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arXiv:2204.13008

Quantum mappings and designs

Grzegorz Rajchel-Mieldzioć

Published Apr 27, 2022• 3 citations
arXivHTMLPDF

Abstract

In order to use quantum devices for computations, it is necessary to understand the intricacies of the theoretical description. To this end, we provide several novel constructions useful for the comprehension of quantum mechanics from the perspective of mappings and designs. The unistochasticity problem, which relates the classical and the quantum domain, is solved in specific cases, e.g. for all matrices of dimension 4. Furthermore, we provide an explicit formula for entangling power in the multipartite case. Most importantly, the thesis presents and elaborates on the path that lead to the recent construction of absolutely maximally entangled state of four subsystems six levels each. Finally, we study cardinality as a measure of "quantumness" of quantum Latin squares and quantum Sudoku (SudoQ). Arrays of the highest cardinality yield families of quantum measurements of special properties. A connection between SudoQ designs and mutually unbiased bases is demonstrated.

DOI
10.48550/arxiv.2204.13008

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In order to use quantum devices for computations, it is necessary to understand the intricacies of the theoretical description. To this end, we provide several novel constructions useful for the comprehension of quantum mechanics from the perspective of mappings and designs. The unistochasticity problem, which relates the classical and the quantum domain, is solved in specific cases, e.g. for all matrices of dimension 4. Furthermore, we provide an explicit formula for entangling power in the multipartite case. Most importantly, the thesis presents and elaborates on the path that lead to the recent construction of absolutely maximally entangled state of four subsystems six levels each. Finally, we study cardinality as a measure of "quantumness" of quantum Latin squares and quantum Sudoku (SudoQ). Arrays of the highest cardinality yield families of quantum measurements of special properties. A connection between SudoQ designs and mutually unbiased bases is demonstrated.
We expand the quantum variant of the popular game Sudoku by introducing the notion of cardinality of a quantum Sudoku (SudoQ), equal to the number of distinct vectors appearing in the pattern. Our considerations are focused on the genuinely quantum solutions---the solutions of size N2{N}^{2}N2 that have cardinality greater than N2{N}^{2}N2, and therefore cannot be reduced to classical counterparts by a unitary transformation. We find the complete parametrization of the genuinely quantum solutions of a $4\ifmmode\times\else\texttimes\fi{}4$ SudoQ game and establish that in this case the admissible cardinalities are 4, 6, 8, and 16. In particular, a solution with the maximal cardinality equal to 16 is presented. Furthermore, the parametrization enabled us to prove a recent conjecture of Nechita and Pillet [I. Nechita and J. Pillet, Quantum Inf. Comput. 21, 781 (2021)] for this special dimension. In general, we proved that for any NNN it is possible to find an ${N}^{2}\ifmmode\times\else\texttimes\fi{}{N}^{2}$ SudoQ solution of cardinality N4{N}^{4}N4, which for a prime NNN is related to a set of NNN mutually unbiased bases of size N2{N}^{2}N2. Such a construction of N4{N}^{4}N4 different vectors of size NNN yields a set of N3{N}^{3}N3 orthogonal measurements.
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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.
We explore the set of unitary matrices characterized by a given structure in the context of their applications in the field of Quantum Information. In the first part of the Thesis we focus on classification of special classes of unitary matrices and possibility of introducing certain internal parameterizations. Several new results and conjectures are discussed. Second part of the Thesis is devoted to the concept of multipartite quantum entanglement. We present a solution to the problem of absolutely maximally entangled states of four subsystems with six levels each. Finally, we analyze the excess of a matrix and the corresponding Bell inequalities. This combination allows us to draw new conclusions related to quantum nonlocality.
We expand the quantum variant of the popular game Sudoku by introducing the notion of cardinality of a quantum Sudoku (SudoQ), equal to the number of distinct vectors appearing in the pattern. Our considerations are focused on the genuinely quantum solutions, which are the solutions of size N2N^2N2 that have cardinality greater than N2N^2N2, and therefore cannot be reduced to classical counterparts by a unitary transformation. We find the complete parameterization of the genuinely quantum solutions of 4×44 \times 44×4 SudoQ game and establish that in this case the admissible cardinalities are 4, 6, 8 and 16. In particular, a solution with the maximal cardinality equal to 16 is presented. Furthermore, the parametrization enabled us to prove a recent conjecture of Nechita and Pillet for this special dimension. In general, we proved that for any NNN it is possible to find an N2×N2N^2 \times N^2N2×N2 SudoQ solution of cardinality N4N^4N4, which for a prime NNN is related to a set of NNN mutually unbiased bases of size N2N^2N2. Such a construction of N4N^4N4 different vectors of size NNN yields a set of N3N^3N3 orthogonal measurements.
We introduce several classes of quantum combinatorial designs, namely quantum Latin squares, cubes, hypercubes, and a notion of orthogonality between them. A further introduced notion, quantum orthogonal arrays, generalizes all previous classes of designs. We show that mutually orthogonal quantum Latin arrangements can be entangled in the same way in which quantum states are entangled. Furthermore, we show that such designs naturally define a remarkable class of genuinely multipartite highly entangled states called kkk-uniform, i.e., multipartite pure states such that every reduction to kkk parties is maximally mixed. We derive infinitely many classes of mutually orthogonal quantum Latin arrangements and quantum orthogonal arrays having an arbitrary large number of columns. The corresponding multipartite kkk-uniform states exhibit a high persistency of entanglement, which makes them ideal candidates to develop multipartite quantum information protocols.