Measurement Contextuality and Planck's Constant

Contextuality is a necessary resource for universal quantum computation and non-contextual quantum mechanics can be simulated efficiently by classical computers in many cases. Orders of Planck's constant, $\hslash$ , can also be used to characterize the classical-quantum divide by expanding quantities of interest in powers of $\hslash$ -all orders higher than ${\hslash }^0$ can be interpreted as quantum corrections to the order ${\hslash }^0$ term. We show that contextual measurements in finite-dimensional systems have formulations within the Wigner-Weyl-Moyal (WWM) formalism that require higher than order ${\hslash }^0$ terms to be included in order to violate the classical bounds on their expectation values. As a result, we show that contextuality as a resource is closely related to orders of $\hslash$ as a resource within the WWM formalism. This offers an explanation for why qubits can only exhibit state-independent contextuality under Pauli observables as in the Peres-Mermin square while odd-dimensional qudits can also exhibit state-dependent contextuality. In particular, qubit states exhibit contextuality when measured by qubit Pauli observables regardless of the state being measured and so the Weyl symbol of these observables lack an order ${\hslash }^0$ contribution altogether. On the other hand, odd-dimensional qudit states exhibit contextuality when measured by qudit observables depending on the state measured and so odd-dimensional qudit observables generally possess non-zero order ${\hslash }^0$ terms, and higher, in their WWM formulation: odd-dimensional qudit states that exhibit measurement contextuality have an order ${\hslash }^1$ contribution in their expectation values with the observable that allows for the violation of classical bounds while states that have insufficiently large order ${\hslash }^1$ contributions do not exhibit measurement contextuality.