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. Firstly, 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. Secondly, for arbitrary optimal (N;M)-POVMs conditions on their existence are derived, which exhibit a close connection to the existence of sets of (M - 1)N isospectral, traceless, orthonormal, hermitian operators with particular common spectra. The specific form of the two possible types of spectra depends on whether M is smaller than the dimension of the quantum system under consideration or not. For the special case of M = 2 and dimensions, which are powers of two, these sets of operators always exist and can be expressed in terms of the Clifford algebra of tensor products of Pauli operators.