Nonnegative subtheories and quasiprobability representations of qubits

Abstract

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.

17 pages, 8 figures, comments very welcome; v2 published version. Note that the statement and proof of Theorem III.2 in the published version are incorrect (an erratum has been submitted), and this arXiv version (v2) presents the corrected theorem and proof. The conclusions of the paper are unaffected by this correction

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