the degree of non-locality of a quantum box formula_44 can also provide lower bounds on the hilbert space dimension of the local systems accessible to alice and bob. this problem is equivalent to deciding the existence of a matrix with low completely positive semidefinite rank. finding lower bounds on the hilbert space dimension based on statistics happens to be a hard task, and current general methods only provide very low estimates. however, a bell scenario with five inputs and three outputs suffices to provide arbitrarily high lower bounds on the underlying hilbert space dimension. quantum communication protocols which assume a knowledge of the local dimension of alice and bob’s systems, but otherwise do not make claims on the mathematical description of the preparation and measuring devices involved are termed semi-device independent protocols. currently, there exist semi-device independent protocols for quantum key distribution and randomness expansion.