Relationship between covariance of Wigner functions and transformation noncontextuality

We relate two notions of classicality for quantum transformations often arising in popular subtheories of quantum mechanics: covariance of the Wigner representation of the theory and the existence of a transformation noncontextual ontological model of the theory. We show that covariance implies transformation noncontextuality. The converse holds provided that the underlying ontological model is the one given by the Wigner representation. In addition, we investigate the relationships of covariance and transformation noncontextuality with the existence of a positivity preserving quasiprobability distribution for the transformations of the theory. We conclude that covariance implies transformation noncontextuality, which implies positivity preservation. Therefore the violation of the latter is a stronger notion of nonclassicality than the violation of the former.

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This work was supported (in part) by the Fetzer Franklin Fund of the John E. Fetzer Memorial Trust.