We present an effective set of necessary and sufficient criteria for block-positivity of matrices of order 4 over C. The approach is based on Sturm sequences and quartic polynomial positivity conditions presented in recent literature. The procedure allows us to test whether a given 4 4 complex matrix corresponds to an entanglement witness, and it is exact when the matrix coefficients belong to the rationals, extended by i. The method can be generalized to H₂d systems for d>2 to provide necessary but not sufficient criterion for block-positivity. We also outline an alternative approach to the problem relying on Gröbner bases.
Grzelka et al. (Tue,) studied this question.