  When a quantum system is distributed to spatially separated parties, it is
natural to consider how the system evolves when the parties perform local
quantum operations with classical communication (LOCC). However, the structure
of LOCC channels is exceedingly complex leaving many important physical
problems unsolved. In this paper we consider generalized resource theories of
entanglement based on different relaxations to the class of LOCC. The behavior
of various entanglement measures is studied under non-entangling channels, as
well as the newly introduced classes of dually non-entangling and
PPT-preserving channels. In an effort to better understand the nature of LOCC
bound entanglement, we study the problem of entanglement distillation in these
generalized resource theories. We first show that unlike LOCC, general
non-entangling maps can be superactivated, in the sense that two copies of the
same non-entangling map can nevertheless be entangling. On the single-copy
level, we demonstrate that every NPT entangled state can be converted into an
LOCC-distillable state using channels that are both dually non-entangling and
having a PPT Choi representation and that every state can be converted into an
LOCC-distillable state using operations belonging to any family of polytopes
that approximate LOCC. We then turn to the stochastic convertibility of
multipartite pure states and show that any two states can be interconverted by
any polytope approximation to the set of separable channels. Finally, as an
analog to $k$-positive maps, we introduce and analyze the set of
$k$-non-entangling channels.
