater than or equal to three.
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