David Reutter

Postdoctoral Fellow

Max Planck Institute for MathematicsVivatsgasse 7, 53111 Bonnemail: [lastname]@mpim-bonn.mpg.de

Bio

I am a postdoctoral fellow at the Max Planck Institute for Mathematics in Bonn. Previously, I was a DPhil student at the University of Oxford under the supervision of Jamie Vicary. A brief version of my CV can be found here.

I am interested in applications of higher category theory to — and its interplay with — low-dimensional topology, homotopy theory and mathematical physics. I am currently working on various constructions of 4-dimensional topological field theories and related questions in quantum algebra. Another focus of my work is on applying ideas from low-dimensional category theory and homotopy theory to quantum information theory.

My research interests include:

    • higher category theory
    • topological field theory
    • quantum algebra (fusion categories, subfactors, quantum groups, . . .)
    • quantum information theory & its relations to topological field theory and noncommutative mathematics

News

Publications


8) On the uniqueness of unitary structure for unitarizable fusion categories

A unitary fusion category is a fusion category over the complex numbers with a compatible positive dagger structure. Such unitary structures naturally arise in many applications and constructions of fusion categories, most notably in the context of operator algebras, subfactor theory, and mathematical physics. Not every fusion category admits a unitary structure, and a fusion category could, in principle, have more than one unitary structure. Especially in light of the recent powerful operator algebraic classification techniques of unitary fusion categories, the question of uniqueness of unitary structure has become a significant open problem. In this paper, we prove that every unitarizable fusion category admits a unique unitary structure. We prove analogous results for unitarizable braided fusion categories and unitarizable module categories.

arXiv 2019


7) High-level methods for homotopy constructions in associative n-categories

With Jamie Vicary.

The theory of associative n-categories (ANC) is a model for semistrict higher categories, recently developed by Christoph Dorn, Christopher Douglas, and Jamie Vicary, and based on a combinatorial representation of higher string diagrams. In this work, we continue the development of ANCs and introduce the proof assistant homotopy.io. We develop a general contraction principle in ANCs which forms the mathematical heart of homotopy.io, allowing string diagram manipulations in arbitrary dimensions.

arXiv 2019

Proceedings of the 34th Annual ACM/IEEE Symposium on Logic in Computer Science, 2019 (LICS 2019)

6) Fusion 2-categories and a state-sum invariant for 4-manifolds

With Christopher Douglas.

One of the early successes of quantum topology was the construction of a three-dimensional topological field theory by Turaev-Viro, and its generalization by Barrett-Westbury, starting from the data of a spherical fusion category. In this work, we introduce spherical fusion 2-categories and construct a state-sum invariant of oriented singular piecewise-linear 4-manifolds, which categorifies the Turaev-Viro-Barrett-Westbury invariant. There are many examples of fusion 2-categories, including the 2-category of module categories of a braided fusion category, the 2-category of 2-representations of a finite 2-group and twisted linearizations of finite 2-groups.

arXiv 2018


5) The Morita theory of quantum graph isomorphisms

With Benjamin Musto and Dominic Verdon.

Using the machinery developed in `A compositional approach to quantum functions', we classify instances of quantum pseudo-telepathy in the graph isomorphism game by showing that graphs quantum isomorphic to a graph G correspond to certain Frobenius algebras in a monoidal category of quantum graph automorphisms of G. In certain cases, this classification can be expressed in group-theoretical terms.

arXiv 2018

Communications in Mathematical Physics, 365(2):797-845, 2019


4) A compositional approach to quantum functions

With Benjamin Musto and Dominic Verdon.

Pseudo-telepathy is a phenomenon in quantum information, where two non-communicating parties can use pre-shared entanglement to perform a task classically impossible without communication. In this work, we uncover a connection between pseudo-telepathy and quantum symmetry, expressed in the language of fusion categories. We use this connection to classify instances of pseudo-telepathy in the `graph isomorphism game'. In this case, the relevant categories are (co)representation categories of `quantum automorphisms groups' of graphs, studied in compact quantum group theory. Along the way, we develop a 2-categorical framework for finite quantum set theory.

arXiv 2018

Journal of Mathematical Physics, 59(8):081706, 42, 2018


3) Shaded tangles for the design and verification of quantum programs

With Jamie Vicary.

We give a tangle-based graphical language for the description of quantum circuits such that isotopic tangles yield equivalent programs. This leads to several new protocols and topological insight into known phenomena such as error correcting codes and local inversion of cluster states. We analyze 11 known quantum procedures in this way -- including entanglement manipulation, error correction and teleportation -- and in each case present a fully-topological formal verification, yielding generalized procedures in some cases.

arxiv 2018

Proceedings of the Royal Society A, 475(2224):20180338, 2019


2) A classical groupoid model for quantum networks

With Jamie Vicary.

We describe a certain combinatorial 2-category and use it to give a mathematical analysis of a new type of classical computer network architecture, intended as a model of a new technology that has recently been proposed in industry. Our approach is based on groubits, generalizations of classical bits based on groupoids. This network architecture allows the direct execution of a number of protocols that are usually associated with quantum networks, including teleportation, dense coding and secure key distribution.

arXiv 2017

Logical Methods in Computer Science, 15(1), 2019


1) Biunitary constructions in quantum information theory

With Jamie Vicary.

Biunitary connections are central tools in the study and classification of subfactors. At the same time, several important quantum informatic quantities such as Hadamard matrices, unitary error bases and quantum Latin squares can be expressed as special cases of biunitaries. This leads to a range of applications of biunitaries in various 2-categories to quantum and classical information theory. In this paper, we use the interpretation of Hadamard matrices, unitary error bases and quantum Latin squares as biunitaries to obtain many new construction methods for these quantities. An extended abstract can be found here.

arXiv 2016

Higher Structures, 3(1):109–154, 2019

Talks