Combinatorial Constructions Of Multipartite Uniform Quantum States

Entanglement,which is considered to be the most nonclassical manifestations of quantum formalism,is widely utilized in quantum information theory,such as quantum key distribution,quantum secure communication,superdense coding and teleportation etc.In recent years,quantum information has drawn more and more attention from mathematical researchers.Many classical combinatorial configurations in combinatorial designs,such as orthogonal arrays,Hadamard matrices and mutually orthogonal Latin squares(cubes)etc.,are applied to the investigation of entanglement theory.In 2014,Goyeneche and Zyczkowski introduced a special kind of multipartite entan-glement pure state——k-uniform state after tracing out N-k subsystems,whose remaining k subsystems are maximally mixed.The absolutely maximally entangled(AME)state is the extremal case of k-uniform state,when k=(?).They pointed out that a spe-cial kind of orthogonal arrays——irredundant orthogonal arrays(IrOAs)can generate a kind of k-uniform states.In 2018,Goyeneche et al.introduced several classes of quan-tum combinatorial configurations,including quantum orthogonal arrays and mutually orthogonal quantum Latin squares(cubes).They also showed that mutually orthogonal quantum Latin squares(cubes)can be entangled in the same way in which quantum states are entangled.Moreover,they gave the relationship between the quantum com-binatorial configurations and the k-uniform states,where mutually orthogonal quantum Latin squares(cubes)and quantum orthogonal arrays have the same relations with clas-sical mutually orthogonal Latin squares(cubes)and orthogonal arrays,and similar with the irredundant orthogonal arrays,the quantum orthogonal arrays also can generate a kind of k-uniform states.In this thesis,we aim to gain some kinds of 2,3-uniform states by constructing a series of irredundant orthogonal arrays,mutually orthogonal quantum Latin squares(cubes)and quantum orthogonal arrays.This thesis is organized as follows:In Chapter 1,we primarily review some essential concepts in quantum theory.In Chapter 2,we construct some kinds of 2-uniform states by establishing the exis-tence of IrOA?(2,5,d)for any integer d?4,d?6;IrOA?(2,6,d)for any integer d? 2;IrOA?(2,N,d)for some larger N.In Chapter 3,we construct 3-uniform qubits states for any N?8,N?9 via constructing IrOAs and the case of N=9 via adding minus signs mathematically.In Chapter 4,5,we mainly study the existence of the non-classical mutually orthogo-nal quantum Latin squares(cubes).We put forward the concepts of incomplete quantum Latin squares and mutually orthogonal incomplete quantum Latin squares.Furthermore,the constructions of filling in holes and direct product for mutually orthogonal quantum Latin squares(cubes)are builded.In Chapter 6,we define the notion of generalized mutually orthogonal quantum Latin squares(cubes)which are equivalent with quantum orthogonal arrays when k=2,3 and with minimal supporting r=dk respectively.Moreover,we construct some kinds of quantum orthogonal arrays when k=3.In Chapter 7,we show the main conclusions of this thesis and list some problems for further study.