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.
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