On extension of the family of projections to positive operator-valued measure

The problem of constructing a measure on a discrete set $X$ taking values in a positive cone of bounded operators in a Hilbert space is considered. It is assumed that a projectionvalued function defined on a subset of $X_0$ of the original set $X$ is initially given. The aim of the study is to find such a scalar measure $\mu$ on the set $X$ and the continuation of a projector-valued function from $X_0$ to $X$, which results in an operator-valued measure having a projector-valued density relative to $\mu$. In general, the problem is solved for $|X| = 4$ and $|X_0| = 2$. As an example, we consider a function on $X_0$ that takes values in a set of projections on coherent states. For this case, the question of the information completeness of the measurement determined by the constructed measure is investigated. In other words, is it possible to reconstruct a quantum state (a positive unit trace operator) from the values of the matrix trace from the product of a measure with a quantum state. It is shown that for the constructed measure it is possible to restore the quantum state only if it is a projection. A restriction on the probability distribution is also found, at which it can be obtained as a result of measuring a certain quantum state.