Effective criteria for entanglement witnesses in small dimensions
Abstract
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 Gr\"obner bases.
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