  We consider probabilistic theories in which the most elementary system, a
two-dimensional system, contains one bit of information. The bit is assumed to
be contained in any complete set of mutually complementary measurements. The
requirement of invariance of the information under a continuous change of the
set of mutually complementary measurements uniquely singles out a measure of
information, which is quadratic in probabilities. The assumption which gives
the same scaling of the number of degrees of freedom with the dimension as in
quantum theory follows essentially from the assumption that all physical states
of a higher dimensional system are those and only those from which one can
post-select physical states of two-dimensional systems. The requirement that no
more than one bit of information (as quantified by the quadratic measure) is
contained in all possible post-selected two-dimensional systems is equivalent
to the positivity of density operator in quantum theory.
