they imply $$\sum_{{\mathbf{a}}} P({\mathbf{a}}|{\mathbf{x}})=\sum_{{\mathbf{a}}}
P({\mathbf{a}}|{\mathbf{x'}}).$$ Second, although the no-signalling conditions refer only to fiducial measurements, it can be shown that they are sufficient to prevent signalling using any kind of measurement [@barrett].

Box world permits many states that do not have counterparts in quantum theory. An interesting example is this bipartite state: $$\label{PS-state_eqn}
P_{PR} (a_1 a_2 | x_1 x_2) = \left\{
\begin{array}{ccl} \frac{1}{2} & : & a_1
+ a_2 = x_1 x_2\; \textrm{(mod 2)} \\
0& :  & {\mathrm{otherwise}}\end{array} \right. ,$$ where $x_1, x_2, a_1, a_2 \in\{0,1\}$. The correlations generated by performing fiducial measurements on this state are nonlocal, meaning that they violate the Clauser-Horne-Shimony-Holt (CHSH) inequality [@CHSH]. In fact, they are *more nonlocal* than is possible in quantum theory, because they return a value of 4 for the CHSH expression. Tsirelson’s theorem [@tsirelson] shows that quantum correlations always return a value $\leq 2\sqrt{2}$, and CHSH showed that local correlations always return a value $\leq 2$. These superquantum nonlocal correlations have appeared in the literature before [@popescu; @boxes], and they are sometimes referred to as a Popescu-Rohrlich (PR) box. Thus we refer to this state as the PR box state.

What are the allowed measurements in box world? The model 