Construction and properties of a class of private states in arbitrary dimensions

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 $\mathcal{SEP}$, showing that for a fixed dimension of key part $d_k$ the distance increases with $d_s$. We provide explicit examples of PPT states (in $d$ dimensions) which are nearly as far from separable ones as possible. Precisely, the distance from the set of $\mathcal{SEP}$ is $2 - \epsilon$, where $d$ scales with $\epsilon$ as $d \propto 1/\epsilon^3$, as opposed to $d \propto 2^{(log(4/\epsilon))^2}$ obtained in [Badzi\c{a}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 Badzi\c{a}g et al. paper.