  A measure of total correlations cannot increase under deterministic local
operations. We show that, for any number of systems, this condition alone does
not guarantee the existence of maximally correlated states. Namely, there is no
state that simultaneously maximizes all the measures satisfying it. If, in
addition, the measures do not increase with probability unity under local
measurements, then such states exist for two systems. They are the maximally
entangled states. For a larger number of systems, it depends on their Hilbert
space dimensions.
