${\mathbf{M}}$ is a convex combination of total measurement arrays for basic measurements.

The first step in the proof of Theorem \[wiringtheorem\] is to note that the converse also holds. That is, given ${\mathbf{M}}$ and $\{{\mathbf{R}}_r\}$, if ${\mathbf{M}}$ can be written as a convex combination of total measurement arrays for basic measurements, then there is a convex combination of basic measurements with the same total measurement array ${\mathbf{M}}$, and with outcomes corresponding to $\{{\mathbf{R}}_r\}$. To see this, suppose that ${\mathbf{M}}=\sum_i q_i
{\mathbf{M}}_i$, where $0\leq q_i\leq 1$, $\sum_i q_i = 1$ and ${\mathbf{M}}_i$ is the total measurement array for a basic measurement. Construct an appropriate convex combination of basic measurements as follows. With probability $q_i$, let the order of fiducial measurements to perform be that dictated by ${\mathbf{M}}_i$. The probability of measuring ${\mathbf{x}}$ and obtaining outputs ${\mathbf{a}}$ is given by Equation (\[eqn:basic\_meas\_probs\]) (which continues to hold for convex combinations of basic measurements). In the case that $\{ {\mathbf{a}},{\mathbf{x}} \}$ is realized, announce result $r$ with probability $\mathrm{Pr}(r|{\mathbf{a}}, {\mathbf{x}} ) =
R_r({\mathbf{a}}|{\mathbf{x}})/M({\mathbf{a}}|{\mathbf{x}})$ (where if $\{
{\mathbf{a}},{\mathbf{x}} \}$ is realized, the right hand side must be $\geq
0$ and $\leq 1$). The overall probability of obtaining outcome $r$ is now given by $$\begin{aligned}
\mathrm{Pr}(r) 