We exhibit a two-parameter family of bipartite mixed states ${\ensuremath{\rho}}_{\mathrm{bc}},$ in a $d\ensuremath{\bigotimes}d$ Hilbert space, which are negative under partial transposition (NPT), but for which we conjecture that no maximally entangled pure states in $2\ensuremath{\bigotimes}2$ 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 ${\ensuremath{\rho}}_{\mathrm{bc}}$ using a projection on $2\ensuremath{\bigotimes}2.$ 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 a NPT state of the ${\ensuremath{\rho}}_{\mathrm{bc}}$ 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.
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