Tr} (\hat{\Gamma}^2) \rangle &=& \sum_{n=-\infty}^\infty \sum_{n_2,n_4 =0}^\infty v^{2(n_2+n_4)} v^{2|n|} \langle{A}(n)\rangle\nonumber \\
&=& \frac{1}{(1-v^2)^2} \sum_{n=-\infty}^\infty  v^{2|n|} \langle{A}(n)\rangle \label{TrG}\\ 
& = & \frac{1+v^2}{1-v^2} \langle{\rm Tr} (\hat{K}^2)\rangle,\label{TrG2}\end{aligned}$$ as previously reported in (\[GvsK\]). Note that this condition has been derived based on neglecting interfering contributions which are wiped out by appropriate averaging. We have not made any further assumptions about the underlying classical dynamics and the relations (\[GvsK\]) – (\[varGKWig\]) are universal, independent of whether the system dynamics is regular, chaotic or mixed. In particular, the variance of $\hat{\Gamma}$ always exceeds that of $\hat{K}$, only approaching the same value in the limit of strong damping, $v \to 0$.

We note that relation between wave fluctuations (as a function of energy or frequency) and classical decay rates is a well established concept and has been discussed in the context of fluctuations in scattering cross sections [@BS88; @DSF90] or conductance fluctuations for transport through open chaotic cavities [@JBS90], see also [@Bar95]. The relevant classical quantity is here the classical escape rate $\gamma_{\rm esc}$, which in our setting corresponds to the rate of dissipation $\gamma_{esc} =  -\log v$. A generalised treatment involving the full spectrum of the FP operator is given in [@Agam00]. Our result highlights the 