Uniqueness of Noncontextual Models for Stabilizer Subtheories

David Schmid; Haoxing Du; John Selby; Matthew Fairbairn Pusey

doi:10.1103/PhysRevLett.129.120403

Uniqueness of Noncontextual Models for Stabilizer Subtheories

David Schmid, Haoxing Du, John Selby, Matthew Fairbairn Pusey

Mathematics

Research output: Contribution to journal › Letter › peer-review

Abstract

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.

Original language	English
Article number	120403
Journal	Physical Review Letters
Volume	129
Issue number	12
DOIs	https://doi.org/10.1103/PhysRevLett.129.120403
Publication status	Published - 14 Oct 2022

Bibliographical note

© 2022 American Physical Society. This is an author-produced version of the published paper. Uploaded with permission of the publisher/copyright holder. Further copying may not be permitted; contact the publisher for details.

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10.1103/PhysRevLett.129.120403

StabilizerWigner
© 2022 American Physical Society. This is an author-produced version of the published paper. Uploaded with permission of the publisher/copyright holder. Further copying may not be permitted; contact the publisher for details.
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https://arxiv.org/abs/2101.06263

Cite this

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title = "Uniqueness of Noncontextual Models for Stabilizer Subtheories",

abstract = "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{\textquoteright}s discrete Wigner function. This representation is equivalent to Spekkens{\textquoteright} epistemically restricted toy theory, which is consequently singled out as the unique noncontextual ontological model for the stabilizer subtheory. Strikingly, the principle ofnoncontextuality is powerful enough (at least in this setting) to single out one particular classical realist interpretation. Our result explains the practical utility of Gross{\textquoteright}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 anecessary 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.",

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N2 - 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 ofnoncontextuality 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 anecessary 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.

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