Nonnegative subtheories and quasiprobability representations of qubits

Details Nonnegative subtheories and quasiprobability representations of qubits Nonnegative subtheories and quasiprobability representations of qubits

2012-03-12

arxiv.org/abs/1203.2652v1

Physics

Quantum Physics

17 pages, 8 figures, comments very welcome

Scientific paper

Negativity in a quasiprobability representation is typically interpreted as an indication of nonclassical behavior. However, this does not preclude states that are nonnegative from exhibiting phenomena typically associated with quantum mechanics - the single qubit stabilizer states have nonnegative 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 of a qubit can be nonnegative in a quasiprobability distribution, and to identify nontrivial groups of unitary transformations 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 nonnegative for more than 2 bases in any plane of the Bloch sphere. Furthermore, there is a unique set of 4 bases that can be nonnegative 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 distribution and are also closed under a group of unitary transformations. This list includes 2 nontrivial families of 3 bases that both include the single qubit stabilizer states as a special case. For qudits, we prove that there can be no more than 2^{d^2} states in nonnegative 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 different from each other.

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