  Negativity in a quasiprobability representation is typically interpreted as
an indication of nonclassical behavior. However, this does not preclude states
that are non-negative from exhibiting phenomena typically associated with
quantum mechanics - the single qubit stabilizer states have non-negative Wigner
functions and yet play a fundamental role in many quantum information tasks. We
seek to determine what other sets of quantum states and measurements for a
qubit can be non-negative in a quasiprobability representation, and to identify
nontrivial unitary groups that permute the states in such a set. These sets of
states and measurements are analogous to the single qubit stabilizer states. We
show that no quasiprobability representation of a qubit can be non-negative for
more than four bases and that the non-negative bases in any quasiprobability
representation must satisfy certain symmetry constraints. We provide an
exhaustive list of the sets of single qubit bases that are non-negative in some
quasiprobability representation and are also permuted by a nontrivial unitary
group. This list includes two families of three bases that both include the
single qubit stabilizer states as a special case and a family of four bases
whose symmetry group is the Pauli group. For higher dimensions, we prove that
there can be no more than 2^{d^2} states in non-negative bases of a
d-dimensional Hilbert space in any quasiprobability representation.
Furthermore, these bases must satisfy certain symmetry constraints,
corresponding to requiring the bases to be sufficiently complementary to each
other.
