In the current quantum information and computation theory, the notion of entanglement is considered as one of the most important resources. Nevertheless, distinguishing entanglement from separability is very difficult, and known to be NP hard in general. Among various separability criteria, the PPT criterion is very simple to test but powerful: The partial transpose of a separable state must be positive (semi-deﬁnite). Of particular interest are some extremal PPT states, called entangled edge states, because all PPT states are convex sums of separable states and edge states.
I will talk about an explicit construction of a parameterized family of n⊗n PPT states of corank one whose partial transposes have corank 2n-3 for every n≥3. We checked that these states are in fact entangled edge states, for n up to 800. We conjecture that they are entangled edge states for all n≥3. Our states are the first explicit examples of PPT entangled edge states for n≥4.
Based on a joint work with Jinwon Choi and Seung-Hyeok Kye.