$$R({\mathbf{a}}|{\mathbf{x}}) = R_1(a_1|x_1)R_2(a_2|x_2)\ldots R_n(a_n|x_n),$$ where $R_i(a_i|x_i)$ is a valid effect on the $i$th system. An effect is separable if it can be written as a sum of product effects, otherwise it is entangled. (Note that the sum here really is just a sum - not a convex combination, as in the definitions applicable to states.) Any array ${\mathbf{R}}$ with one entry $\in
(0,1]$ and the rest zero represents a product effect. Hence Theorem \[positivity\] yields

\[noenteffect\] There are no entangled effects in box world.

Section \[consequences\] shows that Theorem \[positivity\] also prevents entanglement swapping and teleportation in box world.

A certain class of measurements is particularly simple and will play a special role in what follows.

A measurement is *basic* if it can be implemented by a sequence of fiducial measurements on individual subsystems, where later measurement choices may depend (deterministically) on earlier outcomes, and the final measurement outcome $r$ is given by a deterministic function of the fiducial measurement outcomes ${\mathbf{a}}$.

An example of a basic measurement for two subsystems is given in Figure \[fig:basic\_measurement\], where fiducial measurement $x_1=0$ is performed on the first subsystem and then measurement $x_2=a_1$ is performed on the second, and the final measurement outcome 