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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> </mml:mrow> </mml:math> . 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mn>4</mml:mn> <mml:mo>×</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> complex matrix corresponds to an entanglement witness, and it is exact when the matrix coefficients belong to the rationals, extended by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">i</mml:mi> </mml:mrow> </mml:mrow> </mml:math> . The method can be generalized to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">H</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>⊗</mml:mo> <mml:msub> <mml:mrow> <mml:mi class="MJX-tex-calligraphic">H</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msub> </mml:mrow> </mml:math> systems for d &gt; 2 to provide necessary but not sufficient criterion for block-positivity. We also outline an alternative approach to the problem relying on Gröbner bases.