and $a=a_x$) with at least one zero entry in each block (where $b=b_{xy}$). Similarly, each column of blocks contains a column of components with at least one zero entry in each block.

But case \[nowiring2\] is impossible. The example at the end of this subsection may be helpful in understanding the various stages of the following.

To derive a contradiction, suppose that case \[nowiring2\] holds. Consider a value $x^*$, and the corresponding $a_{x^*}$ and $b_{x^*y}$ for which $M(a_{x^*} b_{x^*y}  |x^* y)=0 \; \forall y$. Consider a product state ${\mathbf{P}}$ such that $P(a_{x^*} b_{x^*y}  |
x^*  y)=1 \; \forall y$, and another product state ${\mathbf{P}}'$ obtained from ${\mathbf{P}}$ by swapping the outputs $a=a_{x^*}$ with $a=a'$ when $x=x^*$ (a local relabelling of the outputs).

From Equation (\[dynamic\_eqn\]) and the definition of a subnormalised total measurement array, it is clear that $${\mathbf{M}}.({\mathbf{P}} - {\mathbf{P}}') = \sum_{a,b,x,y} M(ab|xy) (P(ab|xy)
- P'(ab|xy) ) = 0. \label{diff_eqn}$$ This implies that $$\sum_{y} (M(a_{x^*} b_{x^* y}  |x^* y)- M(a' b_{x^* y}  |x^* y)) =
0,$$ hence $M(a' b_{x^* y}  |x^* y)=0 \; \forall y$. By modifying the chosen parameters $x^*$ and $a'$, this generalises to the result that $M(a b_{xy}  |x y)=0 \;  \forall \, a, x, y$. 