Measuring the local efficiency of symmetric informationally complete positive operator-valued measures in two-level systems with the density of quantum states

We study the local efficiency of symmetric informationally complete positive operator-valued measures (SIC-POVMs) from an information geometrical perspective. The local efficiency of a SIC-POVM is defined as the relative magnitude of the density of distinguishable states acquired via the measurement over the intrinsic density of quantum states. The two densities measure the local distinguishability of quantum states and its corresponding quantum resources, respectively. Strikingly, the efficiency is intrinsic in the sense of invariance under reparametrization. Specific to two-level systems, the efficiency is highly anisotropic and has a tetrahedral symmetry. Especially for a shell of the Bloch sphere, states in directions parallel (antiparallel) to the measurement basis have minimal (maximal) efficiency. Furthermore, the optimality of SIC-POVM versus other equal-length measurements is shown.