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 the problem of extending a system with a finite set of transformations. We also investigate the problem of adding to a translation invariant set of systems a finite set of entangled states and effects, plus all their images by the translation symmetry. We show that determining whether such extensions are consistent with the axioms of GPTs is undecidable: they are computationally equivalent to the halting problem for Turing machines. The source of the undecidability is that these finite extensions generate infinitely many conditions which must be checked, because iterating transformations produces infinitely many new transformations, and similarly, entangled states and effects generate infinitely many new states via the analog 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.