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Perfect Tensors from Multiple Angles and (Quantum) Combinatorial Structures in Category Theory

Paulina L. A. Goedicke

Published Jan 1, 2026• 0 citations
OpenAlex

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.

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A perfect tensor of order $d$ is a state of four $d$-level systems that is maximally entangled under any bipartition. These objects have attracted considerable attention in quantum information and many-body theory. Perfect tensors generalize the combinatorial notion of orthogonal Latin squares (OLS). Deciding whether OLS of a given order exist has historically been a difficult problem. The case $d=6$ proved particularly thorny, and was popularized by Leonhard Euler in terms of a putative constellation of "36 officers". It took more than a century to show that Euler's puzzle has no solution. After yet another century, its quantum generalization was resolved in the affirmative: 36 entangled officers can be suitably arranged. However, the construction and verification of known instances relies on elaborate computer codes. In this paper, we present the first human-made order-$6$ perfect tensors. We decompose the Hilbert space $(\mathbb{C}^6)^{\otimes 2}$ of two quhexes into the direct sum $(\mathbb{C}^3)^{\otimes 2}\oplus(\mathbb{C}^3)^{\otimes 3}$ comprising superpositions of two-qutrit and three-qutrit states. Perfect tensors arise when certain Clifford unitaries are applied separately to the two sectors. Technically, our construction realizes solutions to the perfect functions ansatz recently proposed by Rather. Generalizing an observation of Bruzda and Życzkowski, we show that any solution of this kind gives rise to a two-unitary complex Hadamard matrix, of which we construct infinite families. Finally, we sketch a formulation of the theory of perfect tensors in terms of quasi-orthogonal decompositions of matrix algebras.

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Classical block designs are important combinatorial structures with a wide range of applications in Computer Science and Statistics. Here we give a new abstract description of block designs based on the arrow category construction. We show that models of this structure in the category of matrices and natural numbers recover the traditional classical combinatorial objects, while models in the category of completely positive maps yield a new definition of quantum designs. We show that this generalizes both a previous notion of quantum designs given by Zauner and the traditional description of block designs. Furthermore, we demonstrate that there exists a functor which relates every categorical block design to a quantum one.

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While quantum circuits built from two-particle dual-unitary (maximally entangled) operators serve as minimal models of typically nonintegrable many-body systems, the construction and characterization of dual-unitary operators themselves are only partially understood.A nonlinear map on the space of unitary operators has been proposed in Phys.Rev. Lett.125, 070501 (2020) that results in operators being arbitrarily close to dual unitaries.Here, we study the map analytically for the two-qubit case describing the basins of attraction, fixed points, and rates of approach to dual unitaries.A subset of dual-unitary operators having maximum entangling power are 2-unitary operators or perfect tensors and these are equivalent to four-party absolutely maximally entangled states.It is known that they only exist if the local dimension is larger than d = 2.We use the nonlinear map, and introduce stochastic variants of it, to construct explicit examples of new dual and 2-unitary operators.A necessary criterion for their local unitary equivalence to distinguish classes is also introduced and used to display various concrete results and a conjecture in d = 3.It is known that orthogonal Latin squares provide a "classical combinatorial design" for constructing permutations that are 2-unitary.We extend the underlying design from classical to genuine quantum ones for general dual-unitary operators and give an example of what might be the smallest-sized genuinely quantum design of a 2-unitary in d = 4.

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Quantum combinatorial designs are gaining popularity in quantum information theory. Quantum Latin squares can be used to construct mutually unbiased maximally entangled bases and unitary error bases. Here we present a general method for constructing quantum Latin arrangements from irredundant orthogonal arrays. As an application of the method, many new quantum Latin arrangements are obtained. We also find a sufficient condition such that the improved quantum orthogonal arrays [10] are equivalent to quantum Latin arrangements. We further prove that an improved quantum orthogonal array can produce a quantum uniform state.

