Construction and properties of a class of private states in arbitrary dimensions
Adam J. Rutkowski, Michał Studziński, Piotr Ćwikliński, Michał Horodecki · Physical Review A · 2015
We present a construction of quantum states in dimension $d$ that has at least 1 dit of ideal key, called private dits (pdits), which covers most of the known examples of private bits (pbits) $d=2$. We examine properties of this class of states, focusing mostly on its distance to the set of separable states $\mathcal{S}$, showing that for a fixed dimension of key part ${d}_{k}$, the distance increases with ${d}_{s}$. We provide explicit examples of positive partial transpose states (in $d$ dimensions) which are nearly as far from separable ones as possible. Precisely, the distance from the set of $\mathcal{S}$ is $2\ensuremath{-}\ensuremath{\epsilon}$, where $d$ scales with $\ensuremath{\epsilon}$ as $d\ensuremath{\propto}1/{\ensuremath{\epsilon}}^{3}$, as opposed to $d\ensuremath{\propto}{2}^{{[log(4/\ensuremath{\epsilon})]}^{2}}$ obtained by Badzia\ifmmode \mbox{\c{}}\else \c{}\fi{}g et al. [Phys. Rev. A 90, 012301 (2014)]. We do not use boosting (taking many copies of pdits to boost the distance) as in the Badzia\ifmmode \mbox{\c{}}\else \c{}\fi{}g et al. paper.