Existence of symmetric informationally complete (SIC) measurements in all finite dimensions

Establish whether symmetric informationally complete positive operator-valued measures (SIC-POVMs) exist in every finite Hilbert-space dimension N; specifically, determine for arbitrary N≥2 the existence of a set of N^2 rank‑1 projectors {Πj} satisfying Tr(Πi Πj) = (δi,j N + 1)/(N + 1), thereby resolving the general existence question for SIC measurements.

Background

The text introduces informationally complete (IC) measurements and highlights a special class—symmetric informationally complete (SIC) measurements—characterized by a specific overlap condition, Tr ΠiΠj = (δi,j N + 1)/(N + 1). These POVMs are central to quantum tomography and cryptography because they enable unique reconstruction of quantum states from measurement statistics.

While SIC-POVMs are known to exist in many specific dimensions, the general existence problem across all finite dimensions remains unresolved. The paper explicitly notes this uncertainty, framing it as an open question tied to fundamental aspects of quantum measurement theory and practical applications in quantum information processing.