questions that can be asked in information theory involve more than one system. So how should we describe the state of a multi-partite system in a general operational model? Consider $n$ systems, $A_1,\ldots,A_n$, with a set of fiducial measurements for each. If a fiducial measurement $x_1$ is performed on $A_1$, $x_2$ on $A_2$, and so on, then at the least, the joint state of $A_1,\ldots,A_n$ should determine a joint probability for each combination of outcomes. We assume something further: a specification of the joint probability of each combination of outcomes for each possible combination of fiducial measurements is sufficient to determine completely the joint state. Note that this property does indeed hold in both quantum theory and classical probability theory. But it is not trivial. For example, in an alternative quantum theory, formulated using real instead of complex Hilbert spaces, the property does not hold [@realqm1; @realqm2; @realqm3].

It is convenient to write a multi-partite state of $n$ systems in the form of an $n$-dimensional array, with entries $P({\mathbf{a}}|{\mathbf{x}})$, where ${\mathbf{x}}=(x_1, \ldots, x_n)$ specifies a fiducial measurement for each subsystem, ${\mathbf{a}}$ is a list of the corresponding measurement outcomes, and $P({\mathbf{a}}|{\mathbf{x}})$ is the probability of getting ${\mathbf{a}}$ given ${\mathbf{x}}$. For 