Conditions for the existence of positive operator valued measures

Abstract: Sufficient and necessary conditions are presented for the existence of $(N,M)$-positive operator valued measures ($(N,M)$-POVMs) valid for arbitrary-dimensional quantum systems. A sufficient condition for the existence of $(N,M)$-POVMs is presented. It yields a simple relation determining an upper bound on the continuous parameter of an arbitrary $(N,M)$-POVM, below which all its POVM elements are guaranteed to be positive semidefinite. Necessary conditions are derived for the existence of optimal $(N,M)$-POVMs. One of these necessary conditions exhibits a close connection between the existence of optimal informationally complete $(N,M)$-POVMs and the existence of isospectral, traceless, orthonormal, hermitian operator bases in cases, in which the parameter $M$ exceeds the dimension of the quantum system under consideration. Another necessary condition is derived for optimal $(N,M)$-POVMs, whose parameter $M$ is less than the dimension of the quantum system. It is shown that in these latter cases all POVM elements necessarily are projection operators of equal rank. This significantly constrains the possible parameters for constructing optimal $(N,M)$-POVMs. For the special case of $M=2$ a necessary and sufficient condition for the existence of optimal $(N,2)$-POVMs is presented.