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Presenting necessary conditions for entanglement witnesses in 4x4 complex matrices, suggesting extensions to higher dimensions.
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.
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Grzelka et al. (2025) studied this question.
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