The membership problem for constant-sized quantum correlations is undecidable

TL;DR

This paper proves that determining whether certain fixed-sized quantum correlations can be achieved is fundamentally undecidable, highlighting intrinsic computational limits in understanding quantum entanglement and correlations.

Contribution

It demonstrates that the quantum membership problem remains undecidable even for constant-sized correlations, strengthening previous results and showing limits independent of measurement complexity.

Findings

Undecidability applies to fixed-sized quantum correlations.

The result constrains descriptions of quantum correlation sets.

Proof combines quantum self-testing and undecidability of linear system nonlocal games.

Abstract

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.