Equivalence between contextuality and negativity of the Wigner function for qudits

Abstract

Understanding what distinguishes quantum mechanics from classical mechanics is crucial for quantum information processing applications. In this work, we consider two notions of nonclassicality for quantum systems, negativity of the Wigner function and contextuality for Pauli measurements. We prove that these two notions are equivalent for multi-qudit systems with odd local dimension. For a single qudit, the equivalence breaks down. We show that there exist single qudit states that admit a non-contextual hidden variable model description and whose Wigner functions are negative. function in terms of the hidden variable model (HVM) and we observe that if the HVM is non-contextual then the Wigner function is non-negative. Howard et al required a two-qudit experiment to demonstrate contextuality of a single qudit state. We elucidate this by showing explicitly that in a single qudit experiment, the equivalence between contextuality and negativity does not hold. We show this by constructing quantum states that admit a noncontextual HVM (NCHVM) description although their Wigner functions take negative values.Our proof of this equivalence relies on the choice of a simple definition of contextuality based on value assignments introduced in the work of Kochen and Specker [22], whereas the work of Howard et al is based on the graphical formalism of Cabello et al [39]. The relations between these different notions of non-classicality are depicted in figure 1. Following [22][23][24], we consider only contextuality of stabilizer measurements and HVMs are assumed to be deterministic. This last assumption is also present in the graphical formalism of Cabello et al [39] and in the work of Howard et al [18]. Our argument does not apply to the generalized notion of HVMs considered for instance by Spekkens [38,40].Our work clarifies the relationship between discrete Wigner function and stabilizer contextuality for odddimensional qudits. It cannot be extended to systems of qubits due to the presence of state-independent contextuality [23][24][25]. In addition, no qubit Wigner function that satisfies all the properties of Gross' qudit Wigner function seems to exist [41][42][43].This article is organized as follows. The necessary stabilizer formalism is recalled in section 2. Section 3 introduces a notion contextuality based on value assignment and a proof of the equivalence between this notion and the negativity of Gross' discrete Wigner function, i.e. (ii) ⇔ (iv). The purpose of section 4 is to prove the equivalence between the two notions of contextuality (i) and (ii), completing the square in figure 1.

Contextuality as a Resource for Models of Quantum Computation with Qubits

“…We extended earlier results on odd-prime dimensional qudits [4,5] and rebits [6], and thereby completed establishing contextuality as a resource in QCSI in arbitrary prime dimensions. We conjecture that this result generalizes to all composite dimensions [15] (the composite odd case was recently covered after completion of this work [16]) and to algebraic extensions of QCSI models based on normalizer gates [11,[17][18][19][20]. Further, we demonstrated the applicability of our result to a concrete qubit QCSI scheme that does not exhibit state independent contextuality while retaining tomographic completeness.…”

Resource theory of contextuality

“…39 , the authors show that if non-negative Wigner functions remain non-negative under free measurements, then contextuality and Wigner function negativity are necessary resources for universal quantum computation on these schemes. The result on contextuality is however strictly stronger than the result on Wigner functions, since different from the qudit case 16 , qubit magic states can have negative Wigner functions but still be non-contextual. These results where later generalized in ref.…”

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

“…In Appendix A we will further expand equivalence in our method between this contextuality inequality and value assignment analogous to Equation (2). [21][22][23][24] Furthermore for testing the relation between state-dependent contextuality and the discrete Wigner function multiple qudits are involved in the most methods, [13,14,16,17] even for the test on a single qudit [13] the second auxiliary qudit is needed. The currently known testing methods on a single qutrit with the minimum numbers of measurements consist of five measurements for a state-dependent test [2] and 13 measurements for a state-independent test.…”

“…The negativity of this function receives widespread attention in quantum optics because it shows the nonclassical properties of some special states to a certain extent. [13,14] Hence, it is an intuition that the negativity of Wigner function connects the quantum contextuality and quantum computation resource. It has been proven that the negativity of discrete Wigner function in quantum state with odd prime dimension is a necessary resource for quantum computation.…”

“…It has been proven that the negativity of discrete Wigner function in quantum state with odd prime dimension is a necessary resource for quantum computation. Multiple qudits are involved in most methods of testing the relation between state-dependent contextuality and the discrete Wigner function, [13,14,16,17] even for the case of a single qudit [13] the second auxiliary qudit is needed. [10][11][12] Meanwhile, the states with the negativity of Wigner function have been www.advancedsciencenews.com www.ann-phys.org proven to exhibit state-dependent contextuality under stabilizer measurements.…”