Equivalence of contextuality and Wigner function negativity in continuous-variable quantum optics

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.

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“…Our work also extends the body of known connections between contextuality, negativity, and computation [13][14][15][16][46][47][48][49][50][51].…”

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confidence: 54%

Uniqueness of Noncontextual Models for Stabilizer Subtheories

Schmid

Selby

et al. 2022

Phys. Rev. Lett.

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“…The expectation value of the number operator a † 0 a 0 can be equivalently found using either its Glauber-Sudarshan, Husimi, or Wigner quasiprobability distributions [24]. The latter is particularly interesting, as its negative values certify the quantum nature of the electromagnetic field [26] and may witness contexuality [8][9][10], exactly as anomalous weak values [5][6][7]. For this reason here we derive the conditional Wigner function of the mode of frequency ω 0 :…”

Section: Conditional Wigner Functionmentioning

confidence: 99%

“…In correspondence of such anomalous values, the Wigner function dictating the statistical distribution of the postselected measurements takes negative values [5]. Furthermore, it can be proven that both anomalous weak values [5][6][7], and Wigner-function negativity [8][9][10], are witnesses of contextuality. Here we show that those effects occur in a paradigmatic setting of waveguide quantum electrodynamics where energy exchanges feature anomalous weak values.…”

Section: Introductionmentioning

confidence: 96%

Anomalous energy exchanges and Wigner-function negativities in a single-qubit gate

Maffei

Elouard²,

Goes³

et al. 2023

Phys. Rev. A

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Anomalous weak values and the Wigner function's negativity are well-known witnesses of quantum contextuality. We show that these effects occur when analyzing the energetics of a single-qubit gate generated by a resonant coherent field traveling in a waveguide. The buildup of correlations between the qubit and the field is responsible for bounds on the gate fidelity, but also for a nontrivial energy balance recently observed in a superconducting setup. In the experimental scheme, the field is continuously monitored through heterodyne detection and then postselected over the outcomes of a final qubit's measurement. The postselected data can be interpreted as the field's weak values and can show anomalous values in the variation of the field's energy. We model the joint system dynamics with a collision model, gaining access to the qubit-field entangled state at any time. We find an analytical expression of the quasiprobability distribution of the postselected heterodyne signal, i.e., the conditional Husimi-Q function. The latter grants access to all the field's weak values: we use it to obtain that of the field's energy change and display its anomalous behavior. Finally, we derive the field's conditional Wigner function and show that anomalous weak values and Wigner function negativities arise for the same values of the gate's angle.

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“…Regarding the former, Kochen-Specker contextuality [2] has recently been given a cohomological underpinning [3,4,5]. Regarding the latter, Wigner function negativity-a traditional indicator of nonclassicality in quantum optics [6]-has been shown to be equivalent to contextuality in certain cases [7,8,9,10], and both have been linked to the possibility of quantum computational advantage [11,12,13,14,15].…”

Section: Introductionmentioning

confidence: 99%

The role of cohomology in quantum computation with magic states

Raussendorf

Okay

Zurel

et al. 2023

Quantum

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A web of cohomological facts relates quantum error correction, measurement-based quantum computation, symmetry protected topological order and contextuality. Here we extend this web to quantum computation with magic states. In this computational scheme, the negativity of certain quasiprobability functions is an indicator for quantumness. However, when constructing quasiprobability functions to which this statement applies, a marked difference arises between the cases of even and odd local Hilbert space dimension. At a technical level, establishing negativity as an indicator of quantumness in quantum computation with magic states relies on two properties of the Wigner function: their covariance with respect to the Clifford group and positive representation of Pauli measurements. In odd dimension, Gross' Wigner function – an adaptation of the original Wigner function to odd-finite-dimensional Hilbert spaces – possesses these properties. In even dimension, Gross' Wigner function doesn't exist. Here we discuss the broader class of Wigner functions that, like Gross', are obtained from operator bases. We find that such Clifford-covariant Wigner functions do not exist in any even dimension, and furthermore, Pauli measurements cannot be positively represented by them in any even dimension whenever the number of qudits is n≥2. We establish that the obstructions to the existence of such Wigner functions are cohomological.

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Equivalence of contextuality and Wigner function negativity in continuous-variable quantum optics

Cited by 4 publications

References 57 publications

Uniqueness of Noncontextual Models for Stabilizer Subtheories

Uniqueness of Noncontextual Models for Stabilizer Subtheories

Anomalous energy exchanges and Wigner-function negativities in a single-qubit gate

The role of cohomology in quantum computation with magic states

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