Universität zu Köln

Goedicke, Paulina L. A. (2026). Perfect Tensors from Multiple Angles and (Quantum) Combinatorial Structures in Category Theory. PhD thesis, Universität zu Köln.

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Abstract

This dissertation investigates well-known (quantum) combinatorial structures, such as (quantum) designs and (quantum) orthogonal Latin squares through the lenses of matrix algebras, group theory, and category theory. A central focus is placed on perfect tensors that are examined using representation theory and quasi-orthogonal subalgebras of matrix algebras. In particular, it is shown that the existence of a 2-unitary in dimension d-square is equivalent to the existence of four mutually quasi-orthogonal subalgebras of a d-square dimensional matrix algebra that are each isomorphic to a d-dimensional matrix algebra. This correspondence is further interpreted in terms of groups and their representations, relating the existence of perfect tensors to the existence of a group with a d-square dimensional irreducible representation and four subgroups on which the respective character “factorises“. A search algorithm implemented in the algebra software GAP is presented that was used to find examples of such groups. Furthermore, additional construction schemes for other dimensions, leading to 2-unitary complex Hadamard matrices, are developed using doubly perfect bi-unimodular sequences. Using the same ansatz, an analytical resolution of Euler’s 36 officers problem is provided, which differs from the earlier constructions of perfect tensors in dimension 36 via computer-assisted methods. Beyond this, the thesis introduces an abstract notion of categorical designs formulated via arrow categories. When applied to the categories Mat(N) and CP[FHilb], this framework produces a category-theoretic description of balanced incomplete block designs and of quantum designs, respectively. The construction generalises earlier definitions of quantum designs and extends it to a broader concept that can also be interpreted as combinatorial superoperators. These new concepts are used to define a category of mutually unbiased bases and a category of so-called combinatorial quantum channels.