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We obtain formulas (bot. p. 12)--including $\frac{2}{121}$ and $\frac{4\left(242 \sqrt{3} \pi -1311\right)}{9801}$--for the eight atoms (Fig. 11),summing to 1, which span a 256-dimensional three-set (P, S, PPT)entanglement-probability boolean algebra for the two-qutrit Hiesmayr-Lofflerstates. PPT denotes positive partial transpose, while P and S provide theLi-Qiao necessary and}sufficient conditions for entanglement. The constraintsensuring entanglement are $s> \frac{16}{9} \approx 1.7777$ and $p>\frac{2^{27}}{3^{18} \cdot 7^{15} \cdot13} \approx 5.61324 \cdot 10^{-15}$.Here, $s$ is the square of the sum (Ky Fan norm) of the eight singular valuesof the $8 \times 8$ correlation matrix in the Bloch representation, and $p$,the square of the product of the singular values. In the two-ququartHiesmayr-Loffler case, one constraint is $s>\frac{9}{4} \approx 2.25$, while$\frac{3^{24}}{2^{134}} \approx 1.2968528306 \cdot 10^{-29}$ is an upper boundon the appropriate $p$ value, with an entanglement probability $\approx0.607698$. The $S$ constraints, in both cases, prove equivalent to thewell-known CCNR/realignment criteria. Further, we detect and verify--usingsoftware of A. Mandilara--pseudo-one-copy undistillable (POCU) negative partialtransposed two-qutrit states distributed over the surface of the separablestates. Additionally, we study the best separable approximation problem withinthis two-qutrit setting, and obtain explicit decompositions of separable statesinto the sum of eleven product states. Numerous quantities ofinterest--including the eight atoms--were, first, estimated using a quasirandomprocedure.

8 Boolean Atoms Spanning the 256-Dimensional Entanglement-Probability  Three-Set Algebra of the Two-Qutrit Hiesmayr-Loffler Magic Simplex of Bell  States