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When a quantum system is distributed to spatially separated parties, it isnatural to consider how the system evolves when the parties perform localquantum operations with classical communication (LOCC). However, the structureof LOCC channels is exceedingly complex leaving many important physicalproblems unsolved. In this paper we consider generalized resource theories ofentanglement based on different relaxations to the class of LOCC. The behaviorof various entanglement measures is studied under non-entangling channels, aswell as the newly introduced classes of dually non-entangling andPPT-preserving channels. In an effort to better understand the nature of LOCCbound entanglement, we study the problem of entanglement distillation in thesegeneralized resource theories. We first show that unlike LOCC, generalnon-entangling maps can be superactivated, in the sense that two copies of thesame non-entangling map can nevertheless be entangling. On the single-copylevel, we demonstrate that every NPT entangled state can be converted into anLOCC-distillable state using channels that are both dually non-entangling andhaving a PPT Choi representation and that every state can be converted into anLOCC-distillable state using operations belonging to any family of polytopesthat approximate LOCC. We then turn to the stochastic convertibility ofmultipartite pure states and show that any two states can be interconverted byany polytope approximation to the set of separable channels. Finally, as ananalog to $k$-positive maps, we introduce and analyze the set of$k$-non-entangling channels.

Entanglement manipulation and distillability beyond LOCC