Local unambiguous unidentifiability, entanglement generation, and Hilbert space splitting

We consider collections of mixed states supported on mutually orthogonal subspaces whose rank add up to the total dimension of the underlying Hilbert space. We then ask whether it is possible to find such collections in which no state from the set can be unambiguously identified by local operations and classical communication (LOCC) with non-zero success probability. We show the necessary and sufficient condition for such a property to exist is that the states must be supported in entangled subspaces. In fact, the existence of such a set guarantees the existence of a type of entangling projective measurement other than rank one measurements and vice versa. This projective measurement can create entanglement from any product state picked from the same Hilbert space on which the measurement is applied. Here the form of the product state is not characterized. Ultimately, these sets or the measurements are associated with the splitting of a composite Hilbert space, i.e., the Hilbert space can be written as a direct sum of several entangled subspaces. We then characterize present sets (measurements) in terms of dimensional constraints, maximum-minimum cardinalities (outcomes), etc. The maximum cardinalities of the sets constitute a class of state discrimination tasks where several stronger classes of measurements (like separable measurements, etc.) do not provide any advantage over LOCC. Finally, we discuss genuine local unambiguous unidentifiability and generation of genuine entanglement from completely product states.