In this paper, we present a linear and reversible language with inductive and coinductive types, together with a Curry-Howard correspondence with the positive fragment of the logic µMALL: linear logic extended with least fixed points allowing inductive statements. Linear, reversible computation makes an important sub-class of quantum computation without measurement. In the latter, the notion of purely quantum recursive type is not yet well understood. Moreover, models for reasoning about quantum algorithms only provide complex types for classical datatypes: there are usually no types for purely quantum objects beside tensors of quantum bits. This work is a first step towards understanding purely quantum recursive types.