  The discovery of entangled quantum states from which one cannot distill pure
entanglement constitutes a fundamental recent advance in the field of quantum
information. Such bipartite bound-entangled (BE) quantum states \emph{could}
fall into two distinct categories: (1) Inseparable states with positive partial
transposition (PPT), and (2) States with negative partial transposition (NPT).
While the existence of PPT BE states has been confirmed, \emph{only one} class
of \emph{conjectured} NPT BE states has been discovered so far. We provide
explicit constructions of a variety of multi-copy undistillable NPT states, and
conjecture that they constitute families of NPT BE states. For example, we show
that for every pure state of Schmidt rank greater than or equal to three, one
can construct n-copy undistillable NPT states, for any $n\geq1$. The abundance
of such conjectured NPT BE states, we believe, considerably strengthens the
notion that being NPT is only a necessary condition for a state to be
distillable.
