The Negativity‐to‐Violation Map between Wigner Function and Quantum Contextuality Inequality for a Single Qudit

Abstract

The relation between discrete Wigner function and quantum contextuality based on graph theory has been studied, following the work in [Nature 510,351(2014)]. To do this, non-stabilizer projectors have been introduced to a series of non-contextuality graphs based on stabilizer projectors for a single qudit with odd prime dimension. It has been found that, for a phase space point defined by Wootters, there exists a given set of states for an odd-prime qudit where the negative discrete Wigner function on the phase space point means its quantum contextuality under measurements on the graphs designed by a specific method. To implement this method, a subset of non-stabilizer projectors has been found. In the union of the set of states for all phase space points, there exists a negativity-to-violation map between Wigner function and quantum contextuality inequality. The robustness of the equivalence under depolarizing noise has been analyzed and discussed. For demonstration purposes, the graphs with different independence numbers and the corresponding set of states have been established on a single qutrit. Different to the cited work, this method involves only a single qudit, then is experimentally feasible for a qutrit.

Kenfack – Życzkowski indicator of nonclassicality for two non-equivalent representations of Wigner function of qutrit

“…Despite of a long history of studies of the quasiprobabilty distributions nowadays new aspects of the Wigner function negativity has drawn attention. Particularly, in connection to the quantum computing and theory of quantum information, a special class of so-called discrete Wigner functions [15,16] has been introduced, for which negativity is equivalent to the quantum contextuality [17,18,19].…”