  We performed a high resolution study of $0^{+}$ states in $^{134}$Ba using
the $^{136}$Ba($p,t$) two-neutron transfer reaction. Our experiment shows a
significant portion of the $L = 0$ pair-transfer strength concentrated at
excited $0^+$ levels in $^{134}$Ba. Potential implications in the context of
$^{136}$Xe $\to$ $^{136}$Ba neutrinoless double beta decay matrix element
calculations are briefly discussed.
