there may be a different array ${\mathbf{R}}'_r$ satisfying ${\mathbf{R}}_r\cdot{\mathbf{P}} =
{\mathbf{R}}'_r\cdot{\mathbf{P}} \; \forall \, {\mathbf{P}} \in \mathcal{P}$.

A particular operational model must contain a specification of the set of measurements that can physically be performed on a particular type of system. There is a constraint: any measurement must correspond to a set $\{{\mathbf{R}}_r\}$ such that $$\label{constraint1}
{\mathbf{R}}_r\cdot{\mathbf{P}} \geq 0 \ \ \forall r\ \forall
\,{\mathbf{P}}\in\mathcal{P}$$ and $$\label{constraint2}
\sum_r {\mathbf{R}}_r\cdot{\mathbf{P}} = 1 \ \ \forall
\,{\mathbf{P}}\in\mathcal{P}.$$ Furthermore, if a measurement is performed on one subsystem of a bipartite system, it is possible to calculate the subsequent (“collapsed”) state of the other subsystem, conditioned on a particular outcome. Clearly this should be an allowed state of that subsystem.

Dynamics
--------

In addition to preparations and measurements, it may be possible to perform transformations on a system, i.e., to act on it in such a way that the system is preserved but its state changes. In the most general case, a system can change into a system of a different type. But here we consider only transformations that preserve the type of system. Such a transformation can be represented as a map ${\mathbf{T}}:\mathcal{P}\rightarrow\mathcal{P}$. As with measurements, a consideration of mixed states implies that 