  The characterization of transformations among entangled pure states via local
operations assisted by classical communication (LOCC) is a crucial problem in
quantum information theory for both theoretical and practical reasons. As LOCC
has a highly intricate structure, sometimes the larger set of separable (SEP)
maps is considered, which has a mathematically much simpler description. In the
literature, mainly SEP maps consisting of invertible Kraus operators have been
taken into account. In this paper we show that the consideration of those maps
is not sufficient when deciding whether a state can be mapped to another via
general SEP transformations. This is done by providing explicit examples of
transformations among pure 3- and 5- qubits states, which are feasible via SEP
maps containing singular Kraus operators, however, not possible via SEP maps
containing solely regular Kraus operators. The key point that allows to
construct the SEP maps is to introduce projective measurements that occur with
probability zero on the input state. The fact that it is not sufficient to
consider SEP maps composed out of regular Kraus operators even in the case of
pure state transformations, also affects the results on LOCC transformations
among pure states. However, we show that non-invertible Kraus operators do not
help in state transformations under LOCC with finitely many rounds of classical
communication, i.e. the necessary and sufficient condition for SEP
transformations with invertible Kraus operators is still a necessary condition
for convertibility under finite-round LOCC. Moreover, we show that the results
on transformations via SEP that are not possible with LOCC (including
infinitely many rounds of classical communication) presented in M. Hebenstreit,
C. Spee, and B. Kraus, Phys. Rev. A 93, 012339 (2016) are not affected.
