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 $2-\epsilon$ from the separable states, we need a dimension of $2^{\mathrm{poly}[log\left(\frac{1}{\epsilon }\right)]}$, as opposed to $2^{\mathrm{poly}\left(\frac{1}{\epsilon }\right)}$ 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.

• Received 9 December 2013

DOI:https://doi.org/10.1103/PhysRevA.90.012301