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A note on the optimality of decomposable entanglement witnesses and completely entangled subspaces | Remigiusz Augusiak
; Jordi Tura
; Maciej Lewenstein
; | Date: |
17 Dec 2010 | Abstract: | 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 <e,f|W|e,f>=0 span the corresponding Hilbert
space, (ii) W is optimal, (iii) W=Q^{Gamma} with positive Q supported on a
completely entangled subspace (CES). 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 that product vectors
orthogonal to any CES in C^2otimes C^n span, after partial conjugation, the
whole space. On the other hand, already in the case of C^3otimes 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. | Source: | arXiv, 1012.3786 | Services: | Forum | Review | PDF | Favorites |
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