Problem
Research questions and friction points this paper is trying to address.
Quantum correlations
Computational complexity
Quantum entanglement verification
The membership problem for constant-sized quantum correlations is undecidable
🤖 AI Summary
This paper addresses the quantum membership problem: determining whether correlations obtained by spatially separated parties performing fixed-number measurements with fixed-output cardinality on an unknown entangled state belong to the set of quantum correlations. The authors combine techniques from quantum self-testing, the undecidability construction of linear-system nonlocal games, and tools from operator algebras and computability theory. They establish, for the first time, that this membership problem is uncomputable—even when the measurement settings’ size is held constant. This result refutes parameter-scaling–based explanations of quantum nonlocality and reveals the fundamental non-axiomatizability of the quantum correlation set. Moreover, it establishes a tight computational boundary for Bell experiments, rigorously excluding the possibility that any finite axiomatic system can fully characterize the family of constant-size quantum correlations.
📝 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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Quantum Contextuality
Computational Intractability
Quantum Self-Tests
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