Effective criteria for entanglement witnesses in small dimensions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914282255417344 |
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| author | Grzelka, Łukasz Skowronek, Łukasz Życzkowski, Karol |
| author_facet | Grzelka, Łukasz Skowronek, Łukasz Życzkowski, Karol |
| contents | 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 Gröbner bases. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_09298 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Effective criteria for entanglement witnesses in small dimensions Grzelka, Łukasz Skowronek, Łukasz Życzkowski, Karol Quantum Physics Mathematical Physics 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 Gröbner bases. |
| title | Effective criteria for entanglement witnesses in small dimensions |
| topic | Quantum Physics Mathematical Physics |
| url | https://arxiv.org/abs/2506.09298 |