R_1(a_1 a_2| x_1 x_2) &=& \left( \begin{array}{c|c}
\begin{array}{cc} 0&1 \\ 0&0 \end{array}  &
\begin{array}{cc} 0&0 \\ 0&1  \end{array} \\ \hline
\begin{array}{cc} 0&0 \\ 0&0 \end{array}  &
\begin{array}{cc} 0&0 \\ 0&0 \end{array}
\end{array} \right).\label{r1example}\end{aligned}$$

The *total measurement array* for a measurement $\{ {\mathbf{R}}_r
\}$ is given by $${\mathbf{M}}= \sum_r {\mathbf{R}}_r.$$

From Eq. (\[constraint2\]), ${\mathbf{M}}$ satisfies $$\label{dynamic_eqn}
{\mathbf{M}} \cdot {\mathbf{P}} = 1\ \ \forall {\mathbf{P}} \in \mathcal{P}.$$

Now consider the total measurement array corresponding to a basic measurement. It has a simple form, which can be described iteratively. First, since each ${\mathbf{a}}$ corresponds to a specific $r$, the component $M({\mathbf{a}}|{\mathbf{x}})$ is equal to 1 if and only if it is possible for the pair $\{ {\mathbf{a}}, {\mathbf{x}} \}$ to be realized. The probability of $\{ {\mathbf{a}}, {\mathbf{x}} \}$ being realized is given by $$\label{eqn:basic_meas_probs}
\mathrm{Pr}({\mathbf{a}}, {\mathbf{x}}) = M({\mathbf{a}}|{\mathbf{x}})\times
P({\mathbf{a}}|{\mathbf{x}}).$$ Now, for a single system, $M(a|x)=\delta_{xi}$ for some $i$. For a multi-partite system composed of $n$ subsystems, there must exist some $k$, and some $i$, such that the first step in the basic measurement is to perform the fiducial measurement $x_k=i$ on the $k$th subsystem. Let ${\mathbf{\hat{x}}}_{ k }$ represent a sequence of measurements and ${\mathbf{\hat{a}}}_{ k }$ a sequence of outcomes on 