Uniqueness of Noncontextual Models for Stabilizer Subtheories

We give a complete characterization of the (non)classicality of all stabilizer subtheories. First, we prove that there is a unique nonnegative and diagram-preserving quasiprobability representation of the stabilizer subtheory in all odd dimensions, namely Gross's discrete Wigner function. This representation is equivalent to Spekkens' epistemically restricted toy theory, which is consequently singled out as the unique noncontextual ontological model for the stabilizer subtheory. Strikingly, the principle of noncontextuality is powerful enough (at least in this setting) to single out one particular classical realist interpretation. Our result explains the practical utility of Gross's representation by showing that (in the setting of the stabilizer subtheory) negativity in this particular representation implies generalized contextuality. Since negativity of this particular representation is a necessary resource for universal quantum computation in the state injection model, it follows that generalized contextuality is also a necessary resource for universal quantum computation in this model. In all even dimensions, we prove that there does not exist any nonnegative and diagram-preserving quasiprobability representation of the stabilizer subtheory, and, hence, that the stabilizer subtheory is contextual in all even dimensions.

Automated summary

This study provides a complete characterization of the (non)classicality of all stabilizer subtheories. It proves that there is a unique nonnegative and diagram-preserving quasiprobability representation of the stabilizer subtheory in all odd dimensions, known as Gross's discrete Wigner function. This representation is equivalent to Spekkens' epistemically restricted toy theory, establishing it as the unique noncontextual ontological model for the stabilizer subtheory. On the other hand, in all even dimensions, it is shown that there does not exist any nonnegative and diagram-preserving quasiprobability representation of the stabilizer subtheory, indicating its contextual nature.