In (Spekkens 2005), the author defined an "ontological model" as a model that assumes (page 3, above Eq. 4):
- That every preparation procedure $P$ is associated with a normalised probability density over the ontic state space $\Omega$, that is, some $\mu_P:\Omega\to[0,1]$ such that $\int \mu_P=1$.
- That every measurement procedure $M$ is associated with a set of indicator functions $\{\xi_{M,k}(\lambda)\}_k$ over the ontic states. Here $k$ label the outcomes, and the functions satisfy $\xi_{M,k}:\Omega\to[0,1]$ and $\sum_k \xi_{M,k}(\lambda)=1$.
- Some additional things about the "transformation procedure", which I won't focus on here.
My understanding is that in the quantum case, the "preparation procedures" correspond to the quantum states, and $M$ corresponds to the POVM used to measure the state. In this sense, $\xi_{M,k}$ would seem to just correspond to the rule to obtain outcome probabilities for any given state, that is, $\xi_{M,k}(\lambda)=\operatorname{tr}(M_k\rho_\lambda)$ for some state $\rho_\lambda$ and POVM $M$.
However, what I can't quite figure out is how precisely should I think about $\mu_P$ here.
I kind of thought that I should think of $\lambda$ as the "contextual equivalent" of states $\rho$, but later in the paper, around Eq. (8), they write that "the assumption of preparation noncontextuality in quantum theory is that the probability distribution over ontic states that is associated with a preparation procedure $P$ depends only on the density operator $\rho$ associated with $P$: $\mu_P(\lambda)=\mu_\rho(\lambda)$. This confuses me because then clearly a given state $\rho$ can correspond to different $\lambda$, but at the same time it doesn't seem that $\lambda$ should correspond to measurement outcomes, given the previous definition of $\xi_{M,k}$ function and labels $k$.
But also, when first introduced (second column of page 2), they write that someone who knows that a system was prepared using the preparation procedure $P$ describes the system by a probability density $\mu_P(\lambda)$ over the model variables. So it would seem that $\lambda\in\Omega$ are "model variables" that somehow characterise the state $\rho$.
So then this leaves me confused. States $\rho$ are represented by "model variables" $\lambda$ via a probability distribution $\mu_\rho(\lambda)$. But this distribution is not the probability distribution corresponding to different measurement outcomes. So what is it exactly? An answer to What is a simple example of Spekken's measurement contentextuality? would probably also help in figuring out this question.
