of measurements corresponding to the three Pauli operators, $\sigma_x,
\sigma_y$ and $\sigma_z$. Note that the choice of which measurements to use as fiducial measurements is not unique.

A convenient way of writing down the state is as a vector ${\mathbf{P}}$, with components $P(a|x)$, where this is the probability of obtaining outcome $a$ when fiducial measurement $x$ is performed. Obviously, $P(a|x)$ must be positive and normalised such that $\sum_{a} P(a|x) = 1$. For example, for binary valued $a$ and $x$: $${\mathbf{P}} = \left(\begin{array}{c} P(0|0) \\ P(1|0) \\ \hline P(0|1)
\\ P(1|1) \end{array} \right).$$

Within a particular operational model, it is not necessarily the case that any vector ${\mathbf{P}}$ represents an allowed state (that is, a state which can actually be prepared). In quantum theory there is no qubit state that assigns probability 1 to the +1 outcome for all three Pauli measurements. An operational model must specify, for each type of system, a set $\mathcal{P}$ of states which can physically be prepared. We assume that arbitrary probabilistic mixtures of states can be prepared (e.g., by tossing some coins and preparing either state ${\mathbf{P}}$ or state ${\mathbf{Q}}$ depending on the result). Hence $\mathcal{P}$ is convex.

Multi-partite systems
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Most of the interesting 