  In this work, we investigate the possibility of compressing a quantum system
to one of smaller dimension in a way that preserves the measurement statistics
of a given set of observables. In this process, we allow for an arbitrary
amount of classical side information. We find that the latter can be bounded,
which implies that the minimal compression dimension is stable in the sense
that it cannot be decreased by allowing for small errors. Various bounds on the
minimal compression dimension are proven and an SDP-based algorithm for its
computation is provided. The results are based on two independent approaches:
an operator algebraic method using a fixed point result by Arveson and an
algebro-geometric method that relies on irreducible polynomials and B\'ezout's
theorem. The latter approach allows lifting the results from the single copy
level to the case of multiple copies and from completely positive to merely
positive maps.
