Neumark Operators and Sharp Reconstructions: the Finite Dimensional Case

A commutative positive operator valued POV measure F with real spectrum is characterized by the existence of a projection valued measure E (the sharp reconstruction of F) with real spectrum such that F can be interpreted as a randomization of E. This paper focuses on the relationships between this characterization of commutative POV measures and Neumark’s extension theorem. In particular, we show that in the finite dimensional case there exists a relation between the Neumark operator corresponding to the extension of F and the sharp reconstruction of F. The relevance of this result to the theory of nonideal quantum measurement and to the definition of unsharpness is analyzed.