Enrichment in Categorical Quantum Foundations

Kevin Dunne

Mathieu Huot, Sam Staton

We propose a categorical foundation for the connection between pure and mixed states in quantum information and quantum computation. The foundation is based on distributive monoidal categories. First, we prove that the category of all quantum channels is a canonical completion of the category of pure quantum operations (with ancilla preparations). More precisely, we prove that the category of completely positive trace-preserving maps between finite-dimensional C*-algebras is a canonical completion of the category of finite-dimensional vector spaces and isometries. Second, we extend our result to give a foundation to the topological relationships between quantum channels. We do this by generalizing our categorical foundation to the topologically-enriched setting. In particular, we show that the operator norm topology on quantum channels is the canonical topology induced by the norm topology on isometries.

Peter Nonnenmann

Dan Marsden, Maaike Zwart

Considering resource usage is a powerful insight in the analysis of many phenomena in the sciences. Much of the current research on these resource theories focuses on the analysis of specific resources such quantum entanglement, purity, randomness or asymmetry. However, the mathematical foundations of resource theories are at a much earlier stage, and there has been no satisfactory account of quantitative aspects such as costs, rates or probabilities. We present a categorical foundation for quantitative resource theories, derived from enriched category theory. Our approach is compositional, with rich algebraic structure facilitating calculations. The resulting theory is parameterized, both in the quantities under consideration, for example costs or probabilities, and in the structural features of the resources such as whether they can be freely copied or deleted. We also achieve a clear separation of concerns between the resource conversions that are freely available, and the costly resources that are typically the object of study. By using an abstract categorical approach, our framework is naturally open to extension. We provide many examples throughout, emphasising the resource theoretic intuitions for each of the mathematical objects under consideration.

Peter Sidajaya

Liang Kong, Xiao-Gang Wen, Hao Zheng

Abstract In this work, we use Ising chain and Kitaev chain to check the validity of an earlier proposal in arXiv:2011.02859 that enriched fusion (higher) categories provide a unified categorical description of all gapped/gapless quantum liquid phases, including symmetry-breaking phases, topological orders, SPT/SET orders and CFT-type gapless quantum phases. In particular, we show explicitly that, in each gapped phase realized by these two models, the spacetime observables form a fusion category enriched in a braided fusion category such that its monoidal center is trivial. We also study the categorical descriptions of the boundaries of these models. In the end, we obtain a classification of and the categorical descriptions of all 1-dimensional (spatial dimension) gapped quantum phases with a bosonic/fermionic finite onsite symmetry.