# Jagged Islands of Bound Entanglement and Witness-Parameterized Probabilities

## TL;DR

This paper investigates the probabilities of bound-entangled states within specific quantum state sets, using witness-parameterized families and geometric analysis, revealing precise probabilities and their relationships in two-qutrit and two-ququart systems.

## Contribution

It introduces new witness-parameterized families of bound-entangled probabilities and provides exact calculations for these probabilities in high-dimensional quantum systems.

## Key findings

- Bound-entangled probabilities are approximately 0.00737 (d=3) and 0.00052 (d=4).
- Total entanglement probability is about 0.48148, combining bound and free entanglement.
- PPT probabilities are approximately 0.5374 (d=3) and 0.4050 (d=4).

## Abstract

We report several witness-parameterized families of bound-entangled probabilities. Two pertain to the $d=3$ (two-qutrit) and a third to the $d=4$ (two-ququart) subsets analyzed by Hiesmayr and L{\"o}ffler of "magic" simplices of Bell states. The Hilbert-Schmidt probabilities of positive-partial-transpose (PPT) states--within which we search for bound-entangled states--are $\frac{8 \pi }{27 \sqrt{3}} \approx 0.537422$ ($d=3$) and $\frac{1}{2}+\frac{\log \left(2-\sqrt{3}\right)}{8 \sqrt{3}} \approx 0.404957$ ($d=4$). We obtain bound-entangled probabilities of $-\frac{4}{9}+\frac{4 \pi }{27 \sqrt{3}}+\frac{\log (3)}{6} \approx 0.00736862$ and $\frac{-204+7 \log (7)+168 \sqrt{3} \cos ^{-1}\left(\frac{11}{14}\right)}{1134} \approx 0.00325613$ ($d=3$) and $\frac{8 \log (2)}{27}-\frac{59}{288} \approx 0.00051583$ and $\frac{24 \text{csch}^{-1}\left(\frac{8}{\sqrt{17}}\right)}{17 \sqrt{17}}-\frac{91}{544} \approx 0.00218722$ ($d=4$). Thus, the total entanglement probability appears to equal $(1-\frac{8 \pi }{27 \sqrt{3}})+\frac{2}{81} \left(4 \sqrt{3} \pi -21\right) = \frac{13}{27} \approx 0.481481$.) The families, encompassing these results, are parameterized using generalized Choi and Jafarizadeh-Behzadi-Akbari witnesses. The same bound-entangled probability was achieved with both--the sets ("jagged islands") detected having void intersection. The entanglement (bound and "non-bound"/"free") probability for both was $\frac{1}{6} \approx 0.16667$, while their union and intersection gave $\frac{2}{9} \approx 0.22222$ and $\frac{1}{9} \approx 0.11111$. Further, we examine generalized Horodecki states, as well as estimating PPT-probabilities of approximately 0.39339 (very well-fitted by $\frac{7 \pi}{25 \sqrt{5}} \approx 0.39338962$) and 0.115732 ( for the original (8- [two-qutrit] and 15 [two-ququart]-dimensional) magic simplices themselves.