Suppose two parties, Alice and Bob, who are separated share the "state" $\sigma_{AB}=|0\rangle\langle1|_A\otimes |0\rangle\langle1|_B$. Of course, this not a valid quantum state as it does not satisfy trace 1 and is not Hermitian.
Suppose Alice and Bob have additional qubits 1 and 2 respectively, which are not initially entangled: they are in some separable state $\rho_1\otimes \rho_2$. Can they use LOCC operations as well as access to $|0\rangle\langle1|_A\otimes |0\rangle\langle1|_B$ to entangle qubits 1 and 2?
Naturally if $\sigma_{AB}$ were a valid separable state, then this would not be possible. However, here we relaxing the assumption $\sigma_{AB}$ is a valid state.
It's not totally clear if this question is well defined, as the output state of whatever process you perform will not be a valid quantum state. But if we insist on "ignoring" the state of qubits A and B, maybe one can say something about the separability of the states when only considering qubits 1 and 2?
I am also somewhat aware that one definition of entangled states is the set of states which are not able to be generated from separable states with LOCC. However, it is not clear to me that this definition works in this case.