PIRSA:12110070 Scientific Series

DOI10.48660/12110070

Series: Quantum Foundations

(2012). Quasiprobability representations of qubits. Perimeter Institute. https://pirsa.org/12110070

Quasiprobability representations of qubits. Perimeter Institute, Nov. 13, 2012, https://pirsa.org/12110070 

@misc{ pirsa_PIRSA:12110070,
  doi = {10.48660/12110070},
  url = {https://pirsa.org/12110070},
  author = {},
  keywords = {},
  language = {en},
  title = {Quasiprobability representations of qubits},
  publisher = {Perimeter Institute},
  year = {2012},
  month = {nov},
  note = {PIRSA:12110070 see, \url{https://pirsa.org}}
}

Abstract

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.

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