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arXiv:2501.00196

Group Invariant Quantum Latin Squares

Arnbjörg Soffía Árnadóttir, David E. Roberson

Published Dec 31, 2024• 0 citations
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Abstract

A quantum Latin square is an $n \times n$ array of unit vectors where each row and column forms an orthonormal basis of a fixed complex vector space. We introduce the notion of $(G,G')$-invariant quantum Latin squares for finite groups $G$ and $G'$. These are quantum Latin squares with rows and columns indexed by $G$ and $G'$ respectively such that the inner product of the $a,b$-entry with the $c,d$-entry depends only on $a^{-1}c \in G$ and $b^{-1}d \in G'$. This definition is motivated by the notion of group invariant bijective correlations introduced in [Roberson \& Schmidt (2020)], and every group invariant quantum Latin square produces a group invariant bijective correlation, though the converse does not hold. In this work we investigate these group invariant quantum Latin squares and their corresponding correlations. Our main result is that, up to applying a global isometry to every vector in a $(G,G')$-invariant quantum Latin square, there is a natural bijection between these objects and trace and conjugate transpose preserving isomorphisms between the group algebras of $G$ and $G'$. This in particular proves that a $(G,G')$-invariant quantum Latin square exists if and only if the multisets of degrees of irreducible representations are equal for $G$ and $G'$. Another motivation for this line of work is that whenever Cayley graphs for groups $G$ and $G'$ are quantum isomorphic, then there is a $(G,G')$-invariant quantum correlation witnessing this, and thus it suffices to consider such correlations when searching for quantum isomorphic Cayley graphs. Given a group invariant quantum correlation, we show how to construct all pairs of graphs for which it gives a quantum isomorphism.

DOI
10.48550/arxiv.2501.00196

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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.
Goyeneche et al recently proposed a notion of orthogonality for quantum Latin squares, and showed that orthogonal quantum Latin squares yield quantum codes. We give a simplified characterization of orthogonality for quantum Latin squares, which we show is equivalent to the existing notion. We use this simplified characterization to give an upper bound for the number of mutually orthogonal quantum Latin squares of a given size, and to give the first examples of orthogonal quantum Latin squares that do not arise from ordinary Latin squares. We then discuss quantum Latin isometry squares, generalizations of quantum Latin squares recently introduced by Benoist and Nechita, and define a new orthogonality property for these objects, showing that it also allows the construction of quantum codes. We give a new characterization of unitary error bases using these structures.
ABSTRACT A quantum Latin square of order (denoted as QLS) is an array whose entries are unit column vectors from the ‐dimensional Hilbert space , such that each row and column forms an orthonormal basis. Two unit vectors are regarded as identical if there exists a real number such that ; otherwise, they are considered distinct. The cardinality of a QLS is the number of distinct vectors in the array. In this paper, we use sub‐QLSs to prove that for any integer , and any integer , there is a QLS with cardinality .