Quasiprobability representations of qubits

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Negativity in a quasi-probability representation is typically
interpreted as an indication of nonclassical behavior. 

However, this does not preclude bases that are non-negative from
having interesting applications---the single-qubit

stabilizer states have non-negative Wigner functions and yet
play a fundamental role in many quantum information tasks.

We determine what other sets of quantum states and measurements
of a qubit can be non-negative in a quasiprobability

representation, and identify nontrivial groups of unitary
transformations that permute such states. 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 two bases in any plane
of the Bloch sphere. Furthermore, there is a single family of

sets of four bases that can be non-negative in an arbitrary
quasiprobability representation of a qubit. We provide an 

exhaustive list of the sets of single-qubit bases that are nonnegative
in some quasiprobability representation and are also 

closed under a group of unitary transformations, revealing two
families of such sets of three bases. We also show that not 

all two-qubit Clifford transformations can preserve
non-negativity in any quasiprobability representation that is 

non-negative for the computational basis. This is in stark
contrast to the qutrit case, in which the discrete Wigner 

function is non-negative for all n-qutrit stabilizer states and
Clifford transformations. We also provide some evidence 

that extending the other sets of non-negative single-qubit
states to multiple qubits does not give entangled states.