  Entangled quantum states can be given a separable decomposition if we relax
the restriction that the local operators be quantum states. Motivated by the
construction of classical simulations and local hidden variable models, we
construct `smallest' local sets of operators that achieve this. In other words,
given an arbitrary bipartite quantum state we construct convex sets of local
operators that allow for a separable decomposition, but that cannot be made
smaller while continuing to do so. We then consider two further variants of the
problem where the local state spaces are required to contain the local quantum
states, and obtain solutions for a variety of cases including a region of pure
states around the maximally entangled state. The methods involve calculating
certain forms of cross norm. Two of the variants of the problem have a strong
relationship to theorems on ensemble decompositions of positive operators, and
our results thereby give those theorems an added interpretation. The results
generalise those obtained in our previous work on this topic [New J. Phys. 17,
093047 (2015)].
