Bound entangled states with extremal properties
Piotr Badzia ̧g, Karol Horodecki, Michał Horodecki, Justin Jenkinson, Stanisław J. Szarek · Physical Review A · 2014
Following recent work of Beigi and Shor, we investigate positive partial transpose (PPT) states that are ``heavily entangled.'' We first exploit volumetric methods to show that in a randomly chosen direction, there are PPT states whose distance in trace norm from separable states is (asymptotically) at least $1/4$. We then provide explicit examples of PPT states which are nearly as far from separable ones as possible. To obtain a distance of $2\ensuremath{-}\ensuremath{\epsilon}$ from the separable states, we need a dimension of ${2}^{\mathrm{poly}[log(\frac{1}{\ensuremath{\epsilon}})]}$, as opposed to ${2}^{\mathrm{poly}(\frac{1}{\ensuremath{\epsilon}})}$ given by the construction of Beigi and Shor [J. Math. Phys. 51, 042202 (2010)]. We do so by exploiting the so-called private states, introduced earlier in the context of quantum cryptography. We also provide a lower bound for the distance between private states and PPT states and investigate the distance between pure states and the set of PPT states.