Quantum mechanics with non-negative distribution and measurements

The issues of interaction between the measuring device and the micro system have not yet been completely resolved. Moreover, if at the dawn of quantum mechanics attention was paid to the influence of the device on the state of the system, then later (D.V. Blokhintsev) interest in the inverse problem (and, in essence, more traditional) - on the influence of the micro-object on the state of the macro-system increased. Measuring instruments consist of a number of interconnected links. The quantum theory of measurements asserts that the first cascades can be quantum, but the last should be classical (in the current terminology - an analyzer and a detector). This paper discusses the integration of such an approach to measuring instruments in the formalism of quantum mechanics. Test (auxiliary) functions in quantum mechanics with nonnegative distribution functions (quantum mechanics Kuryshkin-Wodkevich - QDF), in fact, can and should be associated with the measurement process and measuring instruments. Our thesis is that these functions should be related to the measuring instrument used and describe its interaction with the system. It turns out that exactly the non-Hermitian character of the Hamiltonians in QDF expresses this connection. Thus, the QDF, which has long been positioned as an alternative and exotic quantum theory, is increasingly taking shape as a quantum theory of measurements. Only a short time ago, in our works the meaning of this formalism as the theory of quantum measurements was clarified. The analogies of auxiliary functions in the QDF theory with wave functions in the examples of D. Blokhintsev measuring instruments mentioned in this paper, are promising areas of research which, in the framework of the study of quantum entanglement, can give a significant impetus to the development of quantum cryptography. This work, in this sense, should become a turning point in this reversal of the QDF to new applications.