Necessary criteria for Markovian divisibility of linear maps

We describe how to extend the notion of infinitesimal Markovian divisibility from quantum channels general linear maps and compact convex sets generators. give a approach toward proving necessary criteria for (infinitesimal) divisibility. With it, we prove two in any finite dimension d: an upper bound on determinant terms Θ(d)-power smallest singular value product Θ(d) values. These allow us analytically construct, given dimension, set that contains provably non-infinitesimal divisible ones. Moreover, show that, general, no such non-trivial can be derived classical counterpart this scenario.