Quantum Advantage needs Multiple Perspectives

Abstract

The origin of the quantum speedup is a topic of ongoing research. We propose a logical criterion for the possibility of a speedup within the continuous-variable and odd-prime qudit settings. There, Wigner function nonnegativity signals the possibility of efficient classical simulation in the state-injection scenario. We consider representations of quadrature/Weyl measurements in a single von Neumann algebra that reproduce the Born statistics in a particular state. If in that state, the set-theoretic universe of the algebra is ZFC-satisfiable, i.e. every ZFC-theorem holds with unit probability, all measurement contexts can be represented in a single commutative algebra of `beables'. Furthermore, the statistics admit a noncontextual hidden variable model, ensuring Wigner nonnegativity for two or more modes or qudits. For a single mode, a stronger notion of linear representation ensures commutativity, and with it, ZFC-satisfiability independently of the state, as well as Wigner nonnegativity. In consequence, in any setting that cannot be efficiently simulated, it must be impossible to merge the models associated with every context into a single ZFC-satisfiable one: quantum advantage needs multiple perspectives.