The membership problem for constant-sized quantum correlations is undecidable
Honghao Fu, Carl A. Miller, William Slofstra
TL;DR
The paper proves that the membership problem for quantum correlations is undecidable even for constant-sized correlations, showing that a fixed-size description cannot, in general, decide membership in quantum sets like $C_{qa}$ or $C_{qc}$. The authors integrate a suite of techniques—quantum self-testing, undecidability results for linear-system nonlocal games, and deep group-theoretic constructions (KMS groups and solution groups)—to encode the halting problem into a family of fixed-size correlations. They construct computable families $F_n$ with fixed measurement counts and outcomes such that $F_n$ intersects $C_{qa}$ if and only if the input $n$ does not belong to a given RE set, while $F_n$ remains disjoint from $C_{qc}$ for the same inputs, yielding coRE-hardness (and coRE-completeness for qc) of the fixed-size membership problem. Consequently, there can be no uniform polynomial description (e.g., via a finite list of polynomial inequalities) of $C_t(n_A,n_B,m_A,m_B)$ that decides membership, imposing strong limits on how quantum correlation sets can be described algorithmically.
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.