Motivation for the definition of k-distillability

Remember that the partial transpose condition is generally good for detecting entanglement, i.e. a bipartite state $rho$ is certainly entangled if the partial transpose is not non-negative. In other words, if there exists a state $|psirangle$ such that $$ langlepsi|Iotimestext{T}(rho)|psirangle<0, $$ then the state is certainly entangled.

If you want to be able to distil some entanglement from $k$ copies then, crudely, you'd like to look at $k$ copies of the partially transposed state, and if that has a negative eigenvalue, you would be able to extract some entanglement.

With that level of explanation, you'd ask why looking at more than one copy is any use -- the eigenvalues of many copies of $sigma$ are easily related to the eigenvalues of a single copy. However, this is because of the extra condition that $|psirangle$ must be Schmidt rank 1 or less. I presume that this is because you can give an explicit distillation protocol based on the properties of $|psirangle$. Essentially, this is due to the fact that you're trying to project onto a Bell pair which, of course, is Schmidt rank 2.

For a better understanding that the very hand-wavy suggestions I've just given, you'd want to work through page 2 of https://arxiv.org/abs/quant-ph/9801069