Measurements and consistent families of quantum channels

This article introduces the notion of consistent families (Λ ⁽ⁿ⁾) _n<1 of quantum channels. These families correspond to simultaneous observation of different copies of a given quantum system. Here, we are primarily interested in the analysis of measurements connected with them. As usual, the measurement of a quantum system requires the construction of a classical dilation of the corresponding quantum channel. In our case, the quantum systems represented by (Λ ⁽ⁿ⁾) _n<1 are supposed to interact through the measurement instrument only. That is, we construct a classical probability space which allows to have a common dilation for all the Λ ⁽ⁿ⁾'s. Doing this, we introduce and solve a quantum version of the moment problem.