refers to a set of $a$ values (one for each value of $x$). We will use notation such as $x^*$ to refer to a particular $x$-value.

### Single-system measurements

In Ref. [@barrett] it is shown that all single-system measurements in box world are mixtures of fiducial measurements. We include an alternative proof here to illustrate the techniques used in the bi-partite case in a simpler setting.

Consider a (non-zero) subnormalised ${\mathbf{M}}$ associated with a single system characterised by any set of fiducial measurements. One of the following must hold:

1.  It is possible to subtract a (non-zero) subnormalised basic measurement ${\mathbf{M}}_B$ from ${\mathbf{M}}$ to leave a valid subnormalised ${\mathbf{M}}' = {\mathbf{M}}-{\mathbf{M}}_B$. In this case there must be a measurement choice $x^*$ for which $M(a|x^*) >0 \; \forall \, a$, hence it is possible to subtract the subnormalised basic measurement $M_B(a|x) = k \delta_{x,x^*}$ where $k=\min_a (M(a|x^*))$. This generates at least one additional 0 entry in ${\mathbf{M}}'.$ \[yeswiring\]

2.  It is not possible to subtract a (non-zero) subnormalised basic measurement from ${\mathbf{M}}$ without creating a negative component. In this case, there must exist a set $a_x$ such that $M(a_x |x)=0 \; \forall \, x$. \[nowiring\] In terms of the representation of 