Quantum Behaviors Are Not Semialgebraic

Abstract:The conditional probabilities achievable by local measurements on a shared quantum state in a Bell scenario form the set of quantum behaviors, whose structure was studied by Tsirelson. In 1993, Tsirelson asked whether this set is semialgebraic, that is, describable by finite Boolean combinations of polynomial equations and inequalities. This question has remained open. We answer it in the negative: with four binary measurements per party, the set of finite-dimensional quantum behaviors, its closure, and the commuting-operator set are all nonsemialgebraic. More strongly, none admits a finite real-analytic description even locally near a particular classical behavior. These results rule out exact finite semidefinite representations and show that no finite level of the Navascués-Pironio-Acín hierarchy characterizes these sets exactly.

arXiv:2609.18865 [quant-ph]

(or arXiv:2609.18865v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2609.18865 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.