2012/03/31 by Joel J. Wallman, Stephen D. Bartlett · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #Bloch sphere #Hilbert space #Mathematics #Physics #Pure mathematics #Quantum #Quantum Computing Algorithms and Architecture #Quantum Information and Cryptography #Quantum Mechanics and Applications #Quantum mechanics #Quantum state #Qubit #Representation (politics) #Statistical physics #Unitary representation #Unitary state #quant-ph
paper · pdf · doi:10.1103/physreva.85.062121
published as Phys. Rev. A 85, 062121 (2012) · 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
openalex publication_date 2012/06/26 · arxiv created 2012/08/22 · arxiv updated 2012/08/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 of a qubit can be non-negative 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 non-negative for more than two bases in any plane of the Bloch sphere. Furthermore, there is a unique set 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 non-negative in some quasiprobability distribution and are also closed under a group of unitary transformations. This list includes two nontrivial families of three 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^d2 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 different from each other.