  According to usual definitions, entangled states cannot be given a separable
decomposition in terms of products of local density operators. If we relax the
requirement that the local density operators be positive, then an entangled
quantum state may admit a separable decomposition in terms of more general sets
of single-system operators. This form of separability can be used to construct
classical models and simulation methods when only restricted set of
measurements are available. With such motivations in mind, we ask what are the
smallest such sets of local operators such that a pure bipartite entangled
quantum state becomes separable? We find that in the case of maximally
entangled states there are many inequivalent solutions, including for example
the sets of phase point operators that arise in the study of discrete Wigner
functions. We therefore provide a new way of interpreting these operators, and
more generally, provide an alternative method for constructing local hidden
variable models for entangled quantum states under subsets of quantum
measurements.
