**Title**: Environment structures for dagger compact closed categories

**Abstract**: In this talk, I will introduce environment structure for a dagger compact closed category and purification for an environment structure. I will provide an example environment structure in the category of Hilbert spaces and completely positive maps. I will also prove the following theorem:

Suppose C is a dagger compact closed category and C_{pure} is dagger compact closed subcategory of C. (C,C_{pure}) has an environment structure with purification. Then, there exists a dagger functor from a subcategory of C_{pure} – whose objects are endomorphism algebras and completely positive maps – to C. Moreover, the functor is left invertible up to equivalence.

Environment structures are used in the construction and verification of protocols in the category of finite-dimensional Hilbert spaces and completely positive maps.