Measurement Contextuality and Planck's Constant

Abstract

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 equivalent to orders of $\hslash$ as a resource within the WWM formalism. This explains 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, we find that qubit Pauli observables lack an order ${\hslash }^0$ contribution in their Weyl symbol and so exhibit contextuality regardless of the state being measured. On the other hand, odd-dimensional qudit observables generally possess non-zero order ${\hslash }^0$ terms, and higher, in their WWM formulation, and so exhibit contextuality depending on the state measured: odd-dimensional qudit states that exhibit measurement contextuality have an order ${\hslash }^1$ contribution that allows for the violation of classical bounds while states that do not exhibit measurement contextuality have insufficiently large order ${\hslash }^1$ contributions.

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• Noncontextual Pauli Hamiltonians
• Differential Geometry of Contextuality