Consistency of Generalised Probabilistic Theories is Undecidable

Abstract:Generalised Probabilistic Theories (GPTs) provide a unifying framework encompassing classical theories, quantum theories, as well as hypothetical alternatives. We investigate both the problem of extending a system with a finite set of transformations, and the problem of adding to a translation-invariant set of systems a finite set of entangled states and effects, plus all their images under the translation symmetry. We show that determining whether such extensions are consistent with the axioms of GPTs is undecidable. The source of the undecidability is that these finite extensions can generate infinitely many conditions which must be checked: iterating transformations may produce infinitely many new transformations; and similarly, entangled states and effects may generate infinitely many new states via the analogue of teleportation. Our results show that extending GPTs to include dynamics or entanglement encounters fundamental computability obstructions, which can only be circumvented by introducing additional physical or mathematical assumptions.

arXiv:2603.07002 [quant-ph]

(or arXiv:2603.07002v3 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2603.07002 Focus to learn more arXiv-issued DOI via DataCite

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