Infinite dimensionality of the post-processing order of measurements on a general state space

For a partially ordered set $(S,⪯)$, the order (monotone) dimension is the minimum cardinality of total orders (respectively, real-valued order monotone functions) on $S$ that characterize the order $⪯$. In this paper we consider an arbitrary generalized probabilistic theory and the set of finite-outcome measurements on it, which can be described by effect-valued measures, equipped with the classical post-processing orders. We prove that the order and order monotone dimensions of the post-processing order are (countably) infinite if the state space is not a singleton (and is separable in the norm topology). This result gives a negative answer to the open question for quantum measurements posed in [Guff T $\mathit{\text{et}}\mathit{\text{al.}}\mathit{\text{/}}$ 2021 \textit{J.
Phys.\ A: Math.\ Theor.} $\mathbf{\text{54}}$ 225301]. We also consider the quantum post-processing relation of channels with a fixed input quantum system described by a separable Hilbert space $\mathcal{H}$ and show that the order (monotone) dimension is countably infinite when $\dim \mathcal{H}\ge 2$.