Characterization and properties of weakly optimal entanglement witnesses

We present an analysis of the properties and characteristics of weakly optimal entanglement witnesses, that is witnesses whose expectation value vanishes on at least one product vector. Any weakly optimal entanglement witness can be written as the form of Wwopt = σ - cσmaxI, where cσmax is a non-negative number and I is the identity matrix. We show the relation between the weakly optimal witness Wwopt and the eigenvalues of the separable states σ. Further we give an application of weakly optimal witnesses for constructing entanglement witnesses in a larger Hilbert space by extending the result of [P. Badziag et al, Phys. Rev. A 88, 010301(R) (2013)], and we examine their geometric properties.