Fig. 2: Entanglement conditions of order three.

The plot of the value of the third moment p₃ saturating the p₃-PPT (dashed orange curve), the D₃ (dashed green line), and the optimal \({D}_{3}^{{{{\rm{opt}}}}}\) (thick black curve) conditions as a function of the second moment p₂ for a normalized Hermitian matrix. According to the p₃-PPT condition, any state ρ with a value of p₃(ρ^Γ) below the dashed orange curve is entangled. Similarly, the condition D₃ shows that any state ρ with a value of p₃(ρ^Γ) below the dashed green line is entangled. From this plot, it is clear that, for p₂(ρ^Γ) > 1/2, all entangled states detected by the p₃-PPT condition are also detected by D₃, which coincides with \({D}_{3}^{{{{\rm{opt}}}}}\) in this case. When p₂(ρ^Γ) < 1/2, the p₃-PPT condition is then stronger than D₃, and \({D}_{3}^{{{{\rm{opt}}}}}\) represents a slight improvement over the p₃-PPT condition. As illustrated in the applications, this slight improvement can nevertheless be important for the detection of physically relevant states.