1/\sqrt{2} (\sigma_x + \sigma_z)$. On two qubits there is a Bell measurement. By the definition of the fiducial measurements, it must be possible to derive the measurement probabilities for all such measurements from the $P({\mathbf{a}}|{\mathbf{x}})$.

In fact, by considering mixtures of states, it can be shown that the probability $\mathrm{Pr}(r)$ of obtaining a particular outcome $r$ in any measurement must be a linear function of the fiducial measurement probabilities [@lucien; @barrett]. We can therefore associate an *effect* ${\mathbf{R}}_r$ with each measurement outcome, where ${\mathbf{R}}_r$ is an array with components $R_r({\mathbf{a}}|{\mathbf{x}})$, such that $$\mathrm{Pr}(r) = {\mathbf{R_r}} \cdot {\mathbf{P}} \equiv \sum_{{\mathbf{a x}}}
R_r({\mathbf{a}}|{\mathbf{x}}) P({\mathbf{a}}|{\mathbf{x}}).$$ A measurement with various possible outcomes is associated with a set $\{ {\mathbf{R}}_r  \}$. For example, consider again a qubit in quantum theory, with fiducial measurements chosen to be $\sigma_x,
\sigma_y, \sigma_z$. The measurement corresponding to $\sigma_{45^{\circ}}$ is represented by the $1$-dimensional arrays $$\label{eqn:example_meas}
{\mathbf{R}}_{+1} =\left(\begin{array}{r} 2^{-\frac{3}{2}} \\ - 2^{-\frac{3}{2}} \\ \hline 2^{-1} \\
2^{-1} \\ \hline 2^{-\frac{3}{2}} \\ - 2^{-\frac{3}{2}}
\end{array} \right),\ \ \ {\mathbf{R}}_{-1} =\left(\begin{array}{r} -2^{-\frac{3}{2}} \\ 2^{-\frac{3}{2}} \\ \hline 2^{-1} \\
2^{-1} \\ \hline -2^{-\frac{3}{2}} \\  2^{-\frac{3}{2}}
\end{array} \right)$$ Note that the array ${\mathbf{R}}_r$ associated with a measurement outcome is not in general unique, since 