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Entanglement witnesses (EWs) constitute one of the most importantentanglement detectors in quantum systems. Nevertheless, their completecharacterization, in particular with respect to the notion of optimality, isstill missing, even in the decomposable case. Here we show that for anyqubit-qunit decomposable EW (DEW) W the three statements are equivalent: (i)the set of product vectors obeying \bra{e,f}W\ket{e,f}=0 spans thecorresponding Hilbert space, (ii) W is optimal, (iii) W=Q^{\Gamma} with Qdenoting a positive operator supported on a completely entangled subspace (CES)and \Gamma standing for the partial transposition. While, implications$(i)\Rightarrow(ii)$ and $(ii)\Rightarrow(iii)$ are known, here we prove that(iii) implies (i). This is a consequence of a more general fact saying thatproduct vectors orthogonal to any CES in C^{2}\otimes C^{n} span after partialconjugation the whole space. On the other hand, already in the case ofC^{3}\otimes C^{3} Hilbert space, there exist DEWs for which (iii) does notimply (i). Consequently, either (i) does not imply (ii), or (ii) does not imply(iii), and the above transparent characterization obeyed by qubit-qunit DEWs,does not hold in general.

A note on the optimality of decomposable entanglement witnesses and completely entangled subspaces