Entanglement theory

PPT state: A bipartite state with positive partial transpose. For 2x2 and 2x3 dimensions, PPT states are unentangled and NPPT states are entangled. For higher dimensions, NPPT states are entangled, but PPT states need not be. That is, it appears there exists examples of PPT states that are unentangled as well as those that are entangled.

A seperable operation maps a PPT state to PPT state. Implies a PPT state can not be distilled. The set of all states that cannot be distilled is called bound entangled state. All PPT states are also bound entangled. But it is not known whether there exists a NPPT state that cannot be distilled, nor is it proven that all NPPT states can be distilled. Hence, it is unknown whether bound entangled states are the same as PPT states.

Dense subset: Let A be a set with subset B. B is dense subset of A if in the neighbourhood of each element in A there is an element of B. By neighbourhood, we mean an arbitrarily small hypershpere around the element of A. Of course, some notion of distance is being utilized in this definition.