  A unital $C^*$-algebra is called $N$-subhomogeneous if its irreducible
representations are finite dimensional with dimension at most $N$. We extend
this notion to operator systems, replacing irreducible representations by
boundary representations. This is done by considering the matrix state space
associated with an operator system and identifying the boundary representations
as absolute matrix extreme points. We show that two $N$-subhomogeneous operator
systems are completely order equivalent if and only if they are $N$-order
equivalent. Moreover, we show that a unital $N$-positive map into a finite
dimensional $N$-subhomogeneous operator system is completely positive. We apply
these tools to classify pairs of $q$-commuting unitaries up to $*$-isomorphism.
Similar results are obtained for operator systems related to higher dimensional
non-commutative tori.
