Nonnegative subtheories and quasiprobability representations of qubits

TL;DR

This paper characterizes the sets of qubit states and measurements that can be represented non-negatively in quasiprobability frameworks, revealing limitations and symmetries, and extends results to higher-dimensional systems.

Contribution

It provides an exhaustive classification of non-negative bases for qubits in quasiprobability representations and establishes bounds and symmetry conditions for higher dimensions.

Findings

No quasiprobability representation of a qubit can have more than four non-negative bases.

Identifies specific sets of bases including stabilizer states and Pauli group symmetries.

Bounds the number of non-negative states in higher-dimensional systems to 2^{d^2}.

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.