  For many classically chaotic systems, it is believed that in the
semiclassical limit, the matrix elements of smooth observables approach the
phase space average of the observable. In the approach to the limit the matrix
elements can fluctuate around this average. Here we study the variance of these
fluctuations, for the quantum cat map on $\mathbb{T}^4$. We show that for
certain maps and observables, the variance has a different rate of decay, than
is expected for generic chaotic systems.
