  In this short note, we propose a generalization of Atiyah type TQFTs from
pure states to mixed states in the sense that the Hilbert space of pure states
associated to a space manifold is replaced by a quantum coherent space related
to density matrices. Atiyah type TQFT is a symmetric monoidal functor from the
Bord category of manifolds to the category Vec of finite dimensional vector
spaces. In this paper, we define mixed-state TQFTs by replacing the target
category Vec by QCS--the category of quantum coherent spaces, then a
mixed-state TQFT is simply a symmetric monoidal functor from Bord to QCS. We
also discuss how to construct interesting examples from subsystem quantum error
correction codes beyond the trivial ones--all Atiyah type TQFTs.
