  Consider a quantum cat map $M$ associated to a matrix
$A\in\mathop{\mathrm{Sp}}(2n,\mathbb Z)$, which is a common toy model in
quantum chaos. We show that the mass of eigenfunctions of $M$ on any nonempty
open set in the position-frequency space satisfies a lower bound which is
uniform in the semiclassical limit, under two assumptions: (1) there is a
unique simple eigenvalue of $A$ of largest absolute value and (2) the
characteristic polynomial of $A$ is irreducible over the rationals. This is
similar to previous work [arXiv:1705.05019], [arXiv:1906.08923] on negatively
curved surfaces and [arXiv:2103.06633] on quantum cat maps with $n=1$, but this
paper gives the first results of this type which apply in any dimension. When
condition (2) fails we provide a weaker version of the result and discuss
relations to existing counterexamples. We also obtain corresponding statements
regarding semiclassical measures and damped quantum cat maps.
