On positive operator-valued measures generated by a family of one-dimensional projectors

We study positive operator-valued measures generated by projections on one-dimensional subspaces. A special attention is paid to the case in which subspaces are spanned by vectors forming a Riesz basis. It is shown that the measurement fulfilled by such measure is informationally complete for quantum states being a convex hull of projections on subspaces spanned by the system of biorthogonal vectors. We also discuss the properties of different quantum channels associated with a discrete measurement. Finally, we show that our measurement allows to introduce a quantum instrument taking values in the set of two points.