  We exhibit a two-parameter family of bipartite mixed states $\rho_{bc}$, in a
$d\otimes d$ Hilbert space, which are negative under partial transposition
(NPT), but for which we conjecture that no maximally entangled pure states in
$2\otimes 2$ can be distilled by local quantum operations and classical
communication (LQ+CC). Evidence for this undistillability is provided by the
result that, for certain states in this family, we cannot extract entanglement
from any arbitrarily large number of copies of $\rho_{bc}$ using a projection
on $2\otimes 2$. These states are canonical NPT states in the sense that any
bipartite mixed state in any dimension with NPT can be reduced by LQ+CC
operations to an NPT state of the $\rho_{bc}$ form. We show that the main
question about the distillability of mixed states can be formulated as an open
mathematical question about the properties of composed positive linear maps.
