Symmetric informationally complete measurements (SICs) are elegant, celebrated and broadly useful discrete structures in Hilbert space. We introduce a more sophisticated discrete structure compounded by several SICs. A SIC-compound is defined to be a collection of d^3 vectors in d-dimensional Hilbert space that can be partitioned in two different ways: into d SICs and into d^2 orthonormal bases. While a priori their existence may appear unlikely when d>2, we surprisingly answer it in the positive through an explicit construction for d=4. Remarkably this SIC-compound admits a close relation to mutually unbiased bases, as is revealed through quantum state discrimination. Going beyond fundamental considerations, we leverage these exotic properties to construct a protocol for quantum key distribution and analyze its security under general eavesdropping attacks. We show that SIC-compounds enable secure key generation in the presence of errors that are large enough to prevent the success of the generalisation of the six-state protocol.
