  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.
