Abstract
Bell inequalities, considered within quantum mechanics, can be regarded as
non-optimal witness operators. We discuss the relationship between such Bell
witnesses and general entanglement witnesses in detail for the Bell inequality
derived by Clauser, Horne, Shimony, and Holt (CHSH). We derive bounds on how
much an optimal witness has to be shifted by adding the identity operator to
make it positive on all states admitting a local hidden variable model. In the
opposite direction, we obtain tight bounds for the maximal proportion of the
identity operator that can be subtracted from such a CHSH witness, while
preserving the witness properties. Finally, we investigate the structure of
CHSH witnesses directly by relating their diagonalized form to optimal
witnesses of two different classes.
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