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

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R. Augusiak, J. Tura, M. Lewenstein


Journal of Physics A: Mathenatical and Theoretical, vol. 44, n. 21 (2011)

Entanglement witnesses (EWs) constitute one of the most important entanglement detectors in quantum systems. Nevertheless, their complete characterization, in particular with respect to the notion of optimality, is still missing, even in the decomposable case. Here we show that for any qubit–qunit decomposable EW (DEW) W, the three statements are equivalent: (i) the set of product vectors obeying lange, f|W|e, frang = 0 spans the corresponding Hilbert space, (ii) W is optimal, and (iii) W = QΓ, with Q denoting a positive operator supported on a completely entangled subspace (CES) and Γ standing for the partial transposition. While implications (i)⇒(ii) and (ii)⇒(iii) are known, here we prove that (iii) implies (i). This is a consequence of a more general fact saying that product vectors orthogonal to any CES in {\bb C}^{2}\otimes {\bb C}^{n} span after partial conjugation the whole space. On the other hand, already in the case of the {\bb C}^{3}\otimes {\bb C}^{3} Hilbert space, there exist DEWs for which (iii) does not imply (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.