${\mathbf{T}}$ is linear. Thus a transformation can be represented by an array such that $$\label{dynamicsequation}
P'({\mathbf{a}}'|{\mathbf{x}}') = \sum_{{\mathbf{a x}}} T({\mathbf{a}}'|{\mathbf{x}}',
{\mathbf{a}}|{\mathbf{x}}) P({\mathbf{a}}|{\mathbf{x}}).$$ For each type of system, an operational model should specify a set of physically possible transformations. A valid transformation should satisfy $${\mathbf{T}}({\mathbf{P}}) \in \mathcal{P} \ \forall {\mathbf{P}} \in
\mathcal{P}.$$ There are other consistency conditions that the sets of allowed states, measurements and transformations should satisfy, which we will not go into in detail. For example, if a transformation is followed by a fiducial measurement, then this process taken as a whole should correspond to a valid measurement on the initial state.

Box World
=========

Box world is one particular operational model, which has a natural definition in terms of the framework defined above, and which it is interesting to compare with the quantum and classical theories. Box world is defined as follows. Any ${\mathbf{P}}$ satisfying:

1.  Positivity: $P({\mathbf{a}}|{\mathbf{x}})\geq 0$

2.  Normalisation: $|{\mathbf{P}}| = \sum_{{\mathbf{a}}} P({\mathbf{a}}|{\mathbf{0}}) =1 $

3.  No-signalling: $\sum_{a_n} P({\mathbf{a}}|{\mathbf{x}})$ is independent of $x_n$

is an allowed state.[^1]

Two subtleties: first, the normalisation condition is only stated for the measurement choice ${\mathbf{x}}={\mathbf{0}}$. But the no-signalling conditions are then sufficient to ensure normalisation for all measurement choices since 