Constructing symmetric informationally complete positive-operator-valued measures in Bloch space

Abstract

Here we study the construction of symmetric informationally complete positive-operator-valued measures (SIC-POVMs) in the Bloch space. In this space an SIC-POVM corresponds to a regular simplex, that is a set of real equiangular, unitary vectors. Since the Bloch space also contains vectors which do not describe quantum states, it is necessary to add an extra condition to enforce the members of the simplex to describe pure quantum states. We show that in the case of a three-dimensional quantum systems it is possible to find such a simplex in an analytical way. The solution turns out to be unitarily equivalent to a Weyl–Heisenberg covariant SIC-POVM.