In this paper, we extend past work done on the application of the mathematics of category theory to quantum information science. Specifically, we present a realization of a dagger-compact category that can model finite-dimensional quantum systems and explicitly allows for the interaction of systems of arbitrary, possibly unequal, dimensions. Hence our framework can handle generic tensor network states, including matrix product states. Our categorical model subsumes the traditional quantum circuit model while remaining directly and easily applicable to problems stated in the language of quantum information science. The circuit diagrams themselves now become morphisms in a category, making quantum circuits a special case of a much more general mathematical framework. We introduce the key algebraic properties of our tensor calculus diagrammatically and show how they can be applied to solve problems in the field of quantum information.

25 pages, 20 figures. Published version