The Probability of Entanglement

MB

Vladimir N. Chernega, Olga V. Man’ko, Vladimir I. Man’ko

Concept of entangled probability distribution of several random variables is introduced. These probability distributions describe multimode quantum states in probability representation of quantum mechanics. Example of entangled probability distribution is considered.

Paul B. Slater

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{ö}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 $\frac{8 π}{27 \sqrt{3}} \approx 0.537422$ ($d=3$) 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 $-\frac{4}{9}+\frac{4 π}{27 \sqrt{3}}+\frac{\log (3)}{6} \approx 0.00736862$ 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 $(1-\frac{8 π}{27 \sqrt{3}})+\frac{2}{81} \left(4 \sqrt{3} π-21\right) = \frac{13}{27} \approx 0.481481$.) 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 $\frac{7 π}{25 \sqrt{5}} \approx 0.39338962$) and 0.115732 ( for the original (8- [two-qutrit] and 15 [two-ququart]-dimensional) magic simplices themselves.

Garay, Linda Bertel

Chun-Xian Li, Wang Cheng-Zhi, Nie Liu-Ying +1 more

We use non-maximally entangled states (NMESs) to simulate an entangling unitary operator (EUO) with a certain probability. Given entanglement resources, the probability of the success we achieve is a decreasing function of the parameters of the EUO. Given an EUO, for certain entanglement resources the result is optimal, i.e., the probability obtains a maximal value, and for optimal result higher parameters of the EUO match more amount of entanglement resources. The probability of the success we achieve is higher than the known results under some condition.

Liesel K. Hill