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Why state of quantum system is fully defined by density matrix

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.