01577nas a2200133 4500008004100000245008200041210006900123260001400192520114500206100001601351700001701367700002201384856003701406 2021 eng d00aThe membership problem for constant-sized quantum correlations is undecidable0 amembership problem for constantsized quantum correlations is und c1/26/20213 a
When two spatially separated parties make measurements on an unknown entangled quantum state, what correlations can they achieve? How difficult is it to determine whether a given correlation is a quantum correlation? These questions are central to problems in quantum communication and computation. Previous work has shown that the general membership problem for quantum correlations is computationally undecidable. In the current work we show something stronger: there is a family of constant-sized correlations -- that is, correlations for which the number of measurements and number of measurement outcomes are fixed -- such that solving the quantum membership problem for this family is computationally impossible. Thus, the undecidability that arises in understanding Bell experiments is not dependent on varying the number of measurements in the experiment. This places strong constraints on the types of descriptions that can be given for quantum correlation sets. Our proof is based on a combination of techniques from quantum self-testing and from undecidability results of the third author for linear system nonlocal games.
1 aFu, Honghao1 aMiller, Carl1 aSlofstra, William uhttps://arxiv.org/abs/2101.11087