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

Abstract

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-Loffler states. PPT denotes positive partial transpose, while P and S provide the Li-Qiao necessary and}sufficient conditions for entanglement. The constraints ensuring 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 values of the $8 \times 8$ correlation matrix in the Bloch representation, and $p$, the square of the product of the singular values. In the two-ququart Hiesmayr-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 bound on the appropriate $p$ value, with an entanglement probability $\approx 0.607698$. The $S$ constraints, in both cases, prove equivalent to the well-known CCNR/realignment criteria. Further, we detect and verify--using software of A. Mandilara--pseudo-one-copy undistillable (POCU) negative partial transposed two-qutrit states distributed over the surface of the separable states. Additionally, we study the best separable approximation problem within this two-qutrit setting, and obtain explicit decompositions of separable states into the sum of eleven product states. Numerous quantities of interest--including the eight atoms--were, first, estimated using a quasirandom procedure.