is defined so that any set $\{{\mathbf{R}}_r\}$ satisfying conditions (\[constraint1\]) and (\[constraint2\]) above corresponds to a physically possible measurement. In the following, we explore what kinds of measurement this actually allows, and the consequences for information processing. Intuitively, the fact that conditions (\[constraint1\]) and (\[constraint2\]) must be satisfied means that there is a tradeoff between states and measurements. If there is a larger space of states, then there is a smaller set of measurements that are compatible with those states. The space of states in box world is in an obvious sense maximal, so one would expect the possibilities for measurement to be less interesting than in, say, quantum theory. This is indeed the case.

Measurements in box world
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The following property of measurements in box world is proven in Appendix D of [@barrett].

\[positivity\] All effects in box world can be represented using only positive arrays where each component satisfies $$0 \leq R({\mathbf{a}} | {\mathbf{x}}) \leq 1.$$

From hereon, assume that effects are indeed represented this way.

In the case of multi-partite systems, we have already defined product states, and distinguished separable and entangled states. Similar definitions apply to effects. A multi-partite effect is a *product effect* if 