Mixed-state TQFTs

TL;DR

This paper introduces mixed-state topological quantum field theories (TQFTs) by extending pure-state TQFTs to include density matrices, utilizing quantum coherent spaces, and explores their construction via quantum error correction codes.

Contribution

It generalizes Atiyah type TQFTs from pure to mixed states using quantum coherent spaces and proposes methods for constructing examples from quantum error correction codes.

Findings

Defined mixed-state TQFTs as functors to quantum coherent spaces

Extended the framework of Atiyah type TQFTs to mixed states

Discussed construction of examples using subsystem quantum error correction codes

Abstract

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.