I'm going to put my worries about the construction in terms I'm familiar with, quantum mechanics, but which I hope will not be too unfamiliar for you, Erik, instead of in the terms you have been using, which I take to be of stochastic matrices and stochastic vectors representing states.
In the QM context of Hilbert spaces and the representation of states by density matrices, von Neumann entropy, [math]\mathsf{vN}(\hat\rho)=-\mathsf{Tr}[\hat\rho\ln\hat\rho],[/math] is perhaps the commonest measure of information (there are certainly others, but just replace von Neumann by some other measure everywhere in what follows). If we take coarse-graining to be a map, which in general will be nonlinear, from n-by-n density matrices to m-by-m density matrices, [math]X:\mathcal{D}(\mathcal{H}_n)\rightarrow\mathcal{D}(\mathcal{H}_m);\hat\rho\mapsto X(\hat\rho),[/math] then the von Neumann entropy of the coarse-grained system will be [math]\mathsf{vN}(X(\hat\rho))=-\mathsf{Tr}[X(\hat\rho)\ln X(\hat\rho)].[/math] It seems clear(?) that how the von Neumann entropy of the coarse-grained model will be different from the von Neumann entropy of the pre-coarse-grained model will depend both on what choice we make for X and on what density matrices occur as pre-coarse-grained models.
I think my problem is that it's not clear to me that a detailed accounting of where there is more or less information is a specially good way to think about causality or consciousness. In QM, what "causes" what is systematically described by the Hamiltonian, defined as the infinitesimal generator of the evolution of vector states as a function of time; if one works with a Hilbert space of vector states and with density matrices, as Physics now kinda does, that's it.
Now, not very seriously, I'll go off the deep end... There are possibilities for QM to model consciousness insofar as we can construct, not for a toy model of a few dimensions, perhaps, but for a sufficiently large Hilbert space, a measurement operator C that returns "1" if there is at least one conscious agent or "0" if not, as well as more physical measurements such as the intensity of the electric field. The algebraic relationships of C with the Hamiltonian and with whatever coarse-graining operators we use would determine what measurement results would be expected in a given state and at different levels of coarse-graining (which would include, for example, correlations between consciousness and the intensity of the electric field). Ensuring such models are empirically useful is of course not so easy.
