example, for two systems, each with a binary measurement choice $x$, and binary outcomes $a$: $${\mathbf{P}} = \left( \begin{array}{cc|cc} P(00|00) & P(01|00) & P(00|01) & P(01|01) \\
P(10|00) & P(11|00) & P(10|01) & P(11|01) \\ \hline
P(00|10) & P(01|10) & P(00|11) & P(01|11) \\
P(10|10) & P(11|10) & P(10|11) & P(11|11) \end{array} \right)$$ Conditions of positivity and normalisation apply as before.

Finally, all multi-partite states must satisfy the *no-signalling conditions*:

> $\sum_{a_n} P({\mathbf{a}}|{\mathbf{x}})$ is independent of $x_n$ for all $n$.

These ensure that separated parties cannot send messages to one another simply by making measurements on their subsystems. Arguably, if the no-signalling conditions do not hold, then we had no right to be speaking of separate subsystems in the first place.

A multi-partite state is a *product state* if ${\mathbf{P}}$ satisfies $$P({\mathbf{a}}|{\mathbf{x}}) = P_1(a_1|x_1)P_2(a_2|x_2)\ldots P_n(a_n|x_n),$$ where $P_i(a_i|x_i)$ is a valid state of the $i$th system. A state is *separable* if it can be written as a convex combination of product states, otherwise it is *entangled*.

Measurements {#sec:measurement}
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In general, the fiducial measurements will not be the only measurements that one can perform on a system. For example, on a single qubit, there is a measurement corresponding to $\sigma_{45^{\circ}} = 