  We show that probabilities of results of all possible measurements performing
on a quantum system depend on the system's state only through its density
matrix. Therefore all experimentally available information about the state
contains in the density matrix. In this study, we do not postulate that
measurements obey some given formalism (such as observables, positive-operator
valued measures, etc.), and do not use Born rule. The process of measurement is
considered in a fully operational manner---as an interaction of a measured
system with some black-box apparatus. The key point of our approach is the
proof that, for improper mixtures, the expected value of any measurement
depends linearly on the reduced density function. Such a proof is achieved by
considering appropriate thought experiments. We demonstrate that Born rule can
be derived as a natural consequence of our results.
