  We investigate compressibility of the dimension of positive semidefinite
matrices while approximately preserving their pairwise inner products. This can
either be regarded as compression of positive semidefinite factorizations of
nonnegative matrices or (if the matrices are subject to additional
normalization constraints) as compression of quantum models. We derive both
lower and upper bounds on compressibility. Applications are broad and range
from the statistical analysis of experimental data to bounding the one-way
quantum communication complexity of Boolean functions.
