We address the question about the fundamental limitations on the possibility of transmitting messages imposed by the nature of the physical carrier of information. For the presentation of rigorous results for quantum communication channels (including their throughputs), it is necessary to introduce a quantum theory into the statistical structure, which is the basis for the quantum information theory. The minimum requirements to the description of a quantum communication channel are formulated. We use the Kuryshkin-Wodkiewicz model of quantum measurements and construct the expressions for operators dispersions and formulate the problem of finding minimum-dispersion states. The case of an electron in a hydrogen-like atom is considered. © Springer Nature Switzerland AG 2019.