than a wavelength: however, it can be shown using alternative but more involved calculations, not reported here, that these results can also be applied element by element. Finally, any statement involving such traces can be expressed also as a relation between Wigner functions. For example, (\[GvsK\]) can also be expressed in the form, $$\label{varGKWig}
\langle|W_\Gamma(x,p)|^2\rangle 
=\frac{1+v^2}{1-v^2}
\langle|W_K(x,p)|^2\rangle$$ using standard properties of Wigner functions. This relation provides an explicit quantitative statement of the qualitative observations made in Fig. \[fig:kr:k1:a\].

The variance of the diagonal part $\hat{K}$
-------------------------------------------

We consider $\hat{K}$ first and write Eq. (\[K\]) as $$\hat{K} = \sum_{n=0}^\infty v ^{2n}\, \hat{\Gamma}_n = \sum_{n=0}^\infty v ^{2n}\,  \hat{U}^n \hat{\Gamma}_0 \hat{U}^{-n}$$ with $\hat{\Gamma}_n = v^{-2n} \hat{K}_n$ representing the smooth $n$-step correlation function as in Eq. (\[K\]). Using the cyclic permutability of the trace we may write, $$\begin{aligned}
{\rm Tr} (\hat{K}^2) 
&=& \sum_{n,n'=0}^\infty v^{2(n+n')} {\rm Tr} \left[\hat{U}^{n-n'} \hat{\Gamma}_0 \hat{U}^{n'-n} \hat{\Gamma}_0 \right]\\
&=& \sum_{n,n'=0}^\infty v^{2(n+n')} {A}(n-n') ,\end{aligned}$$ where $${A}(n) = {\rm Tr}\left( \hat{\Gamma}_0 \hat{\Gamma}_n\right)$$ is the $n$-step return probability. Note that ${A}(n) = {\rm Tr} \hat{\Gamma}_0 \hat{\Gamma}_n = {\rm Tr} \hat{\Gamma}_n \hat{\Gamma}_0 = {\rm Tr} \hat{\Gamma}_0 \hat{\Gamma}_{-n} = {A}(-n)$. After reordering the double sum, this may be written in the form $$\label{TrK}
{\rm Tr}( \hat{K}^2) 