Enriched Presheaf Model of Quantum FPC

Selinger gave a superoperator model of a first-order quantum programming language and proved that it is fully definable and hence fully abstract. This paper proposes an extension of the superoperator model to higher-order programs based on modules over superoperators or, equivalently, enriched presheaves over the category of superoperators. The enriched presheaf category can be easily proved to be a model of intuitionistic linear logic with cofree exponential, from which one can cave out a model of classical linear logic by a kind of bi-orthogonality construction. Although the structures of an enriched presheaf category are usually rather complex, a morphism in the classical model can be expressed simply as a matrix of completely positive maps. The model inherits many desirable properties from the superoperator model. A conceptually interesting property is that our model has only a state whose "total probability" is bounded by 1, i.e. does not have a state where true and false each occur with probability 2/3. Another convenient property inherited from the superoperator model is a $\omega$CPO-enrichment. Remarkably, our model has a sufficient structure to interpret arbitrary recursive types by the standard domain theoretic technique. We introduce Quantum FPC, a quantum $\lambda$-calculus with recursive types, and prove that our model is a fully abstract model of Quantum FPC.