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, ÿ, can also be used to characterize the classical-quantum divide by expanding quantities of interest in powers of ÿ-all orders higher than ÿ 0 can be interpreted as quantum corrections to the order ÿ 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 ÿ 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 ÿ 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 ÿ 0 contribution altogether. On the other hand, odddimensional 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 ÿ 0 terms, and higher, in their WWM formulation: odd-dimensional qudit states that exhibit measurement contextuality have an order ÿ 1 contribution in their expectation values with the observable that allows for the violation of classical bounds while states that have insufficiently large order ÿ 1 contributions do not exhibit measurement contextuality.

Stationary Phase Method in Discrete Wigner Functions and Classical Simulation of Quantum Circuits

“…Recently, there have also been some more direct approaches [22]. The relationship between contextuality and higher orders of in the WWM formalism has been recently established [16][17][18]. Therefore, a more natural (and historically successful, if continuous systems are included) approach that remains unexplored is to use a semiclassical approach, wherein contextual corrections to non-contextual backbone are added with higher order expansions using the stationary phase method.…”

“…Quantum circuits treated in terms of a path integral formalism require terms at orders greater than 0 for them to attain quantum universality [16][17][18]. In the continuous case, this corresponds to requiring non-Gaussianity as a resource.…”

Approximate stabilizer rank and improved weak simulation of Clifford-dominated circuits for qudits

“…Comparing the algorithms of Bravyi and Gosset and Pashayan should shed more light on the relationship between magic, contextuality and negativity [8,34]. However quasiprobability representations for qubits are distinct from their d-dimensional cousins [24][25][26]. The desire to understand the relationship between magic, contextuality and negativity therefore motivates extension of the algorithm of Bravyi and Gosset to qudits with dimension greater than two.…”

Stationary Phase Method in Discrete Wigner Functions and Classical Simulation of Quantum Circuits

“…For instance, by representing quantum circuits through the path integral perspective, efficient approximation schemes can be formed by only including lowestorder terms in . It is known that quantum circuits require terms at orders greater than 0 for them to attain quantum universality [4][5][6]. In the continuous-variable setting, this corresponds to requiring non-Gaussianity as a resource.…”

“…In continuous systems, a historically successful approach of leveraging contextuality efficiently is to use semiclassical techniques that rely on expansion in orders of . The relationship between contextuality and higher orders of in the Wigner-Weyl-Moyal (WWM) formalism has been recently established [4][5][6]. Previous derivation of WWM in odd dimensions [20][21][22] were not able to accomplish this because they began by using results from the continuous case of the WWM formalism.…”