Analytical $O({\alpha }_s)$ corrections to the beam frame double-spin density matrix elements of $e^{+}{e}^{-}\to t\bar{t}$

We provide analytical results for the $O({\alpha }_s)$ corrections to the double-spin density matrix elements in the reaction $e^{+}{e}^{-}\to t\bar{t}$. These concern the elements $ll$, $lt$, $ln$, $tt$, $tn$, and $nn$ of the double-spin density matrix elements where $l,t,n$ stand for longitudinal, transverse and normal orientations with respect to the beam frame spanned by the electron and the top quark momentum.