Creation of superposition of unknown quantum states

The superposition principle is one of the landmarks of quantum mechanics. The importance of quantum superpositions provokes questions about the limitations that quantum mechanics itself imposes on the possibility of their generation. In this work we systematically study the problem of creation of superpositions of unknown quantum states. First, we prove a no-go theorem that forbids the existence of a universal probabilistic quantum protocol producing a superposition of two unknown quantum states. Secondly, we provide an explicit probabilistic protocol generating a superposition of two unknown states, each having a fixed overlap with the known referential pure state. The protocol is proven to be unique and optimal. Moreover, it can be implemented on arbitrary Hilbert spaces. In the context of quantum optics it can be used to efficiently generate highly nonclassical or nongaussian states.