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Abstract Consider a quantum cat mapMassociated with a matrix $$A\in {{\,\textrm{Sp}\,}}(2n,{\mathbb {Z}})$$ $A\in \text{Sp}(2n,Z)$, which is a common toy model in quantum chaos. We show that the mass of eigenfunctions of Mon 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 ofAof largest absolute value and (2) the characteristic polynomial ofAis irreducible over the rationals. This is similar to previous work (Dyatlov and Jin in Acta Math 220(2):297–339, 2018; Dyatlov et al. in J Am Math Soc 35(2):361–465, 2022) on negatively curved surfaces and (Schwartz in The full delocalization of eigenstates for the quantized cat map, 2021) on quantum cat maps with$$n=1$$ $n=1$, 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.