Braided Categorical Quantum Mechanics I

TL;DR

This paper extends categorical quantum mechanics to braided systems, providing a high-level abstract framework for quantum information in topological quantum computation, including new axioms and graphical calculus.

Contribution

It introduces a categorical framework for braided quantum systems, generalizing existing models and proposing axioms for topological quantum computation.

Findings

Generalized quantum information flow to braided systems

Developed a graphical dagger calculus for braided categories

Proposed new axioms for braided quantum information

Abstract

This is the first paper in a series where we generalize the Categorical Quantum Mechanics program (due to Abramsky, Coecke, et al) to braided systems. In our view a uniform description of quantum information for braided systems has not yet emerged. The picture is complicated by a diversity of examples that lacks a unifying framework for proving theorems and discovering new protocols. We use category theory to construct a high-level language that abstracts the quantum mechanical properties of braided systems. We exploit this framework to propose an axiomatic description of braided quantum information intended for topological quantum computation. In this installment we first generalize the primordial Abramsky-Coecke "quantum information flow" paradigm from compact closed categories to right-rigid strict monoidal categories. We then study dagger structures for rigid and/or braided categories and formulate a graphical dagger calculus. We then propose two generalizations of strongly compact closed categories. Finally we study partial traces in the context of dagger categories.