Device-independent Quantum Key Distribution in the commuting operator framework

Device-independent quantum key distribution (DIQKD) is arguably the gold standard for secure quantum communication, as it aims to rely only on observed input-output statistics of an uncharacterized device which is only assumption to obey the laws of quantum physics. A corresponding security analysis hence demands a description of a quantum experiment from a most general perspective. Under close inspection, existing proof techniques do not always meet this goal as they tend to rely on subtile assumptions on a tensor product structure of the underlying Hilbert space and a 'hidden but finite' dimensionality. In this work, we collect the tools needed for a full analysis of DIQKD in the commuting operator framework, which avoids these subtilities and provides the arguably most general view on a quantum experiment. We rigorously proof the common assumption that in DIQKD measurements can be w.l.o.g. assumed to be projective. Furthermore, we show that task of computing key rates can be casted as a non-commutative polynomial optimization (NPO) problem to which the Navascu\'es-Pironio-Ac\'in (NPA) hierarchy gives a correct and converging relaxation. As a tool, we generalize the integral representation for the relative entropy by Frenkel [Quantum 7, 1102 (2023)] to general von Neumann algebras and apply techniques from Kossmann and Schwonnek [arXiv: 2411.04858] for the approximation in an NPO program.