  A quantitative description of the change in ground-state neutron occupancies
between $^{136}$Xe and $^{136}$Ba, the initial and final state in the
neutrinoless double-$\beta$ decay of $^{136}$Xe, has been extracted from
precision measurements of the cross sections of single-neutron adding and
-removing reactions. Comparisons are made to recent theoretical calculations of
the same properties using various nuclear-structure models. These are the same
calculations used to determine the magnitude of the nuclear matrix elements for
the process, which at present disagree with each other by factors of 2 or 3.
The experimental neutron occupancies show some disagreement with the
theoretical calculations.
