Two and three-uniform states from irredundant orthogonal arrays

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A pure quantum state of N subsystems, each with d levels, is said to be k-uniform if all of its reductions to k qudits are maximally mixed. Only the uniform states obtained from orthogonal arrays (OAs) are considered throughout this work. The Hamming distances of OAs are specially applied to the theory of quantum information. By using difference schemes and orthogonal partitions, we construct a series of infinite classes of irredundant orthogonal arrays (IrOAs), then answer the open questions of whether there exist 3-uniform states of N qubits and 2-uniform states of N qutrits, and whether 3-uniform states of qudits (d > 2) for high values of N can be explicitly constructed. In fact, we obtain 3-uniform states for an arbitrary number of N ≥ 8 qubits and 2-uniform states of N qutrits for every N ≥ 4. Additionally, we provide explicit constructions of the 3-uniform states of N ≥ 8 qutrits, N = 6 and N ≥ 8 ququarts and ququints, N ≥ 6 qudits having d levels for any prime power d > 6, and N = 8 and N ≥ 12 qudits having d levels for non-prime-power d ≥ 6. Moreover, we describe an explicit construction scheme for the 2-uniform states of qudits having d ≥ 4 levels. The proofs of existence of the 2-uniform states of N ≥ 6 qubits are simplified by using a class of OAs. Two special 3-uniform states are obtained from IrOA(32, 10, 2, 3) and IrOA(32, 11, 2, 3) using the interaction column property of OAs.