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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.
A central theme of this thesis is the quantization and generalization of objects from areas of mathematics such as set theory and combinatorics with application to quantum information. We represent mathematical objects such as Latin squares or functions as string diagrams over the category of finite-dimensional Hilbert spaces obeying various diagrammatic axioms. This leads to a direct understanding of how these objects arise in quantum mechanics and gives insight into how quantum analogues can be defined. In Part I of this thesis we introduce quantum Latin squares (QLS), quantum objects which generalize the classical Latin squares from combinatorics. We present a new method for constructing a unitary error basis (UEB) from a quantum Latin square equipped with extra data, which we show simultaneously generalizes the existing shift-and-multiply and Hadamard methods. We introduce two different notions of orthogonality for QLS which we use to construct families of mutually unbiased bases and perfect tensors respectively. We also introduce a further generalization of Latin squares called quantum Latin isometry squares. We use these to produce quantum error codes and to give a new way of characterizing UEBs. In Part II we show that maximal families of mutually unbiased bases (MUBs) are characterized in all dimensions by partitioned UEBs, up to a choice of a family of Hadamards. Furthermore, we give a new construction of partitioned UEBs, and thus maximal families of MUBs, from a finite field, which is simpler and more direct than previous proposals. We introduce new tensor diagrammatic characterizations of maximal families of MUBs, partitioned UEBs, and finite fields as algebraic structures defined over Hilbert spaces. In Part III we introduce quantum functions and quantum sets, which quantize the classical notions. We show that these structures form a 2-category QSet. We extend this framework to introduce quantum graphs and quantum homomorphisms which form a 2-category QGraph. We show that these 2-categories capture several different notions of quantum morphism and noncommutative graph from various previous papers in noncommutative topology, quantum non-local games and quantum information. We later use the correspondence between our quantum graph morphisms and those from quantum non-local games to show that pairs of quantum isomorphic graphs with multiple connected components are built up of quantum isomorphisms on those components.
A perfect tensor of order ddd is a state of four ddd-level systems that is maximally entangled under any bipartition. These objects have attracted considerable attention in quantum information and many-body theory. Perfect tensors generalize the combinatorial notion of orthogonal Latin squares (OLS). Deciding whether OLS of a given order exist has historically been a difficult problem. The case d=6d=6d=6 proved particularly thorny, and was popularized by Leonhard Euler in terms of a putative constellation of "36 officers". It took more than a century to show that Euler's puzzle has no solution. After yet another century, its quantum generalization was resolved in the affirmative: 36 entangled officers can be suitably arranged. However, the construction and verification of known instances relies on elaborate computer codes. In this paper, we present the first human-made order-666 perfect tensors. We decompose the Hilbert space (C6)⊗2(\mathbb{C}^6)^{\otimes 2}(C6)⊗2 of two quhexes into the direct sum (C3)⊗2⊕(C3)⊗3(\mathbb{C}^3)^{\otimes 2}\oplus(\mathbb{C}^3)^{\otimes 3}(C3)⊗2⊕(C3)⊗3 comprising superpositions of two-qutrit and three-qutrit states. Perfect tensors arise when certain Clifford unitaries are applied separately to the two sectors. Technically, our construction realizes solutions to the perfect functions ansatz recently proposed by Rather. Generalizing an observation of Bruzda and Życzkowski, we show that any solution of this kind gives rise to a two-unitary complex Hadamard matrix, of which we construct infinite families. Finally, we sketch a formulation of the theory of perfect tensors in terms of quasi-orthogonal decompositions of matrix algebras.
Classical block designs are important combinatorial structures with a wide range of applications in Computer Science and Statistics. Here we give a new abstract description of block designs based on the arrow category construction. We show that models of this structure in the category of matrices and natural numbers recover the traditional classical combinatorial objects, while models in the category of completely positive maps yield a new definition of quantum designs. We show that this generalizes both a previous notion of quantum designs given by Zauner and the traditional description of block designs. Furthermore, we demonstrate that there exists a functor which relates every categorical block design to a quantum one.
Abstract We introduce a framework to define coalgebra and bialgebra structures on two-dimensional (2D) square lattices, extending the algebraic theory of Hopf algebras and quantum groups beyond the one-dimensional (1D) setting. Our construction is based on defining 2D coproducts through horizontal and vertical maps that satisfy compatibility and associativity conditions, enabling the consistent growth of vector spaces over lattice sites. We present several examples of 2D bialgebras, including group-like and Lie algebra-inspired constructions and a quasi-1D coproduct instance that is applicable to Taft-Hopf algebras and to quantum groups. The approach is further applied to the quantum group U q [ s u ( 2 ) ] , for which we construct 2D generalizations of its generators, analyze q -deformed singlet states, and derive a 2D R-matrix satisfying an intertwining relation in the semiclassical limit. Additionally, we show how tensor network states, particularly projected entangled pair states, naturally induce 2D coalgebra structures when supplemented with appropriate boundary conditions. Our results establish a local and algebraically consistent method to embed quantum group symmetries into higher-dimensional lattice systems, potentially connecting to the emerging theory of fusion 2-categories and categorical symmetries in quantum many-body physics.
While quantum circuits built from two-particle dual-unitary (maximally entangled) operators serve as minimal models of typically nonintegrable many-body systems, the construction and characterization of dual-unitary operators themselves are only partially understood.A nonlinear map on the space of unitary operators has been proposed in Phys.Rev. Lett.125, 070501 (2020) that results in operators being arbitrarily close to dual unitaries.Here, we study the map analytically for the two-qubit case describing the basins of attraction, fixed points, and rates of approach to dual unitaries.A subset of dual-unitary operators having maximum entangling power are 2-unitary operators or perfect tensors and these are equivalent to four-party absolutely maximally entangled states.It is known that they only exist if the local dimension is larger than d = 2.We use the nonlinear map, and introduce stochastic variants of it, to construct explicit examples of new dual and 2-unitary operators.A necessary criterion for their local unitary equivalence to distinguish classes is also introduced and used to display various concrete results and a conjecture in d = 3.It is known that orthogonal Latin squares provide a "classical combinatorial design" for constructing permutations that are 2-unitary.We extend the underlying design from classical to genuine quantum ones for general dual-unitary operators and give an example of what might be the smallest-sized genuinely quantum design of a 2-unitary in d = 4.
Quantum combinatorial designs are gaining popularity in quantum information theory. Quantum Latin squares can be used to construct mutually unbiased maximally entangled bases and unitary error bases. Here we present a general method for constructing quantum Latin arrangements from irredundant orthogonal arrays. As an application of the method, many new quantum Latin arrangements are obtained. We also find a sufficient condition such that the improved quantum orthogonal arrays [10] are equivalent to quantum Latin arrangements. We further prove that an improved quantum orthogonal array can produce a quantum uniform state.