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, hbar, can also be used to characterize this classical-quantum divide by expanding quantities of interest in powers of hbar---all orders higher than hbar⁰ can be interpreted as quantum corrections to the order hbar⁰ term. We show that contextual measurements in finite-dimensional systems have formulations within the Wigner-Weyl-Moyal (WWM) formalism that require higher than order hbar⁰ 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 equivalent to orders of hbar as a resource within the WWM formalism and this explains why qubits can only exhibit exhibit state-independent contextuality under Pauli observables while odd-dimensional qudits can also exhibit state-dependent contextuality. In particular, we find that qubit Pauli observables lack an order hbar⁰ contribution in their Weyl symbol and so exhibit contextuality regardless of the state being measured.