# What proportions of certain sets of PPT-two-retrit states are bound entangled or separable?

For two particular (twelve-and thirteen-dimensional) sets of two-retrit states (corresponding to 9 x 9 density matrices with real off-diagonal entries), I have been able to calculate the Hilbert-Schmidt probabilities that members of the sets have positive partial transposes (the PPT property).

The first set is composed of the two-retrit $$X$$-states--having non-zero diagonal and anti-diagonal entries, and all others zero. For this set, the Hilbert-Schmidt PPT-probability is $$\frac{16}{3 \pi^2} \approx 0.54038$$. (For the rebit-retrit and two-rebit $$X$$-states [https://arxiv.org/abs/1501.02289 p.3], the [now, separability] probability is--somewhat surprisingly--the very same. For still higher [than two-retrit] dimensional states, the PPT-probabilities of their $$X$$-states seem not to be presently known--and also not known for the $$8 \times 8$$ $$X$$-states.)

The second (thirteen-dimensional) set is a one-parameter enlargement of the two-retrit $$X$$-states, in which the (1,2) (and (2,1)) entries are now unrestricted. For this set, the HS PPT-probability increases to $$\frac{65}{36 \pi} \approx 0.574726$$. (It remains an interesting research question of to what extent, if any, does this probability change if other--than the (1,2) (and (2,1)) entries--are chosen to be similarly unrestricted.)

So, now is there any manner, by which I can determine to what extent these two sets of (12- and 13-dimensional) PPT-states are bound entangled or separable?

Also, along similar lines, it would be of interest to try to compute the Hilbert-Schmidt PPT-probability of the eight-dimensional "magic simplex $$\mathcal{W}$$ presented in sec. IV of the paper "The geometry of bipartite qutrits including bound entanglement" https://arxiv.org/abs/0705.1403 . However, at this point, I have not yet been able to fully implement in Mathematica, the steps required.