Measurement outcomes that do not occur and their role in entanglement transformations

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 three- and five-qubit 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 Hebenstreit et al 2016 Phys. Rev. A 93, 012339 are not affected.BK thanks R Brieger and D Sauerwein for discussions related to the characterization of the local unitary symmetries of special graph states. ME, MH, and BK acknowledge financial support from the Austrian Science Fund (FWF) grant DK-ALM: W1259-N27 and the SFB BeyondC (Grant No. F7107). Furthermore, ME and BK acknowledge support of the Austrian Academy of Sciences via the Innovation Fund 'Research, Science and Society' as well as support from the Austrian Science Fund (FWF) grant FG5-L. CS acknowledges support by the Austrian Science Fund (FWF): J 4258-N27 and the ERC (Consolidator Grant 683107/TempoQ). JIdV acknowledges financial support by the Spanish MINECO through Grants MTM2017-84098-P and MTM2017-88385-P and by the Comunidad de Madrid through Grant QUITEMAD-CM P2018/TCS-4342