During my Master thesis, I have worked in the group of Prof. Mauro D’Ariano in Pavia. The topic of the thesis was the universality of the CNOT gate for quantum computation in real-quantum-theory.
The CNOT gate is a logical bipartite gate which is known to be universal in standard quantum theory. Universal means that using CNOT gates, together with local gates, every multipartite gate can be obtained. Real-quantum-theory is an instance of generalized probabilistic theories which is different from quantum mechanics in that it uses real Hilbert spaces instead of complex ones.
What we show in [Belenchia, D’Ariano, Perinotti, EPL 104, 20006 (2013)] is that the CNOT gate is universal also in real quantum theory. However, since now unitary gates are replaced by orthogonal ones, and the orthogonal group has a connected component that does not include the identity, an ancillary re-bit is needed to have full universality. Our proof can be applied also to the standard quantum case and is more direct than the other proofs in the extant literature.