The infinite information gap between mathematical and physical representations

Partial orders have been used to model several experimental setups, going from classical thermodynamics and general relativity to the quantum realm with its resource theories. In order to study such experimental setups, one typically characterizes them via a (numerical) representation, that is, a set of real-valued functions. In the context of resource theory, it is customary to use \textbf{mathematical} representations, i.e. a set of \textbf{measurement outcomes} which characterize the achievable transitions within the experimental setup. However, in line with the minimum energy and maximum entropy principles in classical mechanics and thermodynamics, respectively, one would expect an optimization interpretation for a representation to be called \textbf{physical}. More specifically, a physical representation could consist of a set of competing \textbf{optimization principles} such that a transition happens provided they are all optimized by it. Somewhat surprisingly, we show that this distinction can result in an \textbf{infinite information gap}, with some partial orders having mathematical representations that involve a finite amount of information and requiring infinite information to build a physical representation. We connect this phenomenon with well-known resource-theoretic scenarios like majorization, and develop notions of partial order dimension that run in parallel to the representations that we consider. Our results improve on the classification of preordered spaces in terms of real-valued functions.