Effective criteria for entanglement witnesses in small dimensions

• URL: http://arxiv.org/abs/2506.09298v3
• Date: Mon, 14 Jul 2025 09:20:17 GMT
• Title: Effective criteria for entanglement witnesses in small dimensions
• Authors: Łukasz Grzelka, Łukasz Skowronek, Karol Życzkowski,
• Abstract summary: We present an effective set of necessary and sufficient criteria for block-positivity of order $4$ sequences over $mathbbC$.<n>The method can be generalized to $mathcalHotimesmathcalH_d$ systems for $d>2$ to provide necessary but not sufficient criterion for block-positivity.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-sa/4.0/
• Abstract: We present an effective set of necessary and sufficient criteria for block-positivity of matrices of order $4$ over $\mathbb{C}$. 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 $4\times 4$ complex matrix corresponds to an entanglement witness, and it is exact when the matrix coefficients belong to the rationals, extended by $\mathrm{i}$. The method can be generalized to $\mathcal{H}_2\otimes\mathcal{H}_d$ 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